GEOMETRY OF LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH … · 2012. 4. 11. · 1.1 when Gis...

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GEOMETRY OF LOCALLY COMPACT GROUPS OF

POLYNOMIAL GROWTH AND SHAPE OF LARGE BALLS.

EMMANUEL BREUILLARD

Abstract. We show that any locally compact groupG with polynomial growthis weakly commensurable to some simply connected solvable Lie group S, theLie shadow of G. We then study the shape of large balls and show, generaliz-ing work of P. Pansu, that after a suitable renormalization, they converge toa limiting compact set, which is isometric to the unit ball for a left-invariantsubFinsler metric on the so-called graded nilshadow of S. As by-products, weobtain asymptotics for the volume of large balls, we prove that balls are Folnerand hence that the ergodic theorem holds for all ball averages. Along the waywe also answer negatively a question of Burago and Margulis [7] on asymptoticword metrics and recover some results of Stoll [33] of the rationality of growthseries of Heisenberg groups.

Contents

1. Introduction 12. Quasi-norms and the geometry of nilpotent Lie groups 123. The nilshadow 194. Periodic metrics 235. Reduction to the nilpotent case 276. The nilpotent case 317. Locally compact G and proofs of the main results 398. Coarsely geodesic distances and speed of convergence 479. Appendix: the Heisenberg groups 52References 55

1. Introduction

1.1. Groups with polynomial growth. Let G be a locally compact group withleft Haar measure volG. We will assume that G is generated by a compact sym-metric subset Ω. Classically, G is said to have polynomial growth if there existC > 0 and k > 0 such that for any integer n ≥ 1

volG(Ωn) ≤ C · nk,

Date: April 2012.

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http://arxiv.org/abs/0704.0095v2

2 EMMANUEL BREUILLARD

where Ωn = Ω· . . . · Ω is the n-fold product set. Another choice for Ω would onlychange the constant C, but not the polynomial nature of the bound. One of theconsequences of the analysis carried out in this paper is the following theorem:

Theorem 1.1 (Volume asymptotics). Let G be a locally compact group with poly-nomial growth and Ω a compact symmetric generating subset of G. Then thereexists c(Ω) > 0 and an integer d(G) ≥ 0 depending on G only such that thefollowing holds:

limn→+∞

volG(Ωn)

nd(G)= c(Ω)

This extends the main result of Pansu [27]. The integer d(G) coincides withthe exponent of growth of a naturally associated graded nilpotent Lie group, theasymptotic cone of G, and is given by the Bass-Guivarc’h formula (4) below.The constant c(Ω) will be interpreted as the volume of the unit ball of a sub-Riemannian Finsler metric on this nilpotent Lie group. Theorem 1.1 is a by-product of our study of the asymptotic behavior of periodic pseudodistances on G,that is pseudodistances that are invariant under a co-compact subgroup of G andsatisfy a weak kind of the existence of geodesics axiom (see Definition 4.1).

Our first task is to get a better understanding of the structure of locally compactgroups of polynomial growth. Guivarc’h [21] proved that locally compact groupsof polynomial growth are amenable and unimodular and that every compactlygenerated1 closed subgroup also has polynomial growth.

Guivarc’h [21] and Jenkins [15] also characterized connected Lie groups withpolynomial growth: a connected Lie group has polynomial growth if and only ifit is of type (R), that is if for all x ∈ Lie(S), ad(x) has only purely imaginaryeigenvalues. Such groups are solvable-by-compact and any connected nilpotentLie group is of type (R).

It is much more difficult to characterize discrete groups with polynomial growth,and this was done in a celebrated paper of Gromov [17], proving that they arevirtually nilpotent. Losert [24] generalized Gromov’s method of proof and showedthat it applied with little modification to arbitrary locally compact groups withpolynomial growth. In particular he showed that they contain a normal compactsubgroup modulo which the quotient is a (not necessarily connected) Lie group.We will prove the following refinement.

Theorem 1.2 (Lie shadow). Let G be a locally compact group of polynomialgrowth. Then there exists a connected and simply connected solvable Lie group Sof type (R), which is weakly commensurable to G. We call such a Lie group a Lieshadow of G.

Two locally compact groups are said to be weakly commensurable if, up tomoding out by a compact kernel, they have a common closed co-compact subgroup.More precisely, we will show that, for some normal compact subgroupK, G/K has

1in fact it follows from the Gromov-Losert structure theory that every closed subgroup iscompactly generated.

ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 3

a co-compact subgroup H/K which can be embedded as a closed and co-compactsubgroup of a connected and simply connected solvable Lie group S of type (R).

We must be aware that being weakly commensurable is not an equivalencerelation among locally compact groups (unlike among finitely generated groups).Additionally, the Lie shadow S is not unique up to isomorphism (e.g. Z

3 is aco-compact lattice in both R

3 and the universal cover of the group of motions ofthe plane).

We cannot replace the word solvable by the word nilpotent in the above theo-rem. We refer the reader to Example 7.9 for an example of a connected solvableLie group of type (R) without compact normal subgroups, which admits no co-compact nilpotent subgroup. In fact this is typical for Lie groups of type (R). Soin the general locally compact case (or just the Lie case) groups of polynomialgrowth can be genuinely not nilpotent, unlike what happens in the discrete case.There are important differences between the discrete case and the general case.For example, we will show that no rate of convergence can be expected in Theorem1.1 when G is solvable not nilpotent, while some polynomial rate always holds inthe nilpotent discrete case [9].

Theorem 1.2 will enable us to reduce most geometric questions about locallycompact groups of polynomial growth, and in particular the proof of Theorem1.1, to the connected Lie group case. Observe also that Theorem 1.2 subsumesGromov’s theorem on polynomial growth, because it is not hard to see that aco-compact lattice in a solvable Lie group of polynomial growth must be virtuallynilpotent (see Remark 7.8). Of course in the proof we make use of Gromov’stheorem, in its generalized form for locally compact groups due to Losert. The restof the proof combines ideas of Y. Guivarc’h, D. Mostow and a crucial embeddingtheorem of H.C. Wang. It is given in Paragraph 7.1 and is largely independent ofthe rest of the paper.

1.2. Asymptotic shapes. The main part of the paper is devoted to the asymp-totic behavior of periodic pseudodistances on G. We refer the reader to Definition4.1 for the precise definition of this term, suffices it to say now that it is a class ofpseudodistances which contains both left-invariant word metrics on G and geodesicmetrics on G that are left-invariant under co-compact subgroup of G.

Theorem 1.2 enables us to assume that G is a co-compact subgroup of a simplyconnected solvable Lie group S, and rather than looking at pseudodistances on G,we will look at pseudodistances on S that are left-invariant under a co-compactsubgroup H. More precisely a direct consequence of Theorem 1.2 is the following:

Proposition 1.3. Let G be a locally compact group with polynomial growth and ρa periodic metric on G. Then (G, ρ) is (1, C)-quasi-isometric to (S, ρS) for somefinite C > 0, where S is a connected and simply connected solvable Lie group oftype (R) and ρS some periodic metric on S.

Recall that two metric spaces (X, dX ) and (Y, dY ) are called (1, C)-quasi-isometricif there exists a map φ : X → Y such that any y ∈ Y is at distance at most Cfrom some element in the image of φ and if |dY (φ(x), φ(x

′)) − dX(x, x′)| ≤ C forall x, x′ ∈ X.


In the case when S is Rd and H is Zd, it is a simple exercise to show that anyperiodic pseudodistance is asymptotic to a norm on R

d, i.e. ρ(e, x)/ ‖x‖ → 1 asx → ∞, where ‖x‖ = lim 1

nρ(e, nx) is a well defined norm on Rd. Burago in [6]

showed a much finer result, namely that if ρ is coarsely geodesic, then ρ(e, x)−‖x‖is bounded when x ranges over Rd.When S is a nilpotent Lie group andH a latticein S, then Pansu proved in his thesis [27], that a similar result holds, namely thatρ(e, x)/ |x| → 1 for some (unique only after a choice of a one-parameter group ofdilations) homogeneous quasi-norm |x| on the nilpotent Lie group. However, weshow in Section 8, that it is not true in general that ρ(e, x) − |x| stays bounded,even for finitely generated nilpotent groups, thus answering a question of Burago(see also Gromov [20]). Our main purpose here will be to extend Pansu’s resultto solvable Lie groups of polynomial growth.

As was first noticed by Guivarc’h in his thesis [21], when dealing with geometricproperties of solvable Lie groups, it is useful to consider the so-called nilshadow ofthe group, a construction first introduced by Auslander and Green in [2]. Accord-ing to this construction, it is possible to modify the Lie product on S in a naturalway, by so to speak removing the semisimple part of the action on the nilradical,in order to turn S into a nilpotent Lie group, its nilshadow SN . The two Liegroups have the same underlying manifold, which is diffeomorphic to R

n, only adifferent Lie product. They also share the same Haar measure. This “semisimplepart” is a commutative relatively compact subgroup T (S) of automorphisms of S,image of S under a homomorphism T : S → Aut(S). The new product g ∗ h isdefined as follows by twisting the old one g · h by means of T (S),

(1) g ∗ h := g · T (g−1)h

The two groups S and SN are easily seen to be quasi-isometric, and this is why anylocally compact group of polynomial growth G is quasi-isometric to some nilpotentLie group. In particular, their asymptotic cones are bi-Lipschitz. The asymptoticcone of a nilpotent Lie group is a certain associated graded nilpotent Lie groupendowed with a left invariant geodesic distance (or Carnot group). The gradedgroup associated to SN will be called the graded nilshadow of S. Section 3 will bedevoted to the construction and basic properties of the nilshadow and its gradedgroup.

In this paper, we are dealing with a finer relation than quasi-isometry. We willbe interested in when do two left invariant (or periodic) distances are asymptotic2

(in the sense that d1(e,g)d2(e,g)

→ 1 when g → ∞). In particular, for every locally

compact group G with polynomial growth, we will identify its asymptotic coneup to isometry and not only up to quasi-isometry or bi-Lipschitz equivalence (seeCorollary 1.9 below). One of our main results is the following:

2Yet a finer equivalence relation is (1, C)-quasi-isometry, i.e. being at bounded distance inGromov-Hausdorff metric; classifying periodic metrics up to this kind of equivalence is muchharder.


Theorem 1.4 (Main theorem). Let S be a simply connected solvable Lie groupwith polynomial growth. Let ρ(x, y) be periodic pseudodistance on S which is in-variant under a co-compact subgroup H of S (see Def. 4.1). On the manifold S,one can put a new Lie group structure, which turns S into a stratified nilpotent Liegroup, the graded nilshadow of S, and a subFinsler metric d∞(x, y) on S which isleft-invariant for this new group structure such that

ρ(e, g)

d∞(e, g)→ 1

as g → ∞ in S. Moreover every automorphism in T (H) is an isometry of d∞.

The reader who wishes to see a simple illustration of this theorem can go directlyto subsection 8.1, where we have treated in detail a specific example of periodicmetric on the universal cover of the groups of motions of the plane.

The new stratified nilpotent Lie group structure on S given by the gradednilshadow comes with a one-parameter family of so-called homogeneous dilationsδtt>0. It also comes with an extra group of automorphisms, namely the imageof H under the homomorphism T . This yields automorphisms of S for both theoriginal group structure on S and the new graded nilshadow group structure.Moreover the dilations δtt>0 are automorphisms of the graded nilshadow andthey commute with T (H).

A subFinsler metric is a geodesic distance which is defined exactly as subRie-mannian (or Carnot-Caratheodory) metrics on Carnot groups are defined (see e.g.[25]), except that the norm used to compute the length of horizontal paths is notnecessarily a Euclidean norm. We refer the reader to Section 2.1 for a precisedefinition.

In Theorem 1.4 the subFinsler metric d∞ is left invariant for the new Lie struc-ture on S and it is also invariant under all automorphisms in T (H) (these forma relatively compact commutative group of automorphisms). Moreover it satisfiesthe following pleasing scaling law:

d∞(δt(x), δt(y)) = td∞(x, y) ∀t > 0.

The proof of Theorem 1.4 splits in two important steps. The first is a reductionto the nilpotent case and is performed in Section 5. Using a double averagingof the pseudodistance ρ over both K := T (H) and S/H, we construct an asso-ciated pseudodistance, which is periodic for the nilshadow structure on S (i.e.left-invariant by a co-compact subgroup for this structure), and we prove that itis asymptotic to the original ρ. This reduces the problem to nilpotent Lie groups.The key to this reduction is the following crucial observation: that unipotent au-tomorphisms of S induce only a sublinear distortion, forcing the metric ρ to beasymptotically invariant under T (H). The second step of the proof assumes thatS is nilpotent. This part is dealt with in Section 6 and is essentially a reformula-tion of the arguments used by Pansu in [27].


Incidently, we stress the fact that the generality in which Section 6 is treated(i.e. for general coarsely geodesic, and even asymptotically geodesic periodic met-rics) is necessary to prove even the most basic case (i.e. word metrics) of Theorem1.4 for non-nilpotent solvable groups. So even if we were only interested in theasymptotics of left invariant word metrics on a solvable Lie group of polynomialgrowth S, we would still need to understand the asymptotics of arbitrary coarselygeodesic left invariant distances (and not only word metrics!) on nilpotent Liegroups. This is because the new pseudodistance obtained by averaging, see (30),is no longer a word metric.

The subFinsler metric d∞(e, x) in the above theorem is induced by a certainT (H)-invariant norm on the first stratum m1 of the graded nilshadow (whichis T (H)-invariant complementary subspace of the commutator subalgebra of thenilshadow). This norm can be described rather explicitly as follows.

Recall that we have3 a canonical map π1 : S → m1, which is a group homomor-phism for both the nilshadow and graded nilshadow structures. Then:

v ∈ m1, ‖v‖∞ ≤ 1 =⋂

F⊂S

CvxHull

π1(h)

ρ(e, h), h ∈ H\F

,

where the right hand side is the intersection over all compact subsets F of S ofthe closed convex hull of the points π1(h)/ρ(e, h) for h ∈ H\F .

Figure 1 gives an illustration of the limit shape corresponding to the wordmetric on the 3-dimensional discrete Heisenberg group with standard generators.We explain in the Appendix how one can compute explicitly the geodesics of thelimit metric and the limit shape in this example.

When S itself is nilpotent to begin with and ρ is (in restriction to H) theword metric associated to a symmetric compact generating set Ω of H (namelyρΩ(e, h) := infn ∈ N;h ∈ Ωn), the above norm takes the following simple form:

(2) v ∈ m1, ‖v‖∞ ≤ 1 = CvxHull π1(ω), ω ∈ Ω

For instance, in the special case when H is a torsion-free finitely generated nilpo-tent group with generating set Ω and S is its Malcev closure, the unit ballv ∈ m1, ‖v‖∞ ≤ 1 is a polyhedron in m1. This was Pansu’s description in[27].

However when S is not nilpotent, and is equipped with a word metric ρΩ ona co-compact subgroup, then the determination of the limit shape, i.e. the de-termination of the limit norm ‖ · ‖∞ on the abelianized nilshadow, is much moredifficult. Clearly ‖ · ‖∞ is K-invariant and it is a simple observation that the unitball for ‖ · ‖∞ is always contained in the convex hull of the K-orbit of π1(Ω).

3The subspace m1 can be identified with the abelianized nilshadow (or abelianized gradednilshadow) by first identifying the nilshadow with its Lie algebra via the exponential map andthen projecting modulo the commutator subalgebra. The map does not depend on the choiceinvolved in the construction of the nilshadow. See also Remark 3.7.


Nevertheless the unit ball is typically smaller than that (unless Ω was K-invariantto begin with).

In general it would be interesting to determine whether there exists a simpledescription of the limit shape of an arbitrary word metric on a solvable Lie groupwith polynomial growth. We refer the reader to Section 8 and Paragraph 8.2 for anexample of a class of word metrics on the universal cover of the group of motionsof the plane, for which we were able to compute the limit shape.

Another by-product of Theorem 1.4 is the following result.

Corollary 1.5 (Asymptotic shape). Let S be a simply connected solvable Liegroup with polynomial growth and H a co-compact subgroup. Let ρ be an H-periodic pseudodistance on S. Then in the Hausdorff metric,

limt→+∞

δ 1

t(Bρ(t)) = C,

where C is a T (H)-invariant compact neighborhood of the identity in S, Bρ(t) isthe ρ-ball of radius t in S and δtt>0 is a one-parameter group of dilations on S(equipped with the graded nilshadow structure). Moreover, C = g ∈ S, d∞(e, g) ≤ 1is the unit ball of the limit subFinsler metric from Theorem 1.4.

Proof. By Theorem 1.4, for every ε > 0 we have Bd∞(t−εt) ⊂ Bρ(t) ⊂ Bd∞(t+εt)if t is large enough. Since δ 1

t(Bd∞(t)) = C, for all t > 0, we are done.

Combining this with Theorem 1.2, we also get the following corollary, of whichTheorem 1.1 is only a special case with ρ the word metric associated to the gen-erating set Ω.

Corollary 1.6 (Volume asymptotics). Suppose that G is a locally compact groupwith polynomial growth and ρ is a periodic pseudodistance on G. Let Bρ(t) bethe ρ-ball of radius t in G, i.e. Bρ(t) = x ∈ G, ρ(e, x) ≤ t, then there exists aconstant c(ρ) > 0 such that the following limit exists:

(3) limt→+∞

volG(Bρ(t))

td(G)= c(ρ)

Here d(G) is the integer d(SN ), the so-called homogeneous dimension of thenilshadow SN of a Lie shadow S of G (obtained by Theorem 1.2), and is given bythe Bass-Guivarc’h formula:

(4) d(SN ) =∑

k≥0

dim(Ck(SN ))

where Ck(SN )k is the descending central series of SN .The limit c(ρ) is equal to the volume volS(C) of the limit shape C from Corollary

1.5 once we make the right choice of Haar measure on a Lie shadow S of G. Letus explain this choice. Recall that according to Theorem 1.2, G/K admits a co-compact subgroup H/K which embeds co-compactly in S. Starting with a Haarmeasure volG on G, we get a Haar measure on G/K after fixing the Haar measureof K to be of total mass 1, and we may then choose a Haar measure on H/K sothat the compact quotient G/H has volume 1. Finally we choose the Haar measure


z

yx

Figure 1. The asymptotic shape of large balls in the Cayley graphof the Heisenberg group H(Z) = 〈x, y|[x, [x, y]] = [y, [x, y]] = 1〉viewed in exponential coordinates.

on S so that the other compact quotient S/(H/K) has volume 1. This gives thedesired Haar measure volS such that c(ρ) = volS(C).

Note that Haar measure on S is also invariant under the group of automor-phisms T (S) and is thus left invariant for the nilshadow structure on S. It is alsoleft invariant for the graded nilshadow structure. In both exponential coordinatesof the first kind (on SN ) and of the second kind (as in Lemma 3.10), Haar measureis just Lebesgue measure.

In the case of the discrete Heisenberg group of dimension 3 equipped with theword metric given by the standard generators, it is possible to compute the con-stant c(ρ) and the volume of the limit shape as shown in Figure 1. In this case thevolume is 31

72 (see the Appendix). The 5-dimensional Heisenberg group can also beworked out and the volume of its limit shape (associated to the word metric given

by standard generators) is equal to 200921870 + log 2

32805 . The fact that this number istranscendental implies that the growth series of this group, i.e. the formal powerseries

∑n≥0 |Bρ(n)|z

n is not algebraic in the sense that it is not a solution of a

polynomial equation with rational functions in C(z) as coefficients (see [33, Prop.


