On Lie group structures of homogeneous metric spacesusers.jyu.fi/~vikakivi/prujut/2020-03-30.pdf ·...

IntroductionExistence of Lie group structures

Finding all Lie group structures: CMIReal shadow

On Lie group structures of homogeneous metricspaces

Ville Kivioja

Jyväskylä Geometry seminar, March 30, 2020

These slides are available at my homepagehttp://users.jyu.fi/~vikakivi/

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http://users.jyu.fi/~vikakivi/



1 Introduction

2 Existence of Lie group structures

3 Finding all Lie group structures: CMI

4 Real shadow

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Definition

A metric space is said to be homogeneous, if the action of itsisometry group is transitive, i.e., given any two points on themanifold, there exists an isometry mapping first to the second.

Examples

Euclidean spaces and Carnot groups with CC-distances

Spheres Sn

Rank-one symmetric spaces of non-cpt type HnR, HnC, HnQ, H2CaFlat torus

Cartesian products of these

Non-examples

Physical torus

Ellipsoids

Saddle surface3 / 18



Definition

A metric space is said to be homogeneous, if the action of itsisometry group is transitive, i.e., given any two points on themanifold, there exists an isometry mapping first to the second.

Important class: Lie groups with left-invariant distances:

GL ⊆ Isome(G , d) GL y G transitively

Examples are plentiful:

〈·, ·〉 on TeG ρg (X ,Y ) := 〈dLg−1X , dLg−1Y 〉

Euclidean spaces and Carnot groups with CC-dist, Lie groups

Spheres Sn, NOT Lie groups (except n ∈ {1, 3})Rank-one symmetric spaces HnR, HnC, HnQ, H2Ca, Lie groupsFlat torus, Lie group

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Metric space (M, d) admits a structure of a Lie group, if any ofthe following equivalent conditions hold.

M is isometric to a Lie group with a left-invariant admissibledistance.

There is a Lie group acting on M simply transitively byisometries.

There is a group operation ∗ : M ×M → M that makes M aLie group and the distance d left-invariant.

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Existence, uniqueness and classification of Lie group structures onmetric space.

Given a homogeneous metric space M we wish to

know if there exists a Lie group structure on M, and if yes

know if there are more than one Lie group structure on M,and if yes,

know what are all the possible Lie group structures on M.

know what can happen after perturbations retaining the largescale structure (quasi-isometries)?

That is:How, and how well, do the Lie groups model homogeneousmetric spaces?

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Examples of these phenomena:

The metric space (S2, dE ) does not have a Lie group struc-ture

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Metric space:

span(∂x − y∂z , ∂y + x∂z) subbundle in R3 −→ CC-distance

The only Lie group structure: Heisenberg group

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The Euclidean space R3 admits exactly two Lie group structures:the Abelian one, and the one of S̃E(2), i.e., the universal cover ofthe isometry group of the plane, i.e., R2 o S1.

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Up to arbitrarily small perturbation, every locally compacthomogeneous metric space is a Lie group:

Theorem (CKLNO/Cornulier–De La Harpe/folklore)

Let (M, d) be a locally compact and connected homogeneousmetric space. Then there is a Lie group G , and for all ε > 0 thereis a left-invariant distance dG on G , and a surjective mapF : G → M, so that

dG (g , h)− ε ≤ d(F (g),F (h)) ≤ dG (g , h)

In particular, (M, d) is (1, ε)-quasi-isometric to (G , dG ).

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dG (g , h)− ε ≤ d(F (g),F (h)) ≤ dG (g , h)


Note: S2 . . .

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dG (g , h)− ε ≤ d(F (g),F (h)) ≤ dG (g , h)


Note: S2 is not (1, ε)-QI to singleton. It is (1, ε)-QI to SO(3) witha certain distance.

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dG (g , h)− ε ≤ d(F (g),F (h)) ≤ dG (g , h)


Here G = Iso(M, d) and F is the orbit map F (ϕ) = ϕ(o). Theinteresting question is, who is dG?

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Assume (M, d) loc cpt, homog, ε > 0, o ∈ M, G := Iso(M, d)

Claim ∃dG s.t. F : (G , dG )→ (M, d), F (ϕ) = ϕ(o) is (1, ε)-QI.

Proof Define Busemann gauge

ρ(o, p) := `min{n ∈ N | p ∈ Vn(o, `)}

where B̄(o, 2`) is compact and

Vn(o, `) :=⋃

p∈Vn−1(o,`)

B̄(p, `)

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Assume (M, d) loc cpt, homog, ε > 0, o ∈ M, G := Iso(M, d)

Claim ∃dG s.t. F : (G , dG )→ (M, d), F (ϕ) = ϕ(o) is (1, ε)-QI.

Proof Define Busemann gauge

ρ(o, p) := `min{n ∈ N | p ∈ Vn(o, `)}

The distance that does the job is

dG (ϕ,ψ) = supq∈M

d(ϕ(q), ψ(q))e−ρ(o,q)/ε

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”Many of homogeneous metric spaces admit Lie group structures:”

Theorem (CKLNO)

Let (M, d) be a locally compact and connected homogeneousmetric space. Assume M admits a dilation, i.e., there is a bijectionδ : M → M and a number λ > 1 so that

d(δ(x), δ(y)) = λd(x , y) ∀x , y ∈ M

Then (M, d) admits a structure of a nilpotent (gradable) Lie group,so that d becomes left-invariant and δ becomes an automorphism.

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”Many of homogeneous metric spaces admit Lie group structures:”

Theorem (Heintze)

Let (M, g) be homogeneous Riemannian manifold, with strictlynegative sectional curvatures. Then M admits a structure of asolvable Lie group.

Conjecture (Alekseevskii)

Let (M, g) be homogeneous Riemannian manifold, that isnon-compact, non-flat and satisfies

Ricg = λg (Einstein)

for some λ ∈ R, then M admits a structure of a solvable Lie group.

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If a Lie group structure exist, how to find all the other ones?A bit too hard, but we take a bit different, related, approach:

Definition

We say two Lie groups G and H can be made isometric, write

GCMI∼ H, if there are left-invariant distance functions dG on G and

dH on H, inducing the respective manifold topologies, such that

(G , dG )isome∼= (H, dH)

The following are equivalent with the condition GCMI∼ H

There is a metric space M and two simply transitivesubgroups of Iso(M) isomorphic to G and H respectively.

There is a left-invariant admissible distance funtion dG on Gand a simply transitive action H y G by isometries.

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Definition





Approach is now

Suppose a metric space is given, and a Lie group structure isfound.

We aim to find all the possible other Lie groups, that can bemade isometric to the given Lie group, even if not with thegiven distance.

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Definition





Sometimes we can prove the uniqueness in this stronger way:

Theorem (K.–Le Donne 2017)

Connected nilpotent Lie groups can be made isometric only if theyare isomorphic.

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Proof: If two nilpotent Riemannian Lie groups are isometric, thenthey are isomorphic (Wolf, 60s). Then use the following:

Theorem (K.–Le Donne)

If (G , d) is a connected Lie group with left-invariant distance dinducing the manifold topology, then

Isome(G , d) is a Lie group acting smoothly.

Stabilisers of Isome(G , d) are compact.

there is a left-invariant Riemannian metric ρ on G such thatIsome(G , d) < Isome(G , ρ).

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In particular, suppose F : (G , dG )→ (H, dH) is an isometry.Take ρ Riemannian metric in G so that

Isome(G , d) < Isome(G , ρ)

Now F∗ρ is a Riemannian metric on H. Need to show it isleft-invariant:

Isome(G , ρ) Isome(H,F∗ρ)

Isome(G , dG