3.3.]). This was observed by Stoll in [33] by more direct combinatorial means.Stoll also shows there the interesting fact that the growth series can be rationalfor some other choices of generating sets in the 5-dimensional Heisenberg group.So rationality of the growth series depends on the generating set.

Another interesting feature is asymptotic invariance:

Corollary 1.7 (Asymptotic invariance). Let S be a simply connected solvableLie group with polynomial growth and ρ a periodic pseudodistance on S. Let ∗ bethe new Lie product on S given by the nilshadow group structure (or the gradednilshadow group structure). Then ρ(e, g ∗ x)/ρ(e, x) → 1 as x → ∞ for everyg ∈ S.

This follows immediately from Theorem 1.4, when ∗ is the graded nilshadowproduct, and from Theorem 6.2 below in the case ∗ is the nilshadow group struc-ture.

It is worth observing that we may not in general replace ∗ by the ordinaryproduct on S. Indeed, let for instance S = R ⋉ R

2 be the universal cover of thegroup of motions of the Euclidean plane, then S, like its nilshadow R

3, admitsa lattice Γ ≃ Z

3. The quotient S/Γ is diffeomorphic to the 3-torus R3/Z3 and

it is easy to find Riemannian metrics on this torus so that their lift to R3 is not

invariant under rotation around the z-axis. Hence this metric, viewed on the Liegroup S will not be asymptotically invariant under left translation by elements ofS. Nevertheless, if the metric is left-invariant and not just periodic, then we havethe following corollary of the proof of Theorem 1.4.

Corollary 1.8 (Left-invariant pseudodistances are asymptotic to subFinsler met-rics). Let S be a simply connected solvable Lie group of polynomial growth and ρbe a periodic pseudodistance on S which is invariant under all left-translations byelements of S (e.g. a left-invariant coarsely geodesic metric on S). Then thereis a left-invariant subFinsler metric d on S which is asymptotic to ρ in the sense

that ρ(e,g)d(e,g) → 1 as g → ∞.

We already mentioned above that determining the exact limit shape of a wordmetric on S is a difficult task. Consequently so is the task of telling when twodistinct word metrics are asymptotic. The above statement says that in any caseevery word metric on S is asymptotic to some left-invariant subFinsler metric. Sothe set of possible limit shapes is no richer for word metrics than for left-invariantsubFinsler metrics.

We note that in the case of nilpotent Lie groups (where K is trivial), Theorem1.4 shows that every periodic metric is asymptotic to a left-invariant metric. It isstill an open problem to determine whether every coarsely geodesic periodic metricis at a bounded distance from a left-invariant metric (this is Burago’s theorem inRn, more about it below).

Theorems 1.2 and 1.4 allow us to describe the asymptotic cone of (G, ρ) for anyperiodic pseudodistance ρ on any locally compact group with polynomial growth.


Corollary 1.9 (Asymptotic cone). Let G be a locally compact group with polyno-mial growth and ρ a periodic pseudodistance on G. Then the sequence of pointedmetric spaces (G, 1nρ, e)n≥1 converges in the Gromov-Hausdorff topology. Thelimit is the metric space (N, d∞, e), where N is a graded simply connected nilpo-tent Lie group and d∞ a left invariant subFinsler metric on N . Moreover the Liegroup N is (up to isomorphism) independent of ρ. The space (N, d∞) is isometricto “the asymptotic cone” associated to (G, ρ). This asymptotic cone is independentof the choice of ultrafilter used to define it.

This corollary is a generalization of Pansu’s theorem ((10) in [27]). We referthe reader to the book [18] for the definitions of the asymptotic cone and theGromov-Hausdorff convergence. We discuss in Section 8 the speed of convergence(in the Gromov-Hausdorff metric) in this theorem and its corollaries about volumegrowth. In particular there is a major difference between the discrete nilpotent caseand the solvable non nilpotent case. In the former, one can find a polynomial rateof convergence [9], while in the latter no such rate exist in general (see Theorem8.1).

1.3. Folner sets and ergodic theory. A consequence of Corollary 1.6 is thatsequences of balls with radius going to infinity are Folner sequences, namely:

Corollary 1.10. Let G be a locally compact group with polynomial growth andρ a periodic pseudodistance on G. Let Bρ(t) be the ρ-ball of radius t in G. ThenBρ(t)t>0 form a Folner family of subsets of G namely, for any compact set Fin G, we have (∆ denotes the symmetric difference)

(5) limt→+∞

volG(FBρ(t)∆Bρ(t))

volG(Bρ(t))= 0

Proof. Indeed FBρ(t)∆Bρ(t) ⊂ Bρ(t + c)\Bρ(t) for some c > depending on F .Hence (5) follows from (3).

This settles the so-called “localization problem” of Greenleaf for locally compactgroups of polynomial growth (see [16]), i.e. determining whether the powers ofa compact generating set Ωnn form a Folner sequence. At the same time itimplies that the ergodic theorem for G-actions holds along any sequence of ballswith radius going to infinity.

Theorem 1.11. (Ergodic Theorem) Let be given a locally compact group G withpolynomial growth together with a measurable G-space X endowed with a G-invariant ergodic probability measure m. Let ρ be a periodic pseudodistance onG and Bρ(t) the ρ-ball of radius t in G. Then for any p, 1 ≤ p < ∞, and anyfunction f ∈ L

p(X,m) we have

limt→+∞

1

volG(Bρ(t))

∫

Bρ(t)f(gx)dg =

∫

Xfdm

for m-almost every x ∈ X and also in Lp(X,m).


In fact, Corollary 1.10 above, was the “missing block” in the proof of the ergodictheorem on groups of polynomial growth. So far and to my knowledge, Corollary1.10 and Theorem 1.11 were known only along some subsequence of balls Bρ(tn)nchosen so that (5) holds (see for instance [10] or [34]). This issue was drawn tomy attention by A. Nevo and was my initial motivation for the present work. Werefer the reader to the A. Nevo’s survey paper [26] Section 5.

It later turned out that the mere fact that balls are Folner in a given polynomialgrowth locally compact group can also be derived from the fact these groups aredoubling metric spaces (which is an easier result than the precise asymptoticsvol(Ωn) ∼ cΩn

d(G) proved in this paper and only requires lower and upper bounds

of the form c1nd(G) ≤ vol(Ωn) ≤ c2n

d(G)). This was observed by R. Tessera[35] who rediscovered a cute argument of Colding and Minicozzi [11, Lemma 3.3.]showing that the volume of spheres Ωn+1 \ Ωn is at most some O(n−δ) times thevolume of the ball Ωn, where δ > 0 is a positive constant depending only on thedoubling constant the word metric induced by Ω in G.

In [9], we give a better upper bound (which depends only on the nilpotency classand not on the doubling constant) for the volume of spheres in the case of finitelygenerated nilpotent groups. This is done by showing the following error term inthe asymptotics of the volume of balls: we have vol(Ωn) = cΩn

d(G)+O(nd(G)−αr ),where αr > 0 depends only on the nilpotency class r of G. We refer the reader toSection 8 and to the preprint [9] for more information on this. We only note herethat although the above Colding-Minicozzi-Tessera upper bound on the volume ofspheres holds generally for all locally compact groups G with polynomial growth,unless G is nilpotent, there is no error term in general in the asymptotics of thevolume of balls. An example with arbitrarily small speed is given in §8.1.

1.4. A conjecture of Burago and Margulis. In [7] D. Burago and G. Margulisconjectured that any two word metrics on a finitely generated group which are

asymptotic (in the sense that ρ1(e,γ)ρ2(e,γ)

tends to 1 at infinity) must be at a bounded

distance from one another (in the sense that |ρ1(e, γ) − ρ2(e, γ)| = O(1)). Thisholds for abelian groups. An analogous result was proved by Abels and Margulisfor word metrics on reductive groups [1]. S. Krat [23] established this propertyfor word metrics on the Heisenberg group H3(Z). However using Theorem 1.4(which in this particular case of finitely generated nilpotent groups is just Pansu’stheorem [27]) we will show in Section 8.3, that there are counter-examples andexhibit two word metrics on H3(Z) × Z which are asymptotic and yet are not ata bounded distance. For more on this counter-example, and how to adequatelymodify the conjecture of Burago and Margulis, we refer the interested reader to[9].

1.5. Organization of the paper. Sections 2-4 are devoted to preliminaries. InSection 2 we present the basic nilpotent theory as can be found in Guivarc’h’sthesis [21]. In particular, a full proof of the Bass-Guivarc’h formula is given. InSection 3, we recall the construction of the nilshadow of a solvable Lie group.


In Section 4 we set up the axioms and basic properties of the (pseudo)distancefunctions that are studied in this paper.

Sections 5-7 contain the core of the proof of the main theorems. In Section 5, weassume that G is a simply connected solvable Lie group and reduce the problem tothe nilpotent case. In Section 6, we assume that G is a simply connected nilpotentLie group and prove Theorem 1.4 in this case following the strategy used by Pansuin [27]. In Section 7, we prove Theorem 1.2 for general locally compact groupsand reduce the proof of the results of the introduction to the Lie case.

In the last section we make further comments about the speed of convergence.In particular we give examples answering negatively the aforementioned questionof Burago and Margulis.

The Appendix is devoted to the discrete Heisenberg groups of dimension 3 and5. We compute their limit balls, explain Figure 1, and recover the main result ofStoll [33].

The reader who is mainly interested in the nilpotent group case can read directlySection 6 while keeping an eye on Sections 2 and 4 for background notations andelementary facts.

Finally, let us mention that the results and methods of this paper were largelyinspired by the works of Y. Guivarc’h [21] and P. Pansu [27].

1.6. Nota Bene. A version of this article circulated since 2007. The present ver-sion contains essentially the same material, only the exposition has been improvedand several somewhat sketchy arguments have been replaced by full fledged proofs(in particular in Sections 3 and 7). This delay is due to the fact that I was plan-ning for a long time to improve Section 6 and show an error term in the volumeasymptotics of balls in nilpotent groups. E. Le Donne and I recently managed toachieve this and it has now become an independent joint paper [9].

2. Quasi-norms and the geometry of nilpotent Lie groups

In this section, we review the necessary background material on nilpotent Liegroups. In paragraph 2.4, we give some crucial properties of homogeneous quasinorms and reproduce some lemmas originally due to Y. Guivarc’h which will beused in the sequel. Meanwhile, we prove the Bass-Guivarc’h formula for the de-gree of polynomial growth of nilpotent Lie groups, following Guivarc’h’s originalargument.

2.1. Carnot-Caratheodory metrics. Let G be a connected Lie group with Liealgebra g and let m1 be a vector subspace of g. We denote by ‖·‖ a norm on m1.

We now recall the definition of a left-invariant Carnot-Caratheodory metric alsocalled subFinsler metric on G. Let x, y ∈ G. We consider all possible piecewisesmooth paths ξ : [0, 1] → G going from ξ(0) = x to ξ(1) = y. Let ξ′(u) be thetangent vector which is pulled back to the identity by a left translation, i.e.

(6)dξ

du= ξ(u) · ξ′(u)


where ξ′(u) ∈ g and the notation ξ(u) · ξ′(u) means the image of ξ′(u) underthe differential at the identity of the left translation by the group element ξ(u).We say that the path ξ is horizontal if the vector ξ′(u) belongs to m1 for allu ∈ [0, 1]. We denote by H the set of piecewise smooth horizontal paths. TheCarnot-Caratheodory metric associated to the norm ‖·‖ is defined by:

d(x, y) = inf

∫ 1

0

∥∥ξ′(u)∥∥ du, ξ ∈ H, ξ(0) = x, ξ(1) = y

where the infimum is taken over all piecewise smooth paths ξ : [0, 1] → N withξ(0) = x, ξ(1) = y that are horizontal in the sense that ξ′(u) ∈ m1 for all u. If‖ · ‖ is a Euclidean norm, the metric d(x, y) is also called subRiemannian. In thispaper however the norm ‖ · ‖ will typically not be Euclidean (it can be polyhedrallike in the case of word metrics on finitely generated nilpotent groups) and d(x, y)will only be subFinsler. If m1 = g, and ‖·‖ is a Euclidean (resp. arbitrary) normon g, then d is simply the usual left-invariant Riemannian (resp. Finsler) metricassociated to ‖·‖ .

Chow’s theorem (e.g. see [19] or [25]) tells us that d(x, y) is finite for all x andy in G if and only if the vector subspace m1, together with all brackets of elementsof m1, generates the full Lie algebra g. If this condition is satisfied, then d is adistance on G which induces the original topology of G.

In this paper, we will only be concerned with Carnot-Caratheodory metrics ona simply connected nilpotent Lie group N . In the sequel, whenever we speak of aCarnot-Caratheodory metric on N, we mean one that is associated to a norm ‖·‖on a subspace m1 such that n = m1 ⊕ [n, n] where n = Lie(N). It is easy to checkthat any such m1 generates the Lie algebra n.

Remark 2.1. Let us observe here that for such a metric d on N, we have thefollowing description of the unit ball for ‖·‖

v ∈ m1, ‖v‖ ≤ 1 =

π1(x)

d(e, x), x ∈ N\e

where π1 is the linear projection from n (identified with N via exp) to m1 withkernel [n, n]. Indeed, π1 gives rise to a homomorphism from N to the vector spacem1. And if ξ(u) is a horizontal path from e to x, then applying π1 to (6) we

get dduπ1(ξ(u)) = ξ′(u), hence π1(x) =

∫ 10 ξ

′(u)du. Hence ‖π1(x)‖ ≤ d(e, x) withequality if x ∈ m1.

2.2. Dilations on a nilpotent Lie group and the associated graded group.We now focus on the case of simply connected nilpotent Lie groups. Let N besuch a group with Lie algebra n and nilpotency class r. For background aboutanalysis on such groups, we refer the reader to the book [12]. The exponentialmap is a diffeomorphism between n and N . Most of the time, if x ∈ n, we willabuse notation and denote the group element exp(x) simply by x. We denoteby Cp(n)p the central descending series for n, i.e. Cp+1(n) = [n, Cp(n)] withC0(n) = n and Cr(n) = 0.


Let (mp)p≥1 be a collection of vector subspaces of n such that for each p ≥ 1,

(7) Cp−1(n) = Cp(n)⊕mp.

Then n = ⊕p≥1mp and in this decomposition, any element x in n (or N by abuseof notation) will be written in the form

x =∑

p≥1

πp(x)

where πp(x) is the linear projection onto mp.To such a decomposition is associated a one-parameter group of dilations (δt)t>0.

These are the linear endomorphisms of n defined by

δt(x) = tpx

for any x ∈ mp and for every p. Conversely, the one-parameter group (δt)t≥0

determines the (mp)p≥1’s since they appear as eigenspaces of each δt, t 6= 1. Thedilations δt do not preserve a priori the Lie bracket on n. This is the case if andonly if

(8) [mp,mq] ⊆ mp+q

for every p and q (where [mp,mq] is the subspace spanned by all commutators ofelements of mp with elements of mq). If (8) holds, we say that the (mp)p≥1 form astratification of the Lie algebra n, and that n is a stratified (or homogeneous) Liealgebra. It is an exercise to check that (8) is equivalent to require [m1,mp] = mp+1

for all p.If (8) does not hold, we can however consider a new Lie algebra structure on

the real vector space n by defining the new Lie bracket as [x, y]∞ = πp+q([x, y])if x ∈ mp and y ∈ mq. This new Lie algebra n∞ is stratified and has the sameunderlying vector space as n. We denote by N∞ the associated simply connectedLie group. Moreover the (δt)t>0 form a one-parameter group of automorphisms ofn∞. In fact the original Lie bracket [x, y] on n can be deformed continuously to[x, y]∞ through a continuous family of Lie algebra structures by setting

(9) [x, y]t = δ 1

t([δtx, δty])

and letting t → +∞. Note that conversely, if the δt’s are automorphisms of n,then [x, y] = πp+q([x, y]) for all x ∈ mp and y ∈ mq, and n = n∞.

The graded Lie algebra associated to n is by definition

gr(n) =⊕

p≥0

Cp(n)/Cp+1(n)

endowed with the Lie bracket induced from that of n. The quotient map mp →Cp(n)/Cp+1(n) gives rise to a linear isomorphism between n and gr(n), which isa Lie algebra isomorphism between the new Lie algebra structure n∞ and gr(n).Hence stratified Lie algebra structures induced by a choice of supplementary sub-spaces (mp)p≥1 as in (7) are all isomorphic to gr(n).


On N∞ the left-invariant subFinsler metrics d∞ associated to a choice of normon m1 are of special interest. The one-parameter group of dilations δtt is anautomorphism of N∞ and that

(10) d∞(δtx, δty) = td∞(x, y)

for any x, y ∈ N∞. The metric space (N∞, d∞) is called a Carnot group.If on the other hand the simply connected nilpotent Lie groupN is not stratified,

then the group of dilations (δt)t associated to a choice of supplementary vectorsubspaces mi’s as in (7) will not consist of automorphisms of N and the relation(10) will not hold.

Note also that if we are given two different choices of supplementary subspacesmi’s and m′

i’s as in (7), then the left-invariant Carnot-Caratheodory metrics onthe corresponding stratified Lie groups are isometric if and only if (m1, ‖·‖) and(m′

1, ‖·‖′) are isometric (a linear isomorphism from m1 to m′

1 that sends ‖·‖ to‖·‖′ extends to an isometry of the two Carnot groups).

2.3. The Campbell-Hausdorff formula. The exponential map exp : n → Nis a diffeomorphism. In the sequel, we will often abuse notation and identify Nand n without further notice. In particular, for two elements x and y of n (orN equivalently) xy will denote their product in N , while x + y denotes the sumin n. Let (δt)t be a one-parameter group of dilations associated to a choice ofsupplementary subspaces mi’s as in (7). We denote the corresponding stratifiedLie algebra by n∞ as above and the Lie group by N∞. The product on N∞ isdenoted by x ∗ y. On N∞ the dilations (δt)t are automorphisms.

The Campbell-Hausdorff formula (see [12]) allows to give a more precise form ofthe product in N. Let (ei)1≤i≤d be a basis of n adapted to the decomposition intomi’s, that is mi = spanej , ej ∈ mi. Let x = x1e1 + ...+ xded the correspondingdecomposition of an element x ∈ n. Then define the degree di = deg(ei) to be thelargest j such that ei ∈ Cj−1(n). If α = (α1, ..., αd) ∈ N

d is a multi-index, then letdα = deg(e1)α1 + ...+ deg(ed)αd.

The Campbell-Hausdorff formula yields

(11) (xy)i = xi + yi +∑

Cα,βxαyβ

where Cα,β are real constants and the sum is over all multi-indices α and β suchthat dα + dβ ≤ deg(ei), dα ≥ 1 and dβ ≥ 1.

From (9), it is easy to give the form of the associated stratified Lie group law:

(12) (x ∗ y)i = xi + yi +∑

Cα,βxαyβ

where the sum is restricted to those α’s and β’s such that dα + dβ = deg(ei),dα ≥ 1 and dβ ≥ 1.

2.4. Homogeneous quasi-norms and Guivarc’h’s theorem on polynomialgrowth. Let n be a finite dimensional real nilpotent Lie algebra and consider adecomposition

n = m1 ⊕ ...⊕mr


by supplementary vector subspaces as in (7). Let (δt)t>0 be the one parametergroup of dilations associated to this decomposition, that is δt(x) = tix if x ∈ mi.We now introduce the following definition.

Definition 2.2 (Homogeneous quasi-norm). A continuous function | · | : n → R+

is called a homogeneous quasi-norm associated to the dilations (δt)t, if it satisfiesthe following properties:

(i) |x| = 0 ⇔ x = 0.(ii) |δt(x)| = t|x| for all t > 0.

Example 2.3. (1) Quasi-norms of supremum type, i.e. |x| = maxp ‖πp(x)‖1/pp

where ‖·‖p are ordinary norms on the vector space mp and πp is the projection onmp as above.

(2) |x| = d∞(e, x), where d∞ is a Carnot-Caratheodory metric on a stratifiednilpotent Lie group (as the relation (10) shows).

Clearly, a quasi-norm is determined by its sphere of radius 1 and two quasi-norms (which are homogeneous with respect to the same group of dilations) arealways equivalent in the sense that

(13)1

c|·|1 ≤ |·|2 ≤ c |·|1

for some constant c > 0 (indeed, by continuity, | · |2 admits a maximum on the“sphere” |x|1 = 1). If the two quasi-norms are homogeneous with respect totwo distinct semi-groups of dilations, then the inequalities (13) continue to holdoutside a neighborhood of 0, but may fail near 0.

Homogeneous quasi-norms satisfy the following properties:

Proposition 2.4. Let | · | be a homogeneous quasi-norm on n, then there areconstants C,C1, C2 > 0 such that

(a) |xi| ≤ C · |x|deg(ei) if x = x1e1 + ...+ xnen in an adapted basis (ei)i.(b) |x−1| ≤ C · |x|.(c) |x+ y| ≤ C · (|x|+ |y|)(d) |xy| ≤ C1(|x|+ |y|) + C2.

Properties (a), (b) and (c) are straightforward from the fact that |x| = maxp ‖πp(x)‖1/pp

is a homogeneous quasi-norm and from (13). Property (d) justifies the term “quasi-norm” and follows from Lemma 2.5 below. It can be a problem that the constantC1 in (d) may not be equal to 1. In fact, this is why we use the word quasi-norminstead of just norm, because we do not require the triangle inequality axiom tohold. However the following lemma of Guivarc’h is often a good enough remedyto this situation. Let ‖·‖p be an arbitrary norm on the vector space mp.

Lemma 2.5. (Guivarc’h, [21] lemme II.1) Let ε > 0. Up to rescaling each ‖·‖pinto a proportional norm λp ‖·‖p (λp > 0) if necessary, the quasi-norm |x| =

maxp ‖πp(x)‖1/pp satisfies

(14) |xy| ≤ |x|+ |y|+ ε


for all x, y ∈ N . If N is stratified with respect to (δt)t we can take ε = 0.

This lemma is crucial also for computing the coarse asymptotics of volumegrowth. For the reader’s convenience, we reproduce here Guivarc’h’s argument,which is based on the Campbell-Hausdorff formula (11).

Proof. We fix λ1 = 1 and we are going to give a condition on the λi’s so that (14)holds. The λi’s will be taken to be smaller and smaller as i increases. We set

|x| = maxp ‖πp(x)‖1/pp and let |x|λ = maxp ‖λpπp(x)‖

1/pp for any r-tuple of λi’s.

We want that for any index p ≤ r,

(15) λp ‖πp(xy)‖p ≤ (|x|λ + |y|λ + ε)p

By (11) we have πp(xy) = πp(x) + πp(y) + Pp(x, y) where Pp is a polynomial mapinto mp depending only on the πi(x) and πi(y) with i ≤ p− 1 such that

‖Pp(x, y)‖p ≤ Cp ·∑

l,m≥1,l+m≤p

Mp−1(x)lMp−1(y)

m

where Mk(x) := maxi≤k ‖πi(x)‖1/ii and Cp > 0 is a constant depending on Pp and

on the norms ‖·‖i’s. Since ε > 0, when expanding the right hand side of (15) all

terms of the form |x|lλ|y|mλ with l +m ≤ p appear with some positive coefficient,

say εl,m. The terms |x|pλ and |y|pλ appear with coefficient 1 and cause no troublesince we always have λp ‖πp(x)‖p ≤ |x|pλ and λp ‖πp(y)‖p ≤ |y|pλ. Therefore, for

(15) to hold, it is sufficient that

λpCpMp−1(x)lMp−1(y)

m ≤ εl,m|x|lλ|y|mλ

for all remaining l and m. However, clearly Mk(x) ≤ Λk · |x|λ where Λk :=

maxi≤k1/λ1/ii ≥ 1. Hence a sufficient condition for (15) to hold is

λp ≤ε

CpΛpp−1

where ε = min εl,m. Since Λp−1 depends only on the first p−1 values of the λi’s, itis obvious that such a set of conditions can be fulfilled by a suitable r-tuple λ.

Remark 2.6. The constant C2 in Property (d) above can be taken to be 0 when Nis stratified with respect to the mi’s (i.e. the δt’s are automorphisms), as is easilyseen after changing x and y into their image under δt. And conversely, if C2 = 0for some δt-homogeneous quasi-norm on N, then N admits a stratification. Indeed,from (11) and (12), we see that if the δt’s are not automorphisms, then one can

find x, y ∈ N such that, when t is small enough, |δt(xy) − δt(x)δt(y)| ≥ ct(r−1)/r

for some c > 0. However, combining Properties (c) and Property (d) with C2 = 0above we must have |δt(xy)− δt(x)δt(y)| = O(t) near t = 0. A contradiction.

Guivarc’h’s lemma enables us to show:

Theorem 2.7. (Guivarc’h ibid.) Let Ω be a compact neighborhood of the identityin a simply connected nilpotent Lie group N and ρΩ(x, y) = infn ≥ 1, x−1y ∈ Ωn.


Then for any homogeneous quasi-norm | · | on N, there is a constant C > 0 suchthat

(16)1

C|x| ≤ ρΩ(e, x) ≤ C|x|+ C

Proof. Since any two homogeneous quasi-norms (w.r.t the same one-parametergroup of dilations) are equivalent, it is enough to do the proof for one of them,so we consider the quasi-norm obtained in Lemma 2.5 with the extra property(14). The lower bound in (16) is a direct consequence of (14) and one can takethere C to be max|x|, x ∈ Ω + ε. For the upper bound, it suffices to showthat there is C ∈ N such that for all n ∈ N, if |x| ≤ n then x ∈ ΩCn. Toachieve this, we proceed by induction of the nilpotency length of N. The resultis clear when N is abelian. Otherwise, by induction we obtain C0 ∈ N such thatx = ω1 · ... · ωC0n · z where ωi ∈ Ω and z ∈ Cr−1(N) whenever |x| ≤ n. Hence|z| ≤ |x|+C0n ·max |ω−1

i |+ εC0 · n ≤ C1n for some other constant C1 ∈ N. So wehave reduced the problem to x = z ∈ mr = Cr−1(N) which is central in N. Wehave z = zn

r

1 where |z1| = |z|/n ≤ C1. Since Ω is a neighborhood of the identityin N, the set U of all products of at most dim(mr) simple commutators of lengthr of elements in Ω is a neighborhood of the identity in Cr−1(N) (e.g. see [19],p113). It follows that there is a constant C2 ∈ N such that z1 is in UC2 , hence theproduct of at most C2 dim(mr) simple commutators. Then we are done because zitself will be equal to the same product of commutators where each letter xi ∈ Ωis replaced by xni . This last fact follows from the following lemma:

Lemma 2.8. Let G be a nilpotent group of nilpotency class r and n1, ..., nr bepositive integers. Then for any x1, ..., xr ∈ G

[xn1

1 , [xn2

2 , [..., xnrr ]...] = [x1, [x2, [..., xr]...]

n1·...·nr

To prove the lemma it suffices to use induction and the following obvious fact:if [x, y] commutes to x and y then [xn, y] = [x, y]n.

Finally, we obtain:

Corollary 2.9. Let Ω be a compact neighborhood of the identity in N. Then thereare positive constants C1 and C2 such that for all n ∈ N,

C1nd ≤ volN (Ωn) ≤ C2n

d

where d is given by the Bass-Guivarc’h formula:

(17) d =∑

i≥1

i · dimmi

Proof. By Theorem 2.7, it is enough to estimate the volume of the quasi-normballs. By homogeneity of the quasi-norm, we have volNx, |x| ≤ t = tdvolNx, |x| ≤1.

Remark 2.10. The use of Malcev’s embedding theorem allows, as Guivarc’h ob-served, to deduce immediately that the analogous result holds for virtually nilpotentfinitely generated groups. This fact that was also proven independently by H. Bass


[3] by a direct combinatorial argument. See also Tits’ appendix to Gromov’s pa-per [17]. In fact Guivarc’h’s Theorem 2.7 seems to have been rediscovered severaltimes in the past 40 years, including by Pansu in his thesis [27], the latest exampleof that being [22].

3. The nilshadow

The goal of this section is to introduce the nilshadow of a simply connectedsolvable Lie group G. We will assume that G has polynomial growth, althoughthis last assumption is not necessary for almost everything we do in this section.The only statement which will be used afterwards in the paper (in Section 5) isLemma 3.12 below. The reader familiar with the nilshadow can jump directly tothe statement of this lemma and skip the forthcoming discussion.

3.1. Construction of the nilshadow. The nilshadow of G is a simply connectednilpotent Lie group GN , which is associated to G in a natural way. This notionwas first introduced by Auslander and Green in [2] in their study of flows onsolvmanifolds. They defined it as the unipotent radical of a semi-simple splittingof G. However, we are going to follow a different approach for its construction byworking first at the Lie algebra level. We refer the reader to the book [13] wherethis approach is taken up.

Let g be a solvable real Lie algebra and n the nilradical of g.We have [g, g] ⊂ n.If x ∈ g, we write ad(x) = ads(x) + adn(x) the Jordan decomposition of ad(x) inGL(g). Since ad(x) ∈ Der(g), the space of derivations of g, and Der(g) is the Liealgebra of the algebraic group Aut(g), the Jordan components ads(x) and adn(x)also belong to Der(g). Moreover, for each x ∈ g, ads(x) sends g into n (becauseso does ad(x) and ads(x) is a polynomial in ad(x)). Let h be a Cartan subalgebraof g, namely a nilpotent self-normalizing subalgebra. Recall that the image ofa Cartan subalgebra by a surjective Lie algebra homomorphism is again a Car-tan subalgebra. Now since g/n is abelian, it follows that h maps onto g/n, i.e.h+ n = g. Moreover ads(x)|h = 0 if x ∈ h, because h is nilpotent.

Now pick any real vector subspace v of h in direct sum with n. Then thefollowing two conditions hold:

(i) v⊕ n = g .(ii) ads(x)(y) = 0 for all x, y ∈ v.

From (i) and (ii), it follows easily that ads(x) commutes with ad(y), ads(y) andadn(y), for all x, y in v. We have:

Lemma 3.1. The map v → Der(g) defined by x 7→ ads(x) is a Lie algebrahomomorphism.

Proof. First let us check that this map is linear. Let x, y ∈ v. By the aboveads(y) and ads(x) commute with each other (hence their sum is semi-simple) andcommute with adn(x)+adn(y). From the uniqueness of the Jordan decomposition


it remains to check that adn(x)+adn(y) is nilpotent if x, y in v. To see this, applythe following obvious remark twice to a = adn(x) and V = ad(n) first and then toa = adn(y) and V = spanadn(x), ad((ad(y))

nx), n ≥ 1 : Let V be a nilpotentsubspace of GL(g) and a ∈ GL(g) nilpotent, i.e. V n = 0 and am = 0 for somen,m ∈ N and assume [a, V ] ⊂ V. Then (a+ V )nm = 0.

The fact that this map is a Lie algebra homomorphism follows easily from thefact that all ads(x), x ∈ v commute with one another and with [g, g] ⊂ n.

We define a new Lie bracket on g by setting:

(18) [x, y]N = [x, y]− ads(xv)(y) + ads(yv)(x)

where xv is the linear projection of x on v according to the direct sum v⊕n = g. TheJacobi identity is checked by a straightforward computation where the followingfact is needed: ads (ads(x)(y)) = 0 for all x, y ∈ g. This holds because, as we justsaw, ads(x)(g) ⊂ n for all x ∈ g, and ads(a) = 0 if a ∈ n.

Definition 3.2. Let gN be the vector space g endowed with the new Lie algebrastructure [·, ·]N given by (18). The nilshadow GN of G is defined to be the simplyconnected Lie group with Lie algebra gN .

It is easy to check that gN is a nilpotent Lie algebra. To see this, note first that[gN , gN ]N ⊂ n, and if x ∈ gN and y ∈ n then [x, y]N = (adn(xv) + ad(xn))(y).However, adn(xv) + ad(xn) is a nilpotent endomorphism of n as follows from thesame remark used in the proof of Lemma 3.1. Hence gN is a nilpotent.

The nilshadow Lie product on GN will be denoted by ∗ in order to distinguishit from the original Lie product on G. In the sequel, we will often identify G (resp.GN ) with its Lie algebra g (resp. gN ) via their respective exponential map. Sincethe underlying space of gN was g itself, this gives an identification (although nota group isomorphism) between G and GN . Then the nilshadow Lie product canbe expressed in terms of the original product as follows:

g ∗ h = g · (T (g−1)h)

Here T is the Lie group homomorphism G → Aut(G) induced by the abovechoice of supplementary subspace v as follows.

(19) T (ea)(eb) = exp(eads(av)b) ∀a, b ∈ g.

In other words, T is the unique Lie group homomorphism whose differentialat the identity is the Lie algebra homomorphism deT : g → Der(g) given bydeT (a)(b) = ads(av)b, that is the composition of the map v → Der(g) fromLemma 3.1 with the linear projection g → g/n ≃ v.

It is easy to check that this definition of the new product is compatible withthe definition of the new Lie bracket.

It can also be checked that two choices of supplementary spaces v as above yieldisomorphic Lie structures (see [13, Chap. III]). Hence by abuse of language, we


speak of the nilshadow of g, when we mean the Lie structure on G induced by achoice of v as above.

The following example shows several of the features of a typical solvable Liegroup of polynomial growth.

Example 3.3 (Nilshadow of a semi-direct product). Let G = R ⋉φ Rn where

φt ∈ GLn(R) is some one parameter subgroup given by φt = exp(tA) = ktut whereA is some matrix in Mn(R) and A = As +Au is its Jordan decomposition, givingrise to kt = exp(tAs) and ut = exp(tAu). The group G is diffeomorphic to R

n+1,hence simply connected. If all eigenvalues of As are purely imaginary, then G haspolynomial growth. However G is not nilpotent unless As = 0. So let us assumethat neither As nor Au is zero. Then the nilshadow GN is the semi-direct productR⋉u R

n where ut is the unipotent part of φt.It is easy to compute the homogeneous dimension of G (or GN) in terms of the

dimension of the Jordan blocs of Au. If nk is the number of Jordan blocks of Au

of size k, then

d(G) = 1 +∑

k≥1

k(k + 1)

2nk

3.2. Basic properties of the nilshadow. We now list in the form of a fewlemmas some basic properties of the nilshadow.

Lemma 3.4. The image of T : G → Aut(G) is abelian and relatively compact.Moreover T (T (g)h) = T (h) for any g, h ∈ G.

Proof. Since G has polynomial growth it is of type (R) by Guivarc’h’s theorem.Hence all ads(x) have purely imaginary eigenvalues. It follows that K is compact.Since T factors through the nilradical, its image is abelian. The last equalityfollows from (19) and the fact that ∀x, y ∈ g, ads(ads(x)(y)) = 0.

Lemma 3.5. T (G) also belongs to Aut(GN ) and T is a group homomorphismGN → Aut(GN ).

Proof. The first assertion follows from (19) and the fact that deT is a derivationof gN as one can check from (18) and the fact that ∀x, y ∈ g, ads(ads(x)(y)) = 0.The second assertion then follows from Lemma 3.4.

We denote by K the closure of T (G) in Aut(G) = Aut(g).

Lemma 3.6 (K-action on gN ). K preserves v and acts trivially on it. It alsopreserves the ideals n and the central descending series Ci(gN )i of gN .

Proof. It suffices to check that ads(v) preserves n and Ci(gN ). It preserves n

because ad(x) preserves n for all x ∈ g. It preserves Ci(gN ) because it acts as aderivation of gN as we have already checked in the proof of Lemma 3.5.

Remark 3.7 (Well-definedness of π1). It is also easy to check from the definitionof the nilshadow bracket that the commutator subalgebra [gN , gN ] and in fact eachterm of the central descending series Ci(gN ) is an ideal in g and does not dependon the choice of supplementary subspace v used to defined the nilshadow bracket.


In particular the projection map π1 : gN → gN/[gN , gN ] is a well defined linearmap on g = gN (i.e. independently of the choice involved in the construction ofthe nilshadow Lie bracket).

Lemma 3.8 (Exponential map). The respective exponential maps exp : g → Gand expN : gN → GN coincide on n and on v.

Proof. Since the two Lie products coincide on N = exp(n), so do their exponentialmap. For the second assertion, note that T (e−tv)v = v for every v ∈ v becauseads(x)(y) = 0 for all x, y ∈ ν. It follows that etvt is a one-parameter subgroupfor both Lie structures, hence it is equal to expN (tv)t.

Remark 3.9 (Surjectivity of the exponential map). The exponential map is not

always a diffeomorphism, as the example of the universal cover E of the group Eof motions of the plane shows (indeed any 1-parameter subgroup of E is either atranslation subgroup or a rotation subgroup, but the rotation subgroup is compacthence a torus, so its lift will contain the (discrete) center of E, hence will missevery lift of a non trivial translation). In fact, it is easy to see that if g is the Liealgebra of a solvable (non-nilpotent) Lie group of polynomial growth, then g mapssurjectively on the Lie algebra of E. Hence, for a simply connected solvable andnon-nilpotent Lie group of polynomial growth, the exponential map is never onto.Nevertheless its image is easily seen to be dense.

However, exponential coordinates of the second kind behave nicely. Note that[gN , gN ] ⊂ n.

Lemma 3.10 (Exponential coordinates of the second kind). Let Ci(gN )i≥0

be the central descending series of gN (with C1(gN ) = [gN , gN ]) and pick linearsubspaces mi in gN such that Ci(gN ) = mi ⊕ Ci−1(gN ) for i ≥ 2. Let ℓ be asupplementary subspace of C1(gN ) in n. Define exponential coordinates of thesecond kind by setting

mr ⊕ ...⊕m2 ⊕ ℓ⊕ v → G

(ξr, ..., ξ1, v) 7→ expN (ξr) ∗ . . . ∗ expN (ξ1) ∗ expN (v)

This map is a diffeomorphism. Moreover expN (ξr) ∗ . . . ∗ expN (ξ1) ∗ expN (v) =eξr · ... · eξ1 · ev for all choices of v ∈ v and ξi ∈ mi.

Proof. By Lemma 3.8 the exponential maps of G and GN coincide on n and on v.Moreover g ∗ h = g · h whenever g belongs to the nilradical exp(n) of G. HenceexpN (ξr)∗. . .∗expN (ξ1)∗expN (v) = expN (ξr)·. . .·expN (ξ1)·expN (v) = eξr ·...·eξ1 ·ev.The restriction of the map to n is a diffeomorphism onto exp(n), because this mapand its inverse are explicit polynomial maps (the ξi’s are coordinates of the secondkind, see the book [12]). Now the map n ⊕ v → G sending (n, v) to en · ev is adiffeomorphism, because G is simply connected and hence the quotient groupG/ exp(n) isomorphic to a vector space and hence to exp(v).

Lemma 3.11 (“Bi-invariant” Riemannian metric). There exists a Riemannianmetric on G which is left invariant under both Lie structures.


Proof. Indeed it suffices to pick a scalar product on g which is invariant under thecompact subgroup K = T (G) ⊂ Aut(g).

We identify K = T (g), g ∈ G with its image in Aut(g) under the canonicalisomorphism between Aut(G) and Aut(g). Recall that, according to Lemma 3.6,the central descending series of gN is invariant under ads(x) for all x ∈ v andconsists of ideals of g. The same holds for n. It follows that these linear subspacesalso invariant under K. However since K is compact, its action on g is completelyreducible. Therefore we have proved:

Lemma 3.12 (K-invariant stratification of the nilshadow). Let g be the Lie algebraof a simply connected Lie group G with polynomial growth. Let gN be the nilshadowLie algebra obtained from a splitting g = n⊕v as above (i.e. n is the nilradical and

v satisfies ads(x)(y) = 0 for every x, y ∈ v). Let K := T (g), g ∈ G ⊂ Aut(G),where T is defined by (19). Then there is a choice of linear subspaces mi’s and ℓsuch that

(20) gN = mr ⊕ . . . m2 ⊕ ℓ⊕ v,

where each term is K-invariant, m1 := ℓ⊕ v and the central descending series ofgN satisfies Ci(gN ) = mi ⊕ Ci−1(gN ). Moreover the action on K can be read offon the exponential coordinates of second kind in this decomposition, namely:

k(eξr · ... · eξ0

)= k(eξr) · ... · k(eξ0) = ek(ξr) · ... · ek(ξ0)

= expN (k(ξr)) ∗ ... ∗ expN (k(ξ0))

4. Periodic metrics

In this section, unless otherwise stated, G will denote an arbitrary locally com-pact group.

4.1. Definitions. By a pseudodistance (or metric) on a topological space X, wemean a function ρ : X × X → R+ satisfying ρ(x, y) = ρ(y, x) and ρ(x, z) ≤ρ(x, y) + ρ(y, z) for any triplet of points of X. However ρ(x, y) may be equal to 0even if x 6= y.

We will require our pseudodistances to be locally bounded, meaning that theimage under ρ of any compact subset of G × G is a bounded subset of R+. Toavoid irrelevant cases (for instance ρ ≡ 0) we will also assume that ρ is proper, i.e.the map y 7→ ρ(e, y) is a proper map, namely the preimage of a bounded set isbounded (we do not ask that the map be continuous). When ρ is locally boundedthen it is proper if and only if y 7→ ρ(x, y) is proper for any x ∈ G.

A pseudodistance ρ on G is said to be asymptotically geodesic if for every ε > 0there exists s > 0 such that for any x, y ∈ G one can find a sequence of pointsx1 = x, x2, ..., xn = y in G such that

(21)

n−1∑

i=1

ρ(xi, xi+1) ≤ (1 + ε)ρ(x, y)

and ρ(xi, xi+1) ≤ s for all i = 1, ..., n − 1.


We will consider exclusively pseudodistances on a group G that are invariantunder left translations by all elements of a fixed closed and co-compact subgroupH of G, meaning that for all x, y ∈ G and all h ∈ H, ρ(hx, hy) = ρ(x, y).

Combining all previous axioms, we set the following definition.

Definition 4.1. Let G be a locally compact group. A pseudodistance ρ on G willbe said to be a periodic metric (or H-periodic metric) if it satisfies the followingproperties:

(i) ρ is invariant under left translations by a closed co-compact subgroup H.(ii) ρ is locally bounded and proper.(iii) ρ is asymptotically geodesic.

Remark 4.2. The assumption that ρ is symmetric, i.e. ρ(x, y) = ρ(y, x) is hereonly for the sake of simplicity, and most of what is proven in this paper can bedone without this hypothesis.

4.2. Basic properties. Let ρ be a periodic metric on G and H some co-compactsubgroup of G. The following properties are straighforward.

(1) ρ is at a bounded distance from its restriction to H. This means that if Fis a bounded fundamental domain for H in G and for an arbitrary x ∈ G, if hxdenotes the element of H such that x ∈ hxF, then |ρ(x, y)− ρ(hx, hy)| ≤ C forsome constant C > 0.

(2) ∀t > 0 there exists a compact subset Kt of G such that, ∀x, y ∈ G, ρ(x, y) ≤t ⇒ x−1y ∈ Kt. And conversely, if K is a compact subset of G, ∃t(K) > 0 s.t.x−1y ∈ K ⇒ ρ(x, y) ≤ t(K).

(3) If ρ(x, y) ≥ s, the xi’s in (21) can be chosen in such a way that s ≤ρ(xi, xi+1) ≤ 2s (one can take a suitable subset of the original xi’s).

(4) The restriction of ρ to H × H is a periodic pseudodistance on H. Thismeans that the xi’s in (21) can be chosen in H.

(5) Conversely, given a periodic pseudodistance ρH on H, it is possible to extendit to a periodic pseudodistance on G by setting ρ(x, y) = ρH(hx, hy) where x =hxF for some bounded fundamental domain F for H in G.

4.3. Examples. Let us give a few examples of periodic pseudodistances.

(1) Let Γ be a finitely generated torsion free nilpotent group which is embeddedas a co-compact discrete subgroup of a simply connected nilpotent Lie group N .Given a finite symmetric generating set S of Γ, we can consider the correspondingword metric dS on Γ which gives rise to a periodic metric on N given by ρ(x, y) =dS(γx, γy) where x ∈ γxF and y ∈ γyF if F is some fixed fundamental domain forΓ in N.

(2) Another example, given in [27], is as follows. Let N/Γ be a nilmanifold withuniversal cover N and fundamental group Γ. Let g be a Riemannian metric onN/Γ. It can be lifted to the universal cover and thus gives rise to a Riemannianmetric g on N . This metric is Γ-invariant, proper and locally bounded. Since Γ isco-compact in N, it is easy to check that it is also asymptotically geodesic henceperiodic.


(3) Any word metric on G. That is, if Ω is a compact symmetric generatingsubset of G, let ∆Ω(x) = infn ≥ 1, x ∈ Ωn. Then define ρ(x, y) = ∆Ω(x

−1y).Clearly ρ is a pseudodistance (although not a distance) and it is G-invariant on theleft, it is also proper, locally bounded and asymptotically geodesic, hence periodic.

(4) If G is a connected Lie group, any left invariant Riemannian metric on G.Here again H = G and we obtain a periodic distance. Similarly, any left invariantCarnot-Caratheodory metric on G will do.

Remark 4.3 (Berestovski’s theorem). According to a result of Berestovski [5]every left-invariant geodesic distance on a connected Lie group is a subFinslermetric as defined in Paragraph 2.1.

4.4. Coarse equivalence between invariant pseudodistances. The followingproposition is basic:

Proposition 4.4. Let ρ1 and ρ2 be two periodic pseudodistances on G. Then thereis a constant C > 0 such that for all x, y ∈ G

(22)1

Cρ2(x, y)− C ≤ ρ1(x, y) ≤ Cρ2(x, y) + C

Proof. Clearly it suffices to prove the upper bound. Let s > 0 be the number cor-responding to the choice ε = 1 in (21) for ρ2. From 4.2 (2), there exists a compactsubset Ks in G such that ρ2(x, y) ≤ 2s ⇒ x−1y ∈ K2s, and there is a constantt = t(K2s) > 0 such that x−1y ∈ K2s ⇒ ρ1(x, y) ≤ t. Let C = max2t/s, t, andlet x, y ∈ G. If ρ2(x, y) ≤ s then ρ1(x, y) ≤ t so the right hand side of (22) holds.If ρ2(x, y) ≥ s then, from (21) and 4.2 (3), we get a sequence of xi’s in G from x

to y such that s ≤ ρ2(xi, xi+1) ≤ 2s and∑N

1 ρ2(xi, xi+1) ≤ 2ρ2(x, y). It followsthat ρ1(xi, xi+1) ≤ t for all i. Hence ρ1(x, y) ≤

∑ρ1(xi, xi+1) ≤ Nt ≤ 2

s tρ2(x, y)and the right hand side of (22) holds.

In the particular case when G = N is a simply connected nilpotent Lie group,the distance to the origin x 7→ ρ(e, x) is also coarsely equivalent to any homoge-neous quasi-norm on N. We have,

Proposition 4.5. Suppose N is a simply connected nilpotent Lie group. Let ρ1 bea periodic pseudodistance on N and | · | be a homogeneous quasi-norm, then thereexists C > 0 such that for all x ∈ N

(23)1

C|x−1y| − C ≤ ρ1(x, y) ≤ C|x−1y|+ C

Moreover, if ρ2 is a periodic pseudodistance on the stratified nilpotent group N∞

associated to N, then again, there is a constant C > 0 such that

(24)1

Cρ2(e, x)− C ≤ ρ1(e, x) ≤ Cρ2(e, x) + C

The proposition follows at once from Guivarc’h’s theorem (see Corollary 2.7above), the equivalence of homogeneous quasi-norms, and the fact that left-invariantCarnot-Caratheodory metrics on N∞ are homogeneous quasi norms. However,since the group structures on N and N∞ differ, (24) cannot in general be replacedby the stronger relation (22) as simple examples show.


The next proposition is of fundamental importance for the study of metrics onLie groups of polynomial growth:

Proposition 4.6. Let G be a simply connected solvable Lie group of polynomialgrowth and GN its nilshadow. Let ρ and ρN be arbitrary periodic pseudodistanceson G and GN respectively. Then there is a constant C > 0 such that for allx, y ∈ G

(25)1

CρN (x, y)− C ≤ ρ(x, y) ≤ CρN (x, y) + C

Proof. According to Proposition 4.4, it is enough to show (25) for some choice ofperiodic metrics on G and GN . But in Lemma 3.11 we constructed a Riemannianmetric on G which is left invariant for both G and GN . We are done.

4.5. Right invariance under a compact subgroup. Here we verify that, givena compact subgroup of G, any periodic metric is at bounded distance from anotherperiodic metric which is invariant on the right by this compact subgroup. Let Kbe a compact subgroup of G and ρ a periodic pseudodistance on G. We average ρwith the help of the normalized Haar measure on K to get:

(26) ρK(x, y) =

∫

K×Kρ(xk1, yk2)dk1dk2

Then the following holds:

Lemma 4.7. There is a constant C0 > 0 depending only on ρ and K such thatfor all k1, k2 ∈ K and all x, y ∈ G

(27) |ρ(xk1, yk2)− ρ(x, y)| ≤ C0

Proof. From 4.2 (2), ∃t = t(K) > 0 s.t. ∀x ∈ G, ρ(x, xk) ≤ t. Applying thetriangle inequality, we are done.

Hence we obtain:

Proposition 4.8. The pseudodistance ρK is periodic and lies at a bounded dis-tance from ρ. In particular, as x tends to infinity in G the following limit holds

(28) limx→∞

ρK(e, x)

ρ(e, x)= 1

Proof. From Lemma 4.7 and 4.2 (3), it is easy to check that ρK must be asymp-totically geodesic, and periodic. Integrating (27) we get that ρK is at a boundeddistance from ρ and (28) is obvious.

If K is normal in G, we thus obtain a periodic metric ρK on G/K such thatρK(p(x), p(y)) is at a bounded distance from ρ(x, y), where p is the quotient mapG→ G/K.


5. Reduction to the nilpotent case

In this section, G denotes a simply connected solvable Lie group of polynomialgrowth. We are going to reduce the proof of the theorems of the Introductionto the case of a nilpotent G. This is performed by showing that any periodicpseudodistance ρ on G is asymptotic to some associated periodic pseudodistanceρN on the nilshadow GN . We state this in Proposition 5.1 below.

The key step in the proof is Proposition 5.2 below, which shows the asymptoticinvariance of ρ under the “semisimple part” of G. The crucial fact there is that thedisplacement of a distant point under a fixed unipotent automorphism is negligiblecompared to the distance from the identity (see Lemmas 5.4, 5.5), so that theaction of the semisimple part of large elements can be simply approximated bytheir action by left translation.

5.1. Asymptotic invariance under a compact group of automorphismsof G. The main result of this section is the following. Let G be a connected andsimply connected solvable Lie group with polynomial growth and GN its nilshadow(see Section 3).

Proposition 5.1. Let H be a closed co-compact subgroup of G and ρ an H-periodic pseudodistance (see Definition 4.1) on G. There exist a closed subset HK

containing H which is a co-compact subgroup for both G and GN , and an HK-periodic (for both Lie structures) pseudodistance ρK such that

(29) limx→∞

ρK(e, x)

ρ(e, x)= 1

The closed subgroup HK will be taken to be the closure of the group generatedby all elements of the form k(h), where h belongs to H and k belongs to theclosure K in the group Aut(G) of the image of H under the homomorphismT : G → Aut(G) introduced in Section 3. It is easy to check from the definitionof the nilshadow product (1) that this is indeed a subgroup in both G and itsnilshadow GN .

The new pseudodistance ρK is defined as follows, using a double averagingprocedure:

(30) ρK(x, y) :=

∫

H\HK

∫

Kρ(gk(x), gk(y))dkdµ(g)

Here the measure µ is the normalized Haar measure on the coset space H\HK

and dk is the normalized Haar measure on the compact group K. Recall thatall closed subgroups of S are unimodular (since they have polynomial growth by[21][Lemme I.3.]). Hence the existence of invariant measures on the coset spaces.

An essential part of the proof of Proposition 5.1 is enclosed in the followingstatement:


Proposition 5.2. Let ρ be a periodic pseudodistance on G which is invariantunder a co-compact subgroup H. Then ρ is asymptotically invariant under theaction of K = T (h), h ∈ H ⊆ Aut(G). Namely, (uniformly) for all k ∈ K,

(31) limx→∞

ρ(e, k(x))

ρ(e, x)= 1

The proof of Proposition 5.2 splits into two steps. First we show that it isenough to prove (31) for a dense subset of k’s. This is a consequence of the followingcontinuity statement:

Lemma 5.3. Let ε > 0, then there is a neighborhood U of the identity in K suchthat, for all k ∈ U,

limx→∞ρ(x, k(x))

ρ(e, x)< ε

Then we show that the action of T (g) can be approximated by the conjugationby g, essentially because the unipotent part of this conjugation does not move xvery much when x is far. This is the content of the following lemma:

Lemma 5.4. Let ρ be a periodic pseudodistance on G which is invariant undera co-compact subgroup H. Then for any ε > 0, and any compact subset F in Hthere is s0 > 0 such that

|ρ(e, T (h)x) − ρ(e, hx)| ≤ ερ(e, x)

for any h ∈ F and as soon as ρ(e, x) > s0.

Proof of Proposition 5.2 modulo Lemmas (5.3) and (5.4): As ρ is assumed tobe H-invariant, for every h ∈ H, we have ρ(e, h−1x)/ρ(e, x) → 1. The proof ofthe proposition then follows immediately from the combination of the last twolemmas.

5.2. Proof of Lemmas (5.3) and (5.4). We choose K-invariant subspaces mi’sand ℓ of the nilshadow gN of g as in Lemma 3.12 from Section 3. In particular

gN = mr ⊕ . . . ⊕m2 ⊕ ℓ⊕ v,

where each term is K-invariant, n = [gN , gN ] ⊕ l and Ci(gN ) = mi ⊕ Ci−1(gN ).Moreover δt(x) = tix if x ∈ mi (here m1 = ℓ⊕ v).

We also set v(x) = maxi ‖ξi‖1/dii if x = expN (ξr) ∗ . . . ∗ expN (ξ0) and di = i if

i > 0 and d0 = 1. And we let |x| := maxi ‖xi‖1/di if x = xr + . . . + x1 + x0 in the

above direct sum decomposition.Note that | · | is a δt-homogeneous quasi-norm. Moreover, it is straightforward

to verify (using the Campbell-Hausdorff formula (12) and Proposition 2.4) thatv(x) ≤ C|x|+C for some constant C > 0. In particular ξi/|x|

di remains boundedas |x| becomes large.

Proof of Lemma 5.3. Combining Propositions 4.5 and 4.6, there is a constantC > 0 such that for all x, y ∈ G, ρ(x, y) ≤ C|x∗−1 ∗ y| + C. Therefore we have


reduced to prove the statement for | · | instead of ρ, namely it is enough to showthat |x∗−1 ∗ k(x)| becomes negligible compared to |x| as |x| goes to infinity and ktends to 1.

It follows from the Campbell-Baker-Hausdorff formula (11) and (12) that, if

x, y ∈ GN and |x|, |y| are O(t), then |δ 1

t(x ∗ y) − δ 1

t(x) ∗ δ 1

t(y)| = O(t−1/r), and

similarly |δ 1

t(x1 ∗ . . . ∗ xm) − δ 1

t(x1) ∗ . . . ∗ δ 1

t(xm)| = Om(t−1/r), for m elements

xi with |xi| = O(t). Hence when writing x = expN (ξr) ∗ ... ∗ expN (ξ0), and settingt = |x|, we thus obtain that the following quantity

∣∣∣∣∣∣δ 1

t(x∗−1 ∗ k(x))−

∗∏

0≤i≤r

expN (−t−diξi) ∗

∗∏

0≤i≤r

expN (t−dr−ik(ξr−i))

∣∣∣∣∣∣

is a O(t−1/r). Indeed recall from Lemma 3.12 that k(x) = expN (k(ξr)) ∗ ... ∗expN (k(ξ0)). As x gets larger, each t−diξi remains in a compact subset of mi.Therefore, as k tends to the identity in K, each t−dik(ξi) becomes uniformly closeto t−diξi independently of the choice of x ∈ GN as long as t = |x| is large. Theresult follows.

Proof of Lemma 5.4. Recall that hx = h ∗ T (h)x for all x, h ∈ G (see (1).By the triangle inequality it is enough to bound ρ(y, h ∗ y), where y = T (h)x.From Propositions 4.5 and 4.6, ρ is comparable (up to multiplicative and additiveconstants to the homogeneous quasi-norm | · |. Hence the Lemma follows from thefollowing:

Lemma 5.5. Let N be a simply connected nilpotent Lie group and let | · | be ahomogeneous quasi norm on N associated to some 1-parameter group of dilations(δt)t. For any ε > 0 and any compact subset F of N, there is a constant s2 > 0such that

|x−1gx| ≤ ε|x|

for all g ∈ F and as soon as |x| > s2.

Proof. Recall, as in the proof of the last lemma, that for any c1 > 0 there isa c2 > 0 such that if t > 1 and x, y ∈ N are such that |x|, |y| ≤ c1t, then

|δ 1

t(xy)− δ 1

t(x) ∗ δ 1

t(y)| ≤ c2t

−1/r. In particular, if we set t = |x|, then

∣∣∣δ 1

t(x−1gx)− δ 1

t(x)−1 ∗ δ 1

t(g) ∗ δ 1

t(x)

∣∣∣ ≤ c2t−1/r

On the other hand, as g remains in the compact set F, δ 1

t(g) tends uniformly to

the identity when t = |x| goes to infinity, and δ 1

t(x) remains in a compact set.

By continuity, we see that δ 1

t(x)−1 ∗ δ 1

t(g) ∗ δ 1

t(x) becomes arbitrarily small as t

increases. We are done.


5.3. Proof of Proposition 5.1. First we prove the following continuity state-ment:

Lemma 5.6. Let ρ be a periodic pseudodistance on G and ε > 0. Then thereexists a neighborhood of the identity U in G and s3 > 0 such that

1− ε ≤ρ(e, gx)

ρ(e, x)≤ 1 + ε

as soon g ∈ U and ρ(e, x) > s3.

Proof. Let ρN be a left invariant Riemannian metric on the nilshadow GN .

|ρ(e, x)− ρ(e, gx)| ≤ ρ(x, gx) ≤ ρ(x, g ∗ x) + ρ(g ∗ x, gx)

However ρ(a, b) ≤ CρN(a, b)+C for some C > 0 by Proposition 4.6. Moreover by(1) we have gx = g ∗ T (g)x. Hence

|ρ(e, x) − ρ(e, gx)| ≤ CρN (x, g ∗ x) + CρN (x, T (g)x) + 2C

To complete the proof, we apply Lemmas 5.5 and 5.3 to the right hand side above.

We proceed with the proof of Proposition 5.1. Let L be the set of all g ∈ Gsuch that ρ(e, gx)/ρ(e, x) tends to 1 as x tends to infinity in G. Clearly L is asubgroup of G. Lemma 5.6 shows that L is closed. The H-invariance of ρ insuresthat L contains H. Moreover, Proposition 5.2 implies that L is invariant under K.Consequently L contains HK , the closed subgroup generated by all k(h), k ∈ K,h ∈ H. This, together with Proposition 5.2, grants pointwise convergence of theintegrand in (29). Convergence of the integral follows by applying Lebesgue’sdominated convergence theorem.

The fact that ρK is invariant under left multiplication by H and invariant underprecomposition by automorphisms from K insures that ρK is invariant under ∗-left multiplication by any element h ∈ H, where ∗ is the multiplication in thenilshadow GN . Moreover we check that T (g) ∈ K if g ∈ HK , hence HK is asubgroup of GN . It is clearly co-compact in GN too (if F is compact and HF = Gthen H ∗ FK = G where FK is the union of all k(F ), k ∈ K).

Clearly ρK is proper and locally bounded, so in order to finish the proof, weneed only to check that ρK is asymptotically geodesic. By H-invariance of ρK andsince H is co-compact in G, it is enough to exhibit a pseudogeodesic between eand a point x ∈ H. Let x = z1 · ... ·zn with zi ∈ H and

∑ρ(e, zi) ≤ (1+ε) ·ρ(e, x).

Fix a compact fundamental domain F for H in HK so that integration in (29)over H\HK is replaced by integration over F. Then for some constant CF > 0 wehave |ρ(g, gz) − ρ(e, gz)| ≤ CF for g ∈ F and z ∈ H. Moreover, it follows fromProposition 5.2, Lemma 5.6 and the fact that HK ⊂ L, that

(32) ρ(e, gk(z)) ≤ (1 + ε) · ρ(e, z)

for all g ∈ F, k ∈ K and as soon as z ∈ G is large enough. Fix s large enoughso that CF ≤ εs and so that (32) holds when ρ(e, z) ≥ s. As already observed inthe discussion following Definition 4.1 (property 4.2 (3)) we may take the zi’s so


that s2 ≤ ρ(e, zi) ≤ s. Then nCF ≤ nsε ≤ 3ερ(e, x). Finally we get for ε < 1 and

x large enough∑

ρK(e, zi) ≤ CFn+ (1 + ε)2ρ(e, x)

≤ CFn+ (1 + ε)3ρK(e, x)

≤ (1 + 10ε) · ρK(e, x)

where we have used the convergence ρK/ρ→ 1 that we just proved.

6. The nilpotent case

In this section, we prove Theorem 1.4 and its corollaries stated in the Intro-duction for a simply connected nilpotent Lie group. We essentially follow Pansu’sargument from [27], although our approach differs somewhat in its presentation.Throughout the section, the nilpotent Lie group will be denoted by N, and its Liealgebra by n.

Let m1 be any vector subspace of n such that n = m1 ⊕ [n, n]. Let π1 theassociated linear projection of n onto m1. Let H be a closed co-compact subgroupof N . To every H-periodic pseudodistance ρ on N we associate a norm ‖·‖0 onm1 which is the norm whose unit ball is defined to be the closed convex hull of allelements π1(h)/ρ(e, h) for all h ∈ H\e. In other words,

(33) E := x ∈ m1, ‖x‖0 ≤ 1 = CvxHull

π1(h)

ρ(e, h), h ∈ H\e

The set E is clearly a convex subset of m1 which is symmetric around 0 (since ρis symmetric). To check that E is indeed the unit ball of a norm on m1 it remainsto see that E is bounded and that 0 lies in its interior. The first fact followsimmediately from (23) and Example 2.3. If 0 does not lie in the interior of E,then E must be contained in a proper subspace of m1, contradicting the fact thatH is co-compact in N .

Taking large powers hn, we see that we can replace the set H \e in the abovedefinition by any neighborhood of infinity in H. Similarly, it is easy to see thatthe following holds:

Proposition 6.1. For s > 0 let Es be the closed convex hull of all π1(x)/ρ(e, x)with x ∈ N and ρ(e, x) > s. Then E =

⋂s>0Es.

Proof. Since ρ is H-periodic, we have ρ(e, hn) ≤ nρ(e, h) for all n ∈ N and h ∈ H.This shows E ⊂

⋂s>0Es. The opposite inclusion follows easily from the fact that

ρ is at a bounded distance from its restriction to H, i.e. from 4.2 (1).

We now choose a set of supplementary subspaces (mi) starting with m1 as inParagraph 2.2. This defines a new Lie product ∗ on N so that N∞ = (N, ∗) isstratified. We can then consider the ∗-left invariant Carnot-Caratheodory metricassociated to the norm ‖·‖0 as defined in Paragraph 2.1 on the stratified nilpotentLie group N∞. In this section, we will prove Theorem 1.4 for nilpotent groups inthe following form:


Theorem 6.2. Let ρ be a periodic pseudodistance on N and d∞ the Carnot-Caratheodory metric defined above, then as x tends to infinity in N

(34) limρ(e, x)

d∞(e, x)= 1

Note that d∞ is left-invariant for the N∞ Lie product, but not the original Lieproduct on N .

Before going further, let us draw some simple consequences.(1) In Theorem 6.2 we may replace d∞(e, x) by d(e, x), where d is the left

invariant Carnot-Caratheodory metric on N (rather than N∞) defined by thenorm ‖·‖0 (as opposed to d∞ which is ∗-left invariant). Hence ρ, d and d∞ areasymptotic. This follows from the combination of Theorem 6.2 and Remark 2.1.

(2) Observe that the choice ofm1 was arbitrary. Hence two Carnot-Caratheodorymetrics corresponding to two different choices of a supplementary subspace m1

with the same induced norm on n/[n, n], are asymptotically equivalent (i.e. theirratio tends to 1), and in fact isometric (see Remark 2.1). Conversely, if twoCarnot-Caratheodory metrics are associated to the same supplementary subspacem1 and are asymptotically equivalent, they must be equal. This shows that theset of all possible norms on the quotient vector space n/[n, n] is in bijection withthe set of all classes of asymptotic equivalence of Carnot-Caratheodory metrics onN∞.

(3) As another consequence we see that if a locally bounded proper and asymp-totically geodesic left-invariant pseudodistance on N is also homogeneous withrespect to the 1-parameter group (δt)t (i.e. ρ(e, δtx) = tρ(e, x)) then it has to beof the form ρ(x, y) = d∞(e, x−1y) where d∞ is a Carnot-Caratheodory metric onN∞.

6.1. Volume asymptotics. Theorem 6.2 also yields a formula for the asymptoticvolume of ρ-balls of large radius. Let us fix a Haar measure on N (for exampleLebesgue measure on n gives rise to a Haar measure on N under exp). Since d∞is homogeneous, it is straightforward to compute the volume of a d∞-ball:

vol(x ∈ N, d∞(e, x) ≤ t) = td(N)vol(x ∈ N, d∞(e, x) ≤ 1)

where d(N) =∑

i≥1 dim(Ci(n)) is the homogeneous dimension of N. For a pseu-dodistance ρ as in the statement of Theorem 6.2, we can define the asymptotic vol-ume of ρ to be the volume of the unit ball for the associated Carnot-Caratheodorymetric d∞.

AsV ol(ρ) = vol(x ∈ N, d∞(e, x) ≤ 1)

Then we obtain as an immediate corollary of Theorem 6.2:

Corollary 6.3. Let ρ be periodic pseudodistance on N. Then

limt→+∞

1

td(N)vol(x ∈ N, ρ(e, x) ≤ t) = AsV ol(ρ) > 0

Finally, if Γ is an arbitrary finitely generated nilpotent group, we need to takecare of the torsion elements. They form a normal finite subgroup T and applyingTheorem 6.2 to Γ/T , we obtain:


Corollary 6.4. Let S be a finite symmetric generating set of Γ and Sn the ballof radius n is the word metric ρS associated to S, then

limn→+∞

1

nd(N)#Sn = #T ·

AsV ol(ρS)

vol(N/Γ)> 0

where N is the Malcev closure of Γ = Γ/T , the torsion free quotient of Γ, and dSis the word pseudodistance associated to S, the projection of S in Γ.

Moreover, it is possible to be a bit more precise about AsV ol(ρS). In fact,the norm ‖·‖0 on m1 used to define the limit Carnot-Caratheodory distance d∞associated to ρS is a simple polyhedral norm defined by

‖x‖0 ≤ 1 = CvxHull (π1(s), s ∈ S)

More generally the following holds. Let H be any closed, co-compact subgroupof N. Choose a Haar measure on H so that volN (N/H) = 1. Theorem 6.2 yields:

Corollary 6.5. Let Ω be a compact symmetric (i.e. Ω = Ω−1) neighborhood ofthe identity, which generates H. Let ‖·‖0 be the norm on m1 whose unit ball is

CvxHullπ1(Ω) and let d∞ be the corresponding Carnot-Caratheodory metric onN∞. Then we have the following limit in the Hausdorff topology

limn→+∞

δ 1

n(Ωn) = g ∈ N, d∞(e, g) ≤ 1

and

limn→+∞

volH(Ωn)

nd(N)= volN (g ∈ N, d∞(e, g) ≤ 1)

6.2. Outline of the proof. We first devise some standard lemmas about piece-wise approximations of horizontal paths (Lemmas 6.6, 6.7, 6.10). Then it is shown(Lemma 6.11) that the original product on N and the product in the associatedgraded Lie group are asymptotic to each other, namely, if (δt)t is a 1-parametergroup of dilations of N, then after renormalization by δ 1

t, the product of O(t)

elements lying in some bounded subset of N, is very close to the renormalizedproduct of the same elements in the graded Lie group N∞. This is why all com-plications due to the fact that N may not be a priori graded and the δt’s maynot be automorphisms disappear when looking at the large scale geometry of thegroup. Finally, we observe (Lemma 6.13), as follows from the very definition ofthe unit ball E for the limit norm ‖·‖0 , that any vector in the boundary of E,can be approximated, after renormalizing by δ 1

sby some element x ∈ N lying in

a fixed annulus s(1 − ε) ≤ ρ(e, x) ≤ s(1 + ε). This enables us to assert that anyρ-quasi geodesic gives rise, after renormalization, to a d∞-geodesic (this gives thelower bound in Theorem 6.2). And vice-versa, that any d∞-geodesic can be ap-proximated uniformly by some renormalized ρ-quasi geodesic (this gives the upperbound in Theorem 6.2).


6.3. Preliminary lemmas.

Lemma 6.6. Let G be a Lie group and let ‖·‖e be a Euclidean norm on the Liealgebra of G and de(·, ·) the associated left invariant Riemannian metric on G.Let K be a compact subset of G. Then there is a constant C0 = C0(de,K) > 0such that whenever de(e, u) ≤ 1 and x, y ∈ K

|de(xu, yu)− de(x, y)| ≤ C0de(x, y)de(e, u)

Proof. The proof reduces to the case when u and x−1y are in a small neighborhoodof e. Then the inequality boils down to the following ‖[X,Y ]‖e ≤ c ‖X‖e ‖Y ‖e forsome c > 0 and every X,Y in Lie(G).

Lemma 6.7. Let G be a Lie group, let ‖·‖ be some norm on the Lie algebra ofG and let de(·, ·) be a left invariant Riemannian metric on G. Then for everyL > 0 there is a constant C = C(de, ‖·‖ , L) > 0 with the following property.Assume ξ1, ξ2 : [0, 1] → G are two piecewise smooth paths in the Lie group Gwith ξ1(0) = ξ2(0) = e. Let ξ′i ∈ Lie(G) be the tangent vector pulled back at theidentity by a left translation of G. Assume that supt∈[0,1] ‖ξ

′1(t)‖ ≤ L, and that∫ 1

0 ‖ξ′1(t)− ξ′2(t)‖ dt ≤ ε. Then

de(ξ1(1), ξ2(1)) ≤ Cε

Proof. The function f(t) = de(ξ1(t), ξ2(t)) is piecewise smooth. For small dt wemay write, using Lemma 6.6

f(t+ dt)− f(t) ≤ de(ξ1(t)ξ′1(t)dt, ξ1(t)ξ

′2(t)dt) + de(ξ1(t)ξ

′2(t)dt, ξ2(t)ξ

′2(t)dt)− f(t) + o(dt)

≤∥∥ξ′1(t)− ξ′2(t)

∥∥edt+ C0f(t)

∥∥ξ′2(t)dt∥∥e+ o(dt)

≤ ε(t)dt +C0Lf(t)dt+ o(dt)

where ε(t) = ‖ξ′1(t)− ξ′2(t)‖e . In other words,

f ′(t) ≤ ε(t) + C0Lf(t)

Since f(0) = 0, Gronwall’s lemma implies that f(1) ≤ eC0L∫ 10 ε(s)e

−C0Lsds ≤ Cε.

From now on, we will take G to be the stratified nilpotent Lie group N∞, andde(·, ·) will denote a left invariant Riemannian metric on N∞ while d∞(·, ·) is a leftinvariant Carnot-Caratheodory Finsler metric on N∞ associated to some norm ‖·‖on m1.

Remark 6.8. There is c0 > 0 such that c−10 de(e, x) ≤ d∞(e, x) ≤ c0de(e, x)

1

r in aneighborhood of e. Hence in the situation of the lemma we get d∞(ξ1(1), ξ2(1)) ≤

C1ε1

r for some other constant C1 = C1(L, d∞, de).

Lemma 6.9. Let N ∈ N and dN (x, y) be the function in N∞ defined in thefollowing way:

dN (x, y) = inf

∫ 1

0

∥∥ξ′(u)∥∥ du, ξ ∈ HPL(N), ξ(0) = x, ξ(1) = y


where HPL(N) is the set of horizontal paths ξ which are piecewise linear with atmost N possible values for ξ′. Then we have dN → d∞ uniformly on compactsubsets of N∞.

Proof. Note that it follows from Chow’s theorem (e.g. see [25] or [19]) that thereexists K0 ∈ N such that A := supd∞(e,x)=1 dK0

(e, x) < ∞. Moreover, since piece-

wise linear paths are dense in L1, it follows for example from Lemma 6.7 that foreach fixed x, dn(e, x) → d∞(e, x). We need to show that dN (e, x) → d∞(e, x) uni-formly in x satisfying d∞(e, x) = 1. By contradiction, suppose there is a sequence(xn)n such that d∞(e, xn) = 1 and dn(e, xn) ≥ 1 + ε0 for some ε0 > 0. We mayassume that (xn)n converges to say x. Let yn = x−1 ∗xn and tn = d∞(e, yn). ThendK0

(e, yn) = tndK0(e, δ 1

tn

(yn)) ≤ Atn. Thus dn(e, xn) ≤ dn(e, x) + dn(e, yn) ≤

dn(e, x) +Atn as soon as n ≥ K0. As n tends to ∞, we get a contradiction.

This lemma prompts the following notation. For ε > 0, we let Nε ∈ N be thefirst integer such that 1 ≤ dNε(e, x) ≤ 1 + ε for all x with d∞(e, x) = 1. Then wehave:

Lemma 6.10. For every x ∈ N∞ with d∞(e, x) = 1, and all ε > 0 there exists apath ξ : [0, 1] → N∞ in HPL(Nε) with unit speed (i.e. ‖ξ′‖ = 1) such that ξ(0) = eand d∞(x, ξ(1)) ≤ C2ε and ξ′ has at most one discontinuity on any subinterval of[0, 1] of length εr/Nε.

Proof. We know that there is a path in HPL(Nε) connecting e to x with lengthℓ ≤ 1 + ε. Reparametrizing the path so that it has unit speed, we get a pathξ0 : [0, ℓ] → N∞ in HPL(Nε) with d∞(x, ξ0(1)) = d∞(ξ0(ℓ), ξ0(1)) ≤ ε. The deriva-

tive ξ′0 is constant on at most Nε different intervals say [ui, ui+1). Let us removeall such intervals of length ≤ εr/Nε by merging them to an adjacent intervaland let us change the value of ξ′0 on these intervals to the value on the adja-cent interval (it doesn’t matter if we choose the interval on the left or on theright). We obtain a new path ξ : [0, 1] → N∞ in HPL(Nε) with unit speed and

such that ξ′ has at most one discontinuity on any subinterval of [0, 1] of length

εr/Nε. Moreover∫ 10 ‖ξ′(t)− ξ′0(t)‖ dt ≤ εr. By Lemma 6.7 and Remark 6.3, we

have d∞(ξ(1), ξ0(1)) ≤ C1ε, hence

d∞(ξ(1), x) ≤ d∞(x, ξ0(1)) + d∞(ξ0(1), ξ(1)) ≤ (C1 + 1)ε

Lemma 6.11 (Piecewise horizontal approximation of paths). Let x∗y denote theproduct inside the stratified Lie group N∞ and x · y the ordinary product in N .Let n ∈ N and t ≥ n. Then for any compact subset K of N , and any x1, ..., xnelements of K, we have

de(δ 1

t(x1 · ... · xn), δ 1

t(x1 ∗ ... ∗ xn)) ≤ c1

1

t

and

de(δ 1

t(x1 ∗ ... ∗ xn), δ 1

t(π1(x1) ∗ ... ∗ π1(xn))) ≤ c2

1

t


where c1, c2 depend on K and de only.

Proof. Let ‖·‖ be a norm on the Lie algebra of N. For k = 1, ..., n let zk =x1 · ... ·xk−1 and yk = xk+1 ∗ ...∗xn. Since all xi’s belong to K, it follows from (24)that as soon as t ≥ n, all δ 1

t(zk) and δ 1

t(yk) for k = 1, ..., n remain in a bounded set

depending only on K. Comparing (12) and (11), we see that whenever y = O(1)and δ 1

t(x) = O(1), we have

(35)∥∥∥δ 1

t(xy)− δ 1

t(x ∗ y)

∥∥∥ = O(1

t2)

On the other hand, from (12) it is easy to verify that right ∗-multiplication by abounded element is Lipschitz for ‖·‖ and the Lipschitz constant is locally bounded.It follows that there is a constant C1 > 0 (depending only on K and ‖·‖) suchthat for all k ≤ n∥∥∥δ 1

t((zk · xk) ∗ yk)− δ 1

t(zk ∗ xk ∗ yk)

∥∥∥ ≤ C1

∥∥∥δ 1

t(zk · xk)− δ 1

t(zk ∗ xk)

∥∥∥

Applying n times the relation (35) with x = x1 · ... · xk−1 and y = xk, we finallyobtain ∥∥∥δ 1

t(x1 · ... · xn)− δ 1

t(x1 ∗ ... ∗ xn)

∥∥∥ = O(n

t2) = O(

1

t)

where O() depends only on K. On the other hand, using (11), it is another simpleverification to check that if x, y lie in a bounded set, then 1

c2de(x, y) ≤ ‖x− y‖

≤ c2de(x, y) for some constant c2 > 0. The first inequality follows.For the second inequality, we apply Lemma 6.7 to the paths ξ1 and ξ2 starting

at e and with derivative equal on [ kn ,k+1n ) to nδ 1

t(xk) for ξ1 and to nπ1(xk)

t for ξ2.

We get

de(δ 1

t(x1 ∗ ... ∗ xn), δ 1

t(π1(x1) ∗ ... ∗ π1(xn)) = O(

1

t).

Remark 6.12. From Remark 6.3 we see that if we replace de by d∞ in the above

lemma, we get the same result with 1t replaced by t−

1

r .

Lemma 6.13 (Approximation in the abelianized group). Recall that ‖·‖0 is thenorm on m1 defined in (33). For any ε > 0, there exists s0 > 0 such that for everys > s0 and every v ∈ m1 such that ‖v‖0 = 1, there exists h ∈ H such that

(1− ε)s ≤ ρ(e, h) ≤ (1 + ε)s

and ∥∥∥∥π1(h)

ρ(e, h)− v

∥∥∥∥0

≤ ε

Proof. Let ε > 0 be fixed. Considering a finite ε-net in E, we see that there existsa finite symmetric subset g1, ..., gp of H\e such that, if we consider the closedconvex hull of F = fi = π1(gi)/ρ(e, gi)|i = 1, ..., p and ‖·‖ε the associated normon m1, then ‖·‖0 ≤ ‖·‖ε ≤ (1 + 2ε) ‖·‖0. Up to shrinking F if necessary, we mayassume that ‖fi‖ε = 1 for all i’s. We may also assume that the fi’s generate m1 as


a vector space. The sphere x, ‖x‖ε = 1 is a symmetric polyhedron in m1 and toeach of its facets corresponds d = dim(m1) vertices lying in F and forming a vectorbasis of m1. Let f1, ..., fd, say, be such vertices for a given facet. If x ∈ m1 is of the

form x =∑d

i=1 λifi with λi ≥ 0 for 1 ≤ i ≤ d then we see that ‖x‖ε =∑d

i=1 λi,because the convex hull of f1, ..., fd is precisely that facet, hence lies on the spherex, ‖x‖ε = 1.

Now let v ∈ m1, ‖v‖0 = 1, and let s > 0. The half line tv, t > 0, hits thesphere x, ‖x‖ε = 1 in one point. This point belongs to some facet and thereare d linearly independent elements of F, say f1, ..., fd, the vertices of that facet,such that this point belongs to the convex hull of f1, ..., fd. The point sv then liesin the convex cone generated by π1(g1), ..., π1(gd). Moreover, there is a constantCε > 0 (Cε ≤

d2 max1≤i≤p ρ(e, gi)) such that

∥∥∥∥∥sv −d∑

i=1

niπ1(gi)

∥∥∥∥∥ε

≤ Cε

for some non-negative integers n1, ..., nd depending on s > 0. Hence

1

s

d∑

i=1

niρ(e, gi) =1

s

∥∥∥∥∥

d∑

i=1

niπ1(gi)

∥∥∥∥∥ε

≤1

s(‖sv‖ε + Cε)

≤ 1 + 2ε +Cε

s≤ 1 + 3ε

where the last inequality holds as soon as s > Cε/ε.

Now let h = gn1

1 · ... · gnd

d ∈ H. We have π1(h) =∑d

i=1 niπ1(gi)

ρ(e, h) ≥ ‖π1(h)‖0 ≥ s− Cε ≥ s(1− ε)

Moreover

ρ(e, h) ≤d∑

i=1

niρ(e, gi) ≤ s(1 + 3ε)

Changing ε into say ε5 and for say ε < 1

2 , we get the desired result with s0(ε) =dε max1≤i≤p ρ(e, gi).

6.4. Proof of Theorem 6.2. We need to show that as x→ ∞ in N

1 ≤ limρ(e, x)

d∞(e, x)≤ lim

ρ(e, x)

d∞(e, x)≤ 1

First note that it is enough to prove the bounds for x ∈ H. This follows from(4.2) (1).

Let us begin with the lower bound. We fix ε > 0 and s = s(ε) as in the definitionof an asymptotically geodesic metric (see (21)). We know by 4.2 (3) and (4) thatas soon as ρ(e, x) ≥ s we may find x1, ..., xn in H with s ≤ ρ(e, xi) ≤ 2s such thatx =

∏xi and

∑ρ(e, xi) ≤ (1 + ε)ρ(e, x). Let t = d∞(e, x), then n ≤ 1+ε

s ρ(e, x),

hence n ≤ Cs(ε)t where C is a constant depending only on ρ (see (23)). We may


then apply Lemma 6.11 (and the remark following it) to get, as t ≥ n as soon ass(ε) ≥ C,

d∞(δ 1

t(x), δ 1

t(π1(x1) ∗ ... ∗ π1(xn))) ≤ c′1t

− 1

r

But for each i we have ‖π1(xi)‖0 ≤ ρ(e, xi) by definition of the norm, hence

t = d∞(e, x) ≤∑

‖π1(xi)‖0+ d∞(x, π1(x1) ∗ ... ∗π1(xn)) ≤ (1+ ε)ρ(e, x)+ c′1t1− 1

r

Since ε was arbitrary, letting t→ ∞ we obtain

limρ(e, x)

d∞(e, x)≥ 1

We now turn to the upper bound. Let t = d∞(e, x) and ε > 0. According toLemma 6.10, there is a horizontal piecewise linear path ξ(u)u∈[0,1] with unit

speed such that d∞(δ 1

t(x), ξ(1)) ≤ C2ε and no interval of length ≥ εr

Nεcontains

more than one change of direction. Let s0(ε) be given by Lemma 6.13 and assumet > s0(ε

r)Nε/εr. We split [0, 1] into n subintervals of length u1, ..., un such that

ξ′ is constant equal to yi on the i-th subinterval and s0(εr) ≤ tui ≤ 2s0(ε

r). Wehave ξ(1) = u1y1 ∗ ... ∗ unyn. Lemma 6.13 yields points xi ∈ H such that

∥∥∥∥yi −π1(xi)

tui

∥∥∥∥ ≤ εr

and ρ(e, xi) ∈ [(1 − εr)tui, (1 + εr)tui] (note that tui > s0(εr)). Let ξ be the

piecewise linear path [0, 1] → N∞ with the same discontinuities as ξ and where the

value yi is replaced by π1(xi)tui

. Then according to Lemma 6.7, d∞(ξ(1), ξ(1)) ≤ Cε.

Since ρ(e, xi) ≤ 4s0(εr) for each i, we may apply Lemma 6.11 (and the remark

following it) and see that if y = x1 · ... · xn,

d∞(ξ(1), δ 1

t(y)) ≤ c′1(ε)t

− 1

r

Hence d∞(δ 1

t(x), δ 1

t(y)) ≤ (C2+C)ε+c′1(ε)t

− 1

r and ρ(e, y) ≤∑ρ(e, xi) ≤ (1+εr)t

while ρ(x, y) ≤ C ′td∞(e, δ 1

t(x−1y)) + C ′ ≤ t(Cd∞(δ 1

t(x), δ 1

t(y)) + oε(1)). Hence

ρ(e, x) ≤ t+ oε(t)

Remark 6.14. In the last argument we used the fact that∥∥∥δ 1

t(xu)− δ 1

t(x ∗ u)

∥∥∥ =

O( 1

t1r) if δ 1

t(x) and δ 1

t(u) are bounded, in order to get for y = xu,

d∞(e, δ 1

t(u)) ≤ d∞(δ 1

t(x), δ 1

t(xu)) + d∞(δ 1

t(xu), δ 1

t(x ∗ u))

≤ d∞(δ 1

t(x), δ 1

t(y)) + o(1).


7. Locally compact G and proofs of the main results

In this section, we prove Theorem 1.2 and complete the proof of Theorem 1.4and its corollaries. We begin with the latter.

Proof of Theorem 1.4. It is the combination of Proposition 5.1, which reduces theproblem to nilpotent Lie groups, and Theorem 6.2, which treats the nilpotent case.It only remains to justify the last assertion that d∞ is invariant under T (H).

SinceK = T (H) stabilizesm1 (see Lemma 3.12 for the definition ofm1) and actsby automorphisms of the nilpotent (nilshadow) structure (Lemma 3.5), given anyk ∈ K, the metric d∞(k(x), k(y)) is nothing else but the left invariant subFinslermetric on the nilshadow associated to the norm ‖k(v)‖ for v ∈ m1 (if ‖ · ‖ denotesthe norm associated to d∞).

However, d∞ is asymptotically invariant under K, because of Proposition 5.1.Namely d∞(e, k(x))/d∞(e, x) tends to 1 as x tends to infinity. Finally d∞(e, v) =‖v‖ and d∞(e, k(v)) = ‖k(v)‖ for all v ∈ m1. Two asymptotic norms on a vectorspace are always equal. It follows that the norms ‖ · ‖ and ‖k(·)‖ on m1 coincide.Hence d∞(e, k(x)) = d∞(e, k(x)) for all x ∈ S as claimed.

Proof of Corollary 1.8. First some initial remark (see also Remark 2.1). If d isa left-invariant subFinlser metric on a simply connected nilpotent Lie group Ninduced by a norm ‖ · ‖ on a supplementary subspace m1 of the commutatorsubalgebra, then it follows from the very definition of subFinsler metrics (seeParagraph 2.1) that π1 is 1-Lipschitz between the Lie group and the abelianizationof it endowed with the norm ‖·‖, namely ‖π1(x)‖ ≤ d(e, x), with equality if x ∈ m1.From this and considering the definition of the limit norm in (33), we concludethat ‖ · ‖ coincides with the limit norm of d. In particular Theorem 6.2 impliesthat d is asymptotic to the ∗-left invariant subFinsler metric d∞ induced by thesame norm ‖ · ‖ on the graded Lie group (N∞, ∗).

We can now prove Corollary 1.8. By the above remark, the limit metric d∞on the graded nilshadow of S is asymptotic to the subFinsler metric d inducedby the same norm ‖ · ‖ on the same (K-invariant) supplementary subspace m1

of the commutator subalgebra of the nilshadow, and which is left invariant forthe nilshadow structure on S. However, it follows from Theorem 1.4 that d∞and the norm ‖ · ‖ are K-invariant. This implies that d is also left-invariantwith respect to the original Lie group structure of S. Indeed, by (1), we canwrite d(gx, gy) = d(g ∗ (T (g)x), g ∗ (T (g)y)) = d(T (g)x, T (g)y) = d(x, y), where ∗denotes this time the nilshadow product structure. We are done.

Proof of Corollary 1.7. This follows immediately from Theorem 1.4, when ∗ de-notes the graded nilshadow product. If ∗ denotes the nilshadow group structure,then it follows from Theorem 6.2 and the remark we just made in the proof ofCorollary 1.8 (see also Remark 2.1).

7.1. Proof of Theorem 1.2. Let G be a locally compact group of polynomialgrowth. We will show that G has a compact normal subgroup K such that G/K


contains a closed co-compact subgroup, which can be realized as a closed co-compact subgroup of a connected and simply connected solvable Lie group of type(R) (i.e. of polynomial growth). The proof will follow in several steps.

(a) First we show that up to moding out by a normal compact subgroup, we mayassume that G is a Lie group whose connected component of the identity has nocompact normal subgroup. Indeed, it follows from Losert’s refinement of Gromov’stheorem ([24] Theorem 2) that there exists a normal compact subgroup K of Gsuch that G/K is a Lie group. So we may now assume that G is a Lie group (notnecessarily connected) of polynomial growth. The connected component G0 of Gis a connected Lie group of polynomial growth. Recall the following classical fact:

Lemma 7.1. Every connected Lie group has a unique maximal compact normalsubgroup. By uniqueness it must be a characteristic Lie subgroup.

Proof. Clearly if K1 and K2 are compact normal subgroups, then K1K2 is againa compact normal subgroup. Considering G/K, where K is a compact normalsubgroup of maximal dimension, we may assume that G has no compact normalsubgroup of positive dimension. But every finite normal subgroup of a connectedgroup is central. Hence the closed group generated by all finite normal subgroups iscontained in the center ofG. The center is an abelian Lie subgroup, i.e. isomorphicto a product of a vector space R

n, a torus Rm/Zm, a free abelian group Zk and a

finite abelian group. In such a group, there clearly is a unique maximal compactsubgroup (namely the product of the finite group and the torus). It is also normal,and maximal in G.

The maximal compact normal subgroup of G0 is a characteristic Lie subgroupof of G0. It is therefore normal in G and we may mod out by it. We therefore haveshown that every locally compact (compactly generated) group with polynomialgrowth admits a quotient by a compact normal subgroup, which is a Lie group Gwhose connected component of the identity G0 has polynomial growth and con-tains no compact normal subgroup. We will now show that a certain co-compactsubgroup of G has the embedding property of Theorem 1.2.

(b) Second we show that, up to passing to a co-compact subgroup, we mayassume that the connected component G0 is solvable. For this purpose, let Q bethe solvable radical of G0, namely the maximal connected normal Lie subgroupof G0. Note that it is a characteristic subgroup of G0 and therefore normal in G.Moreover G0/Q is a semisimple Lie group. Since G0 has polynomial growth, itfollows that G0/Q must be compact. Consider the action of G by conjugation onG0/Q, namely the map φ : G→ Aut(G0/Q). Since G0/Q is compact semisimple,its group of automorphisms is also a compact Lie group. In particular, the kernelker φ is a co-compact subgroup of G.

The connected component of the identity of Aut(G0/Q) is itself semisimple andhence has finite center. However the image of the connected component (ker φ)0of ker φ in G0/Q modulo Q is central. Therefore it must be trivial. We haveshown that (ker φ)0 is contained in Q and hence is solvable. Moreover (ker φ)0


has no compact normal subgroup, because otherwise its maximal normal compactsubgroup, being characteristic in (ker φ)0, would be normal in G (note that (ker φ)0is normal in G).

Changing G into the co-compact subgroup kerφ, we can therefore assume thatG0 is solvable, of polynomial growth, and has no non trivial compact normal sub-group. The group G/G0 is discrete, finitely generated, and has polynomial growth.By Gromov’s theorem, it must be virtually nilpotent, in particular virtually poly-cyclic.

(c) We finally prove the following proposition.

Proposition 7.2. Let G be a Lie group such that its connected component of theidentity G0 is solvable, admits no compact normal subgroup, and with G/G0 virtu-ally polycyclic. Then G has a closed co-compact subgroup, which can be embeddedas a closed co-compact subgroup of a connected and simply connected solvable Liegroup.

The proof of this proposition is mainly an application of a theorem of H.C.Wang, which is a vast generalization of Malcev’s embedding theorem for torsionfree finitely generated nilpotent groups. Wang’s theorem [36] states that any S-group can be embedded as a closed co-compact subgroup of a simply connected reallinear solvable Lie group with only finitely many connected components. Wangdefines a S-group to be any real Lie group G, which admits a normal subgroupA such that G/A is finitely generated abelian and A is a torsion-free nilpotentLie group whose connected components group is finitely generated. In particularany S-group has a finite index (hence co-compact) subgroup which embeds as aco-compact subgroup in a connected and simply connected solvable Lie group.In order to prove Proposition 7.2, it therefore suffices to establish that G has aco-compact S-group.

We first recall the following simple fact:

Lemma 7.3. Every closed subgroup F of a connected solvable Lie group S istopologically finitely generated.

Proof. We argue by induction on the dimension of S. Clearly there is an epi-morphism π : S → R. By induction hypothesis F ∩ ker π is topologically finitelygenerated. The image of F is a subgroup of R. However every subgroup of Rcontains either one or two elements, whose subgroup they generate has the sameclosure as the original subgroup. We are done.

Next we show the existence of a nilradical.

Lemma 7.4. Let G be as in Proposition 7.2. Then G has a unique maximalnormal nilpotent subgroup GN .

Proof. The subgroup generated by any two normal nilpotent subgroups of anygiven group is itself nilpotent (Fitting’s lemma, see e.g. [30][5.2.8]). Let GN bethe closure of the subgroup generated by all nilpotent subgroups of G. We needto show that GN is nilpotent. For this it is clearly enough to prove that it is


topologically finitely generated (because any finitely generated subgroup of GN isnilpotent by the remark we just made). Since G/G0 is virtually polycyclic, everysubgroup of it is finitely generated ([29][4.2]). Hence it is enough to prove thatGN ∩G0 is topologically finitely generated. This follows from Lemma 7.3.

Incidently, we observe that the connected component of the identity (GN )0coincides with the nilradicalN of G0 (it is the maximal normal nilpotent connectedsubgroup of G0).

We now claim the following:

Lemma 7.5. The quotient group G/GN is virtually abelian.

The proof of this lemma is inspired by the proof of the fact, due to Malcev,that polycyclic groups have a finite index subgroup with nilpotent commutatorsubgroup (e.g. see [30][ 15.1.6]).

Proof. We will show that G has a finite index normal subgroup whose commutatorsubgroup is nilpotent. This clearly implies the lemma, for this nilpotent subgroupwill be normal, hence contained in GN .

First we observe that the group G admits a finite normal series Gm ≤ Gm−1 ≤. . . ≤ G1 = G, where each Gi is a closed normal subgroup of G such that Gi/Gi+1

is either finite, or isomorphic to either Zn, Rn or Rn/Zn. This see it pick one of theGi’s to be the connected component G0 and then treat G/G0 and G0 separately.The first follows from the definition of a polycyclic group (G/G0 has a normalpolycyclic subgroup of finite index). While for G0, observe that its nilradical N isa connected and simply connected nilpotent Lie group and it admits such a seriesof characteristic subgroups (pick the central descending series), and G0/N is anabelian connected Lie group, hence isomorphic to the direct product of a torusRn/Zn and a vector group R

n. The torus part is characteristic in G0/N , hence itspreimage in G0 is normal in G.

The group G acts by conjugation on each partial quotient Qi := Gi/Gi+1. Thisyields a map G → Aut(Qi). Now note that in order to prove our lemma, it isenough to show that for each i, there is a finite index subgroup of G whose com-mutator subgroup maps to a nilpotent subgroup of Aut(Qi). Indeed, taking theintersection of those finite index subgroup, we get a finite index normal subgroupswhose commutator subgroup acts nilpotently on each Qi, hence is itself nilpotent(high enough commutators will all vanish).

Now Aut(Qi) is either finite (if Qi is finite), or isomorphic to GLn(Z) (in caseQi is either Z

n or Rn/Zn) or to GLn(R) (when Qi ≃ R

n). The image of G inAut(Qi) is a solvable subgroup. However, every solvable subgroup of GLn(R)contains a finite index subgroup, whose commutator subgroup is unipotent (hencenilpotent). This follows from Kolchin’s theorem for example, that a connectedsolvable algebraic subgroup of GLn(C) is triangularizable. We are done.

In the sequel we assume that G/G0 is torsion-free polycyclic. It is legitimateto do so in the proof of Proposition 7.2, because every virtually polycyclic grouphas a torsion-free polycyclic subgroup of finite index (see e.g. [29][Lemma 4.6]).

We now claim the following:


Lemma 7.6. GN is torsion-free.

Proof. Since G/G0 is torsion-free, it is enough to prove that GN ∩ G0 is torsion-free. However the set of torsion elements in GN forms a subgroup of GN (if x, y aretorsion, then xy is too because 〈x, y〉 is nilpotent). Clearly it is a characteristicsubgroup of GN . Hence its intersection with G0 is normal in G0. Taking theclosure, we obtain a nilpotent closed normal subgroup T of G0 which contains adense set of torsion elements. Recall that G0 has no normal compact subgroup.From this it quickly follows that T is trivial, because first it must be discrete (theconnected component T0 is compact and normal in G0), hence finitely generated(by Lemma 7.3), hence made of torsion elements. But a finitely generated torsionnilpotent group is finite. Again since G0 has no compact normal subgroup, T mustbe trivial, and GN is torsion-free.

Now observe that the group of connected components of GN , namely GN/(GN )0is finitely generated. Indeed, since G/G0 is finitely generated (as any polycyclicgroup), it is enough to prove that (G0 ∩GN )/(GN )0 is finitely generated, but thisfollows from the fact that G0∩GN is topologically finitely generated (Lemma 7.3).

Now we are almost done. Note thatG is topologically finitely generated (Lemma7.3), therefore so is G/GN . By Lemma 7.5 G/GN is virtually abelian, hence hasa finite index normal subgroup isomorphic to Z

n × Rm. It follows that G/GN

has a co-compact subgroup isomorphic to a free abelian group Zn+m. Hence after

changing G by a co-compact subgroup, we get that G is an extension of GN (atorsion-free nilpotent Lie group with finitely generated group of connected com-ponents) by a finitely generated free abelian group. Hence it is an S-group in theterminology of Wang [36]. We apply Wang’s theorem and this ends the proof ofProposition 7.2.

(d) We can now conclude the proof of Theorem 1.2. By (a) and (b) G hasa quotient by a compact group which admits a co-compact subgroup satisfyingthe assumptions of Proposition 7.2. Hence to conclude the proof it only remainsto verify that the simply connected solvable Lie group in which a co-compactsubgroup of G/K embeds has polynomial growth (i.e. is of type (R)). But thisfollows from the following lemma (see [21][Thm. I.2]).

Lemma 7.7. Let G be a locally compact group. Then G has polynomial growth ifand only if some (resp. any) co-compact subgroup of it has polynomial growth.

Proof. First one checks that G is compactly generated if and only if some (resp.any) co-compact subgroup is. This is by the same argument which shows thatfinite index subgroups of a finitely generated group are finitely generated. Inparticular, if Ω is a compact symmetric generating set of G and H is a co-compactsubgroup, then there is n0 ∈ N such that Ωn0H = G. Then H ∩ Ω3n0 generatesH.

If G has polynomial growth and H is any compactly generated closed subgroup,then H has polynomial growth. Indeed (see [21][Thm I.2]), if ΩH denotes a com-pact generating set for H, and K a compact neighborhood of the identity in G,


then

volG(K)volH(ΩnH) ≤ volH(KK−1 ∩H)volG(Ω

nHK).

This inequality follows by integrating over a left Haar measure of G the functionφ(x) :=

∫Ωn

H1K(h−1x)dh, where dh is a left Haar measure on H. This integral

equals the left handside of the above displayed equation, while it is pointwisebounded by volH(xK−1 ∩H) inside HK and by zero outside HK.

In the other direction, if H has polynomial growth, then G also has, becauseone can write Ωn ⊂ Ωn

HK for some compact generating set ΩH of H and somecompact neighborhood K of the identity in G (see Proposition 4.4). Then theresult follows from the following inequality

volH(ΩH)volG(ΩnHK) ≤ volH(Ωn+1

H )volG(Ω−1H K),

which itself is a direct consequence of the fact that the function

ψ(x) :=

∫

Ωn+1

H

1Ω−1

HK(h−1x)dh,

where dh is a left Haar measure onH, satisfies∫G ψ(x)dx = volH(Ωn+1

H )volG(Ω−1H K)

on the one hand and is bounded below by volH(ΩH) for every x ∈ ΩnHK on the

other hand.

Note that the above proof would be slightly easier if we already knew that bothG and H were unimodular, in which case G/H has an invariant measure. Butwe know this only a posteriori, because the polynomial growth condition impliesunimodularity ([21]).

Similar considerations show that G has polynomial growth if and only if G/Khas polynomial growth, given any normal compact subgroup K (e.g. see [21]).

We end this paragraph with a remark and an example, which we mentioned inthe Introduction.

Remark 7.8 (Discrete subgroups are virtually nilpotent). Suppose Γ is a discretesubgroup of a connected solvable Lie group of type (R) (i.e. of polynomial growth).Then Γ is virtually nilpotent. Indeed, a similar argument as in Lemma 7.3 showsthat every subgroup of Γ is finitely generated. It follows that Γ is polycyclic. How-ever Wolf [37] proved that polycyclic groups with polynomial growth are virtuallynilpotent.

Example 7.9 (A group with no nilpotent co-compact subgroup). Let G be theconnected solvable Lie group G = R ⋉ (R2 × R

2), where R acts as a dense one-parameter subgroup of SO(2,R) × SO(2,R). Then G is of type (R). It has nocompact subgroup. And it has no nilpotent co-compact subgroup. Indeed suppose His a closed co-compact nilpotent subgroup. Then it has a non-trivial center. Hencethere is a non identity element whose centralizer is co-compact in G. However asimple examination of the possible centralizers of elements of G shows that noneof them is co-compact.


7.2. Proof of Corollary 1.6 and Theorem 1.1. Let G be an arbitrary locallycompact group of polynomial growth and ρ a periodic pseudodistance on G.

Claim 1: Corollary 1.6 holds for a co-compact subgroup H of G, if and onlyif it holds for G. By Lemma 7.7, the groups G and H are unimodular, and henceG/H bears a G-invariant Radon measure volG/H , which is finite since H is co-compact. Now let F be a bounded Borel fundamental domain for H inside G. Andlet ρ be the periodic pseudodistance on G induced by the restriction of ρ to H, thatis ρ(x, y) := ρ(hx, hy) where hx is the unique element of H such that x ∈ hxF. By4.2 (1) and (4), ρ and ρ are at a bounded distance from each other. In particular,Bρ(r−C) ⊂ Bρ(r) ⊂ Bρ(r+C). Hence if the limit (3) holds for ρ, it also holds forρ with the same limit. However, Bρ(r) = x ∈ G, ρ(e, hx) ≤ r = BρH (r)F whereρH is the restriction of ρ to H. Hence volG(Bρ(r)) = volH(BρH (r)) · volG/H(F ).By 4.2 (4), ρH is a periodic pseudodistance on H. So the result holds for (H, ρH)if and only if it holds for (G, ρ). Conversely, if ρ0 is a periodic pseudodistanceon H, then ρ0(x, y) := ρ0(hx, hy) is a periodic pseudodistance on G, hence againvolG(Bρ0(r)) = volH(Bρ0(r)) · volG(F ) and the result will hold for (H, ρ0) if andonly if it holds for (G, ρ0).

Claim 2: If Corollary 1.6 holds for G/K, where K is some compact normalsubgroup, then it holds for G as well. Indeed, if ρ is a periodic pseudodistance onG, then the K-average ρK , as defined in (26), is at a bounded distance from G

according to Lemma 4.7. Now ρK induces a periodic pseudodistance ρK on G/Kand BρK (r) = B

ρK(r)K. Hence, volG(BρK (r)) = volG/K(B

ρK(r)) · volK(K). And

if the limit (3) holds for ρK , it also holds for ρK , hence for ρ too.

Thus the discussion above combined with Theorem 1.2 reduces Corollary 1.6 tothe case when G is simply connected and solvable, which was treated in Section 5and 6.

7.3. Proof of Proposition 1.3 and Corollary 1.9.

Proof of Proposition 1.3. We say that two metric spaces (X, dX ) and (Y, dY ) areat a bounded distance if they are (1, C)-quasi-isometric for some finite C. This isan equivalence relation. Now if ρ is H-periodic with H co-compact, then (G, ρ)is at a bounded distance from (H, ρ|H). Hence we may assume that H = G, i.e.that ρ is left invariant on G.

Now Theorem 1.2 gives the existence of a normal compact subgroup K, a co-compact subgroup H containing K and a simply connected solvable Lie group Ssuch that H/K is isomorphic to a co-compact subgroup of S.

Lemma 4.7 shows that (G, ρ) is at a bounded distance from (G, ρK), where ρK

is defined as in (26). Now ρK induces a left invariant periodic metric on G/K,and (G/K, ρK ) is clearly at a bounded distance from (G, ρK). Now by 4.2, itsrestriction to H/K is at a bounded distance and is left invariant. Now we setρS(s1, s2) = ρK(h1, h2), where (given a bounded fundamental domain F for the


left action of H/K on S) hi is the unique element of H/K such that si ∈ hiF .Clearly then (S, ρS) is at a bounded distance from (H/K, ρK). We are done.

We note that our construction of S here depends on the stabilizer of ρ in G.Certainly not every choice of Lie shadow can be used for all periodic metrics (thinkthat R3 is a Lie shadow of the universal cover of the group of motions of the plane).Perhaps a single one can be chosen for all, but we have not checked that.

Proof of Corollary 1.9. Proposition 1.3 reduces the proof to a periodic metric ρon a simply connected solvable Lie group S. Let d∞ the subFinsler metric on S(left invariant for the graded nilshadow group structure SN ) as given by Theorem1.4. Let δtt is the group of dilations in the graded nilshadow SN of S as definedin Section 3. By definition of the pointed Gromov-Hausdorff topology (see [18]),it is enough to prove theClaim. The following quantity

|1

nρ(s1, s2)− d∞(δ 1

n(s1), δ 1

n(s2))|

converges to zero as n tends to +∞ uniformly for all s1, s2 in a ball of radius O(n)for the metric ρ.

Now this follows in three steps. First ρ is at a bounded distance from itsrestriction to the (co-compact) stabilizer H of ρ (cf. 4.2 (1), 4.2 (4)). Thenfor h1, h2 ∈ H, we can write ρ(h1, h2) = ρ(e, h−1

1 h2). However Proposition 5.1implies the existence of another periodic distance ρK on S, which is invariantunder left translations by elements of H for both the original Lie structure and

the nilshadow Lie structure on S, such that ρ(e,x)ρK(e,x) tends to 1 as x tends to

∞. Hence ρK(e, h−11 h2) = ρK(h1, h2) = ρK(e, h∗−1

1 h2), where ∗ is the nilshadow

product on S. Hence | 1nρ(h1, h2)−1nρK(e, h∗−1

1 h2)| tends to zero uniformly as h1and h2 vary in a ball of radius O(n) for ρ.

Finally Theorem 6.2 implies that | 1nρK(e, h∗−11 h2) −

1nd∞(e, h∗−1

1 h2)| tends tozero and the claim follows, as one verifies from the Campbell Hausdorff formulaby comparing (11) and (12) as we did in (35), that

|d∞(δ 1

n(h1), δ 1

n(h2))− d∞(e, δ 1

n(h∗−1

1 h2)|

converges to zero.The fact that the graded nilpotent Lie group does not depend (up to isomor-

phism) on the periodic metric ρ but only on the locally compact group G fol-lows from Pansu’s theorem [28] that if two Carnot groups (i.e. a graded simplyconnected nilpotent Lie group endowed with left-invariant subRiemannian metricinduced by a norm on a supplementary subspace to the commutator subalgebra)are bi-Lipschitz, the underlying Lie groups must be isomorphic. This deep factrelies on Pansu’s generalized Rademacher theorem, see [28]. Indeed, two differ-ent periodic metrics ρ1 and ρ2 on G are quasi-isometric (see Proposition 4.4),and hence their asymptotic cones are bi-Lipschitz (and bi-Lipschitz to any Carnotgroup metric on the same graded group, by (13)).


8. Coarsely geodesic distances and speed of convergence

Under no further assumption on the periodic pseudodistance ρ, the speed ofconvergence in the volume asymptotics can be made arbitrarily small. This iseasily seen if we consider examples of the following type: define ρ(x, y) = |x−y|+|x− y|α on R where α ∈ (0, 1). It is periodic and vol(Bρ(t)) = t− tα + o(tα).

However, many natural examples of periodic metrics, such as word metrics orRiemannian metrics, are in fact coarsely geodesic. A pseudodistance on G is saidto be coarsely geodesic, if there is a constant C > 0 such that any two points canbe connected by a C-coarse geodesic, that is, for any x, y ∈ G there is a mapg : [0, t] → G with t = ρ(x, y), g(0) = x and g(t) = y, such that

|ρ(g(u), g(v)) − |u− v|| ≤ C

for all u, v ∈ [0, t].This is a stronger requirement than to say that ρ is asymptotically geodesic

(see 21). This notion is invariant under coarse isometry. In the case when Gis abelian, D. Burago [6] proved the beautiful fact that any coarsely geodesicperiodic metric on G is at a bounded distance from its asymptotic norm. Inparticular volG(Bρ(t)) = c · td+O(td−1) in this case. In the remarkable paper [32],

M. Stoll proved that such an error term in O(td−1) holds for any finitely generated2-step nilpotent group. Whether O(td−1) is the right error term for any finitelygenerated nilpotent group remains an open question.

The example below shows on the contrary that in an arbitrary Lie group ofpolynomial growth no universal error term can be expected.

Theorem 8.1. Let εn > 0 be an arbitrary sequence of positive numbers tendingto 0. Then there exists a group G of polynomial growth of degree 3 and a compactgenerating set Ω in G and c > 0 such that

(36)volG(Ω

n)

c · n3≤ 1− εn

holds for infinitely many n, although 1c·n3 volG(Ω

n) → 1 as n→ +∞.

The example we give below is a semi-direct product of Z by R2 and the metric

is a word metric. However, many similar examples can be constructed as soon asthe map T : G → K defined in Paragraph 5.1 in not onto. For example, one canconsider left invariant Riemannian metrics on G = R · (R2 × R

2) where R actsby via a dense one-parameter subgroup of the 2-torus S1 × S1. Incidently, thisgroup G is known as the Mautner group and is an example of a wild group inrepresentation theory.

8.1. An example with arbitrarily small speed. In this paragraph we describethe example of Theorem 8.1. Let Gα = Z · R2 where the action of Z is given bythe rotation Rα of angle πα, α ∈ [0, 1). The group Gα is quasi-isometric to R3

and hence of polynomial growth of order 3 and it is co-compact in the analogously

defined Lie group Gα = R ⋉ R2. Its nilshadow is isomorphic to R

3. The point is


−2

−1.0

−2

−0.8

−0.6

−1−1

−0.4

−0.200

0.0

0.2

1

0.4

1

0.6

2

0.8

1.0

x_1

23

x_2

3

Figure 2. The union of the two cones, with basis the disc of radius2, represents the limit shape of the balls Ωn in the group Z ⋉ R

2,where Z acts by an irrational rotation, with generating set Ω =(±1, 0, 0) ∪ (0, x1, x2),

14x

21 + x22 ≤ 1.

that if α is a suitably chosen Liouville number, then the balls in Gα will not bewell approximated by the limit norm balls.

Elements of Gα are written (k, x) where k ∈ Z and x ∈ R2. Let ‖x‖2 = 1

4x21+x

22

be a Euclidean norm on R2, and let Ω be the symmetric compact generating set

given by (±1, 0)∪(0, x), ‖x‖ ≤ 1. It induces a word metric ρΩ on G. It followsfrom Theorem 1.4 and the definition of the asymptotic norm that ρΩ(e, (k, x)) isasymptotic to the norm on R

3 given by ρ0(e, (k, x)) := |k| + ‖x‖0 where ‖x‖0 is

the rotation invariant norm on R2 defined by ‖x‖20 =

14 (x

21 + x22). The unit ball of

‖·‖0 is the convex hull of the union of all images of the unit ball of ‖·‖ under allrotations Rkα, k ∈ Z.

We are going to choose α as a suitable Liouville number so that (36) holds. Let

δn = (4εn)1/3 and choose α so that the following holds for infinitely many n’s:

(37) d(kα,Z +1

2) ≥ 2δn

for all k ∈ Z, |k| ≤ n. This is easily seen to be possible if we choose α of the form∑1/3ni for some suitable lacunary increasing sequence of (ni)i.Note that, since ‖x‖0 ≥ ‖x‖ , we have ρΩ ≥ ρ0. Let Sn be the piece of R2 defined

by Sn = |θ| ≤ δn where θ is the angle between the point x and the vertical axis


Re2. We claim that if x ∈ Sn, ρ0(e, (k, x)) ≤ n and n satisfies (37), then

ρΩ(e, (k, x)) ≥ |k|+ (1 +δ2n4) ‖x‖0

It follows easily from the claim that volG(Ωn) ≤ (1− εn) · volG(Bρ0(n)). Moreover

volG(Bρ0(n)) = c · n3 + O(n2), where c = 4π3 if volG is given by the Lebesgue

measure.Proof of claim. Here is the idea to prove the claim. To find a short path between

the identity and a point on the vertical axis, we have to rotate by a Rkα such thatkα is close to 1

2 , hence go up from (0, 0) to (k, 0) first, thus making the verticaldirection shorter. However if (37) holds, the vertical direction cannot be made asshort as it could after rotation by any of the Rkα with |k| ≤ n.

Note that if ρ0(e, (k, x)) ≤ n then |k| ≤ n and ρΩ(e, (k, x)) ≥ |k|+inf∑

‖Rkiαxi‖where the infimum is taken over all paths x1, ..., xN such that x =

∑xi and all

rotations Rkiα with |ki| ≤ n. Note that if δn is small enough and (37) holdsthen for every x ∈ Sn we have ‖Rkαx‖ ≥ (1 + δ2n) ‖x‖0 . On the other hand‖x‖0 =

∑‖xi‖0 cos(θi) where θi is the angle between xi and the x. Hence

∑‖Rkiαxi‖ ≥

∑

|θi|≤δn

‖Rkiαxi‖+∑

|θi|>δn

‖Rkiαxi‖

≥ (1 + δ2n)∑

|θi|≤δn

‖xi‖0 cos(θi) +1

cos(δn)

∑

|θi|>δn

‖xi‖0 cos(θi)

≥ (1 +δ2n4) · ‖x‖0

8.2. Limit shape for more general word metrics on solvable Lie groupsof polynomial growth. The determination of the limit shape of the word metricin Paragraph 8.1 was possible due to the rather simple nature of the generatingset. In general, using the identity (see (1))

(38) ω1 · . . . · ωm = ω1 ∗ (T (ω1)ω2) ∗ . . . ∗ (T (ωm−1 · . . . · ω1)ωm)

it is easy to check that the unit ball of the limit norm ‖ · ‖∞ inducing the limitsubFinsler metric d∞ on the nilshadow associated to a given word metric withgenerating set Ω is contained in the K-orbit of the convex hull of the projectionof Ω to the abelianized nilshadow, namely the convex hull of K · π1(Ω).

In the example of Paragraph 8.1, we even had equality between the two. How-ever this is not the case in general. For example, the limit shape is always K-invariant, but clearly the limit shape associated to a generating set Ω coincideswith the one associated with a conjugate gΩg−1 of it, while the convex hull of therespective K-orbits may not be the same.

Of course if the generating set Ω is K-invariant to begin with, then Ωn = Ω∗n

and we are back in the nilpotent case, where we know that the unit ball of thelimit norm is just the convex hull of the projection of the generating set to theabelianization. In general however it is a challenging problem to determine the


precise asymptotic shape of a word metric on a general solvable Lie group withpolynomial growth, and there seems to be no simple description analogous to whatwe have in the nilpotent case.

Even in the above example Gα = Z⋉αR2, or in the universal cover of the group

of motions of the plane (in which Gα embeds co-compactly), it is not that simple.In general the shape is determined by solving an optimization problem in whichone has to find the path which maximizes the coordinates of the endpoint. Inorder to illustrate this, we treat without proof the following simple example.

Suppose Ω is a symmetric compact neighborhood of the identity in Gα = Z⋉αR2

of the form Ω = (0,Ω0) ∪ (1,Ω1) ∪ (1,Ω1)−1, where Ω0,Ω1 ⊂ R

2. Then thelimit shape of the word metric ρΩ associated to Ω is the solid body (rotationallysymmetric around the vertical axis as in Figure 2) made of two copies (upper andlower) of a truncated cone with base a disc on (0,R2) of radius maxr0, r1 andtop (resp. bottom) a disc on the plane (1,R2) (resp. (−1,R2)) of radius r2, wherethe radii are given by

r0 = max‖x‖, x ∈ Ω0, r1 =1

2diam(Ω1),

where diam(Ω1) is the diameter of Ω1 and r2 is given by the integral

(39) r2 =

∫ 2π

0maxπθ(Ω1)

dθ

2π,

where πθ(Ω1) is the orthogonal projection on the x-axis of image of Ω1 ⊂ R2 by a

rotation of angle θ around the origin. It is indeed convex (note that r2 ≤ r1).For example if Ω1 is made of only one point, then the limit shape is the same

as in the previous paragraph and as in Figure 2, namely two copies of a cone.However if Ω1 is made of two points a, b, then the upper part of the limit shape

will be a truncated cone with an upper disc of radius r2 = ‖a−b‖π (which is the

result of the computation of the above integral).Let us briefly explain the formula (39). A path of length n reaching the highest

z-coordinate in Gα is a word of the form (1, ω1) · . . . · (1, ωn), with ωi ∈ Ω1. By(38) this word equals

(n,

n∑

1

Ri−1α ωi).

Here ωi can take any value in Ω1. In order to maximize the norm of the secondcoordinate, or equivalently (by rotation invariance) its x-coordinate, one has tochoose ωi ∈ Ω1 at each stage in such a way that the x-coordinate of Ri−1

α ωi ismaximized. Formula (39) now follows from the fact that Ri−1

α 1≤i≤n becomesequidistributed in SO(2,R) as n tends to infinity.

In order to show that maxr0, r1 is the radius of the base disc and moregenerally that the limit shape is no bigger than this double truncated cone, oneneeds to argue further by considering all possible paths of the form (ε1, ω1) · . . . ·(εn, ωn) where εi ∈ 0,±1 and

∑εi is prescribed.


8.3. Bounded distance versus asymptotic metrics. In this paragraph we an-swer a question of D. Burago and G. Margulis (see [7]). Based on the abelian caseand the reductive case (Abels-Margulis [1]), Burago and Margulis had conjecturedthat every two asymptotic word metrics should be at a bounded distance. We givebelow a counterexample to this. We first give an example (A) of a nilpotent Liegroup endowed with two left invariant subFinsler metrics d∞ and d′∞ that areasymptotic to each other, i.e. d∞(e, x)/d′∞(e, x) → 1 as x → ∞ but such that|d∞(e, x)− d′∞(e, x)| is not uniformly bounded. Then we exhibit (B) a word met-ric that is not at a bounded distance from any homogeneous quasi-norm. Finallythese examples also yield (C) two word metrics ρ1 and ρ2 on the same finitelygenerated nilpotent group which are asymptotic but not at a bounded distance.

Note that the group Gα with ρ0 and ρΩ from the last paragraph also providesan example of asymptotic metrics which are not at a bounded distance (but thisgroup was not discrete).

(A) Let N = R × H3(R) where H3 is classical Heisenberg group and Γ =Z ×H3(Z) a lattice in N . In the Lie algebra n = RV ⊕ h3 we pick two differentsupplementary subspaces of [n, n] = RZ, i.e. m1 = spanV,X, Y and m′

1 =spanV + Z,X, Y , where h3 is the Lie algebra of H3(R) spanned by X,Y andZ = [X,Y ].We consider the L1-norm on m1 (resp. m

′1) corresponding to the basis

(V,X, Y ) (resp. (V + Z,X, Y )). Both norms induce the same norm on n/[n, n].They give rise to left invariant Carnot-Caratheodory Finsler metrics on N , sayd∞ (resp. d′∞). We use the coordinates (v, x, y, z) = exp(vV + xX + yY + zZ).

According to Remark (2) after Theorem 6.2, d∞ and d′∞ are asymptotic. Letus show that they are not at a bounded distance. First observe that, since Vis central, d∞(e, (v; (x, y, z))) = |v| + dH3

(e, (x, y, z)) where dH3is the Carnot-

Caratheodory Finsler metric on H3(R) defined by the standard L1-norm on thespanX,Y . Similarly d′∞(e, (v; (x, y, z))) = |v| + dH3

(e, (x, y, z − v))). If d∞ andd′∞ were at a bounded distance, we would have a C > 0 such that for all t > 0

|d∞(e, (t; (0, 0, t))) − t| ≤ C

Hence |dH3(e, (0, 0, t))| ≤ C, which is a contradiction.

(B) Now let Ω = (1; (0, 0, 1))±1 , (1; (0, 0,−1))±1 , (0; (1, 0, 0))±1 , (0; (0, 1, 0))±1be a generating set for Γ and ρΩ the word metric associated to it. Let | · | bea homogeneous quasi-norm on N which is at a bounded distance from ρΩ, i.e.|ρΩ(e, g) − |g|| is bounded. Then | · | is asymptotic to ρΩ, hence is equal to theCarnot-Caratheodory Finsler metric d asymptotic to ρΩ and homogeneous withrespect to the same one parameter group of dilations δtt>0. Let m1 = v ∈ n,δt(v) = tv. Then d is induced by some norm ‖·‖0 on m1, whose unit ball isgiven, according to Theorem 1.4 by the convex hull of the projections to m1 of thegenerators in Ω. There is a unique vector in m1 of the form V +z0Z. Its ‖·‖0-normis 1 and d(e, (1; (0, 0, z0))) = 1. However d(e, (v; (x, y, z))) = |v|+ dH3

(e, (x, y, z −vz0)). Since ρΩ(e, (n; (0, 0, n))) = n, we get

d(e, (n; (0, 0, n))) − ρΩ(e, (n; (0, 0, n))) = dH3(e, (0, 0, n(1 − z0)))


If this is bounded, this forces z0 = 1. But we can repeat the same argument with(n; (0, 0,−n)) which would force z0 = −1. A contradiction.

(C) Let now Ω2 := (1; (0, 0, 0))±1 , (0; (1, 0, 0))±1 , (0; (0, 1, 0))±1 and ρΩ2the

associated word metric on Γ. Then again ρΩ and ρΩ2are asymptotic by Theorem

6.2 because the convex hull of their projection modulo the z-coordinate coincide.However ρΩ2

is a product metric, namely we have ρΩ2(e, (v; (x, y, z))) = |v| +

ρ(e, (x, y, z)), where ρ is the word metric on the discrete Heisenberg group H3(Z)with standard generators (1, 0, 0)±1, (0, 1, 0)±1. In particular

ρΩ(e, (n; (0, 0, n))) − ρΩ2(e, (n; (0, 0, n))) = ρ(e, (0, 0, n))

which is unbounded.

Remark 8.2 (An abnormal geodesic). We refer the reader to [9] for more onthese examples. In particular we show there that ρ1 and ρ2 above are not (1, C)-quasi-isometric for any C > 0. The key phenomenon behind this example isthe presence of an abnormal geodesic (see [25]), namely the one-parameter group(t; (0, 0, 0))t .

Remark 8.3 (Speed of convergence in the nilpotent case). The slow speed phe-nomenon in Theorem 8.1 relied crucially on the presence of a non-trivial semisim-ple part in Gα ; this doesn’t occur in nilpotent groups. In [9], we show that forword metrics on finitely generated nilpotent groups, the convergence in Theorem

6.2 has a polynomial speed with an error term at least as good as O(d∞(e, x)−2

3r ),where r is the nilpotency class. We conjecture there that the optimal exponent is 1

2 .This involves refining quantitatively the estimates of the above proof of Theorem6.2.

9. Appendix: the Heisenberg groups

Here we show how to compute the asymptotic shape of balls in the Heisenberggroups H3(Z) and H5(Z) and their volume, thus giving another approach to themain result of Stoll [33]. The leading term for the growth of H3(Z) is rationalfor all generating sets (Prop. 9.1 below), whereas in H5(Z) with its standardgenerating set, it is transcendental. This explains how our Figure 1 was made(compare with the odd [22] Fig. 1).

9.1. 3-dim Heisenberg group. Let us first consider the Heisenberg group

H3(Z) = 〈a, b|[a, [a, b]] = [b, [a, b]] = 1〉 .

We see it as the lattice generated by a = exp(X) and b = exp(Y ) in the realHeisenberg group H3(R) with Lie algebra h3 generated by X,Y and spanned byX,Y,Z = [X,Y ]. Let ρΩ be the standard word metric on H3(Z) associated tothe generating set Ω = a±1, b±1. According to Theorem 1.4, the limit shape ofthe n-ball Ωn in H3(Z) coincides with the unit ball C3 = g ∈ H3(R), d∞(e, g) ≤1 for the Carnot-Caratheodory metric d∞ induced on H3(R) by the ℓ1-norm‖xX + yY ‖0 = |x|+ |y| on m1 = spanX,Y ⊂ h3.


Computing this unit ball is a rather simple task. Exchanging the roles of Xand Y , we see that C3 is invariant under the reflection z 7→ −z. Then clearly C3 isof the form xX + yY + zZ, with |x|+ |y| ≤ 1 and |z| ≤ z(x, y). Changing X to−X and Y to −Y, we get the symmetries z(x, y) = z(−x, y) = z(x,−y) = z(y, x).Hence when determining z(x, y), we may assume 0 ≤ y ≤ x ≤ 1, x+ y ≤ 1.

The following well known observation is crucial for computing z(x, y). If ξ(t) is ahorizontal path in H3(R) starting from id, then ξ(t) = exp(x(t)X+y(t)Y +z(t)Z),where ξ′(t) = x(t)X + y(t)Y and z(t) is the “balayage” area of the between thepath x(s)X + y(s)Y 0≤s≤t and the chord joining 0 to x(t)X + y(t)Y.

Therefore, z(x, y) is given by the solution to the “Dido isoperimetric problem”(see [25]): find a path in the X,Y -plane between 0 and xX + yY of ‖·‖0-length 1that maximizes the “balayage area”. Since ‖·‖0 is the ℓ1-norm in the X,Y -plane,as is well-known (see [8]), such extremal curves are given by arcs of square withsides parallel to the X,Y -axes. There is therefore a dichotomy: the arc of squarehas either 3 or 4 sides (it may have 1 or 2 sides, but these are included are limitingcases of the previous ones).

If there are 3 sides, they have length ℓ, x and y + ℓ with y + ℓ ≤ x. Hence1 = ℓ+ x+ y + ℓ and z(x, y) = ℓx+ 1

2xy. Therefore this occurs when y ≤ 3x − 1

and we then have z(x, y) = x(1−x)2 .

If there are 4 sides, they have length ℓ, x+ u, y + ℓ and u, with ℓ+ y = x+ u.Hence 1 = 2ℓ + 2u + x + y and z(x, y) = (ℓ + y)(x + u) − xy

2 . This occurs when

y ≥ 3x− 1 and we then have z(x, y) = (1+x+y)2

16 − xy2 .

Hence if 0 ≤ y ≤ x ≤ 1 and x+ y ≤ 1

(40) z(x, y) = 1y≤3x−1x(1− x)

2+ 1y>3x−1

(1 + x+ y)2

16−xy

2

The unit ball C3 drawn in Figure 1 is the solid body C3 = xX + yY + zZ, with|x|+ |y| ≤ 1 and |z| ≤ z(x, y).

A simple calculation shows that vol(C3) =3172 in the Lebesgue measure dxdydz.

Since H3(Z) is easily seen to have co-volume 1 for this Haar measure on H3(R)(actually xX+yY +zZ, x ∈ [0, 1), y ∈ [0, 1), z ∈ [0, 1) is a fundamental domain),it follows that

limn→∞

#(Ωn)

n4= vol(C3) =

31

72

We thus recover a well-known result (see [4], [31] where even the full growth seriesis computed and shown to be rational).

One can also determine exactly which points of the sphere ∂C3 are joined to idby a unique geodesic horizontal path. The reader will easily check that uniquenessfails exactly at the points (x, y,±z(x, y)) with |x| < 1

3 and y = 0, or |y| < 13 and

x = 0, or else at the points (x, y, z) with |x|+ |y| = 1 and |z| < z(x, y).The above method also yields the following result.


Proposition 9.1. Let Ω be any symmetric generating set for H3(Z). Then theleading coefficient in #(Ωn) is rational, i.e.

limn→∞

#(Ωn)

n4= r

is a rational number.

Proof. We only sketch the proof here. We can apply the method above and com-pute r as the volume of the unit CC-ball C(Ω) of the limit CC-metric d∞ de-fined in Theorem 1.4. Since we know what is the norm ‖·‖ in the (x, y)-planem1 = span 〈X,Y 〉 that generates d∞ (it is the polygonal norm given by the con-vex hull of the points of Ω), we can compute C(Ω) explicitly. We need to knowthe solution to Dido’s isoperimetric problem for ‖·‖ in m1, and as is well known(see [8]) it is given by polygonal lines from the dual polygon rotated by 90. Sincethe polygon defining ‖·‖ is made of rational lines (points in Ω have integer coordi-nates), any vector with rational coordinates has rational ‖·‖-length, and the dualpolygon is also rational. The equations defining z(x, y) will therefore have onlyrational coefficients, and z(x, y) will be piecewisely given by a rational quadraticform in x and y, where the pieces are rational triangles in the (x, y)-plane. Thetotal volume of C(Ω) will therefore be rational.

9.2. 5-dim Heisenberg group. The Heisenberg groupH5(Z) is the group gener-ated by a1, b1, a2, b2,c with relations c = [a1, b1] = [a2, b2], a1 and b1 commute witha2 and b2 and c is central. Let Ω = a±1

i , b±1i , i = 1, 2. Let us describe the limit

shape of Ωn. Again, we see H5(Z) as a lattice of co-volume 1 in the group H5(R)with Lie algebra h5 spanned by X1, Y1X2, Y2 and Z = [Xi, Yi]. By Theorem 1.4,the limit shape is the unit ball C5 for the Carnot-Caratheodory metric on H5(R)induced by the ℓ1-norm ‖x1X1 + y1Y1 + x2X2 + y2Y2‖0 = |x1|+ |y1|+ |x2|+ |y2|.

Since X1, Y1 commute with X2, Y2, in any piecewise linear horizontal path inH5(R), we can swap the pieces tangent to X1 or Y1 with those tangent to X2 orY2 without changing the end point of the path. Therefore if ξ(t) = exp(x1(t)X1 +y1(t)Y1 + x2(t)X2 + y2(t)Y2 + z(t)Z) is a horizontal path, then z(t) = z1(t) +z2(t), where zi(t), i = 1, 2, is the “balayage area” of the plane curve xi(s)Xi +yi(s)Yi0≤s≤t.

Since, just like for H3(Z), we know the curve maximizing this area, we cancompute the unit ball C5 explicitly. In exponential coordinates it will take theform C5 = exp(x1X1 + y1Y1 + x2X2 + y2Y2 + zZ), |x1|+ |y1|+ |x2|+ |y2| ≤ 1 and|z| ≤ z(x1, y1, x2, y2). Then z(x1, y1, x2, y2) = sup0≤t≤1zt(x1, y1)+ z1−t(x2, y2),where zt(x, y) is the maximum “balayage area” of a path of length t between 0and xX+yY. It is easy to see that zt(x, y) = t2z(x/t, y/t) where z is given by (40).Hence zt is a piecewise quadratic function of t. Again z(x1, y1, x2, y2) is invariantunder changing the signs of the xi,yi’s, and swapping x and y, or else swapping1 and 2. We may thus assume that the xi,yi’s lie in D = 0 ≤ yi ≤ xi ≤ 1 andx1+y1+x2+y2 ≤ 1, and x2−y2 ≥ x1−y1.We may therefore determine explicitlythe supremum z(x1, y1, x2, y2), which after some straightforward calculations takes


on D the following form:

z(x1, y1, x2, y2) = 1Amaxd1, d2+ 1B maxd1, c1+ 1C maxc1, c2

where d1 =x1y12 + x2

2 (1−x1− y1−x2), c1 =116 (1+x1+ y1−x2− y2)

2+ x2y2−x1y12 ,

and d2 and c2 are obtained from d1 and c1 by swapping the indices 1 and 2. Thesets A,B and C form the following partition of D : A = D ∩ m ≤ x1 − y1,B = D ∩ x1 − y1 < m < x2 − y2 and C = D ∩ x2 − y2 ≤ m, where m =(1 − x1 − x2 − y1 − y2)/2.

Since C5 has such an explicit form, it is possible to compute its volume. The factthat z(x1, y1, x2, y2) is piecewisely given by the maximum of two quadratic formsmakes the computation of the integral somewhat cumbersome but tractable. Ourequations coincide (fortunately!) with those of Stoll (appendix of [33]), where hecomputed the main term of the asymptotics of #(Ωn) by a different method. Stolldid calculate that integral and obtained

limn→∞

#(Ωn)

n6= vol(C5) =

2009

21870+

log(2)

32805

which is transcendental. It is also easy to see by this method that if we changethe generating set to Ω0 = a±1

1 b±11 a±1

2 b±12 , then we get a rational volume. Hence

the rationality of the growth series of H5(Z) depends on the choice of generatingset, which is Stoll’s theorem.

One advantage of our method is that it can also apply to fancier generatingsets. The case of Heisenberg groups of higher dimension with the standard gen-erating set is analogous: the function z(xi, yi) is again piecewisely defined asthe maximum of finitely many explicit quadratic forms on a linear partition of theℓ1-unit ball

∑|xi|+ |yi| ≤ 1.

Acknowledgments. I would like to thank Amos Nevo for his hospitality at theTechnion of Haifa in December 2005, where part of this work was conducted, andfor triggering my interest in this problem by showing me the possible implicationsof Theorem 1.1 to Ergodic Theory. My thanks are also due to V. Losert forpointing out an inaccuracy in my first proof of Theorem 1.2 and for his otherremarks on the manuscript. Finally I thank Y. de Cornulier, M. Duchin, E. LeDonne, Y. Guivarc’h, A. Mohammadi, P. Pansu and R. Tessera for several usefulconversations.

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E-mail address: [email protected]

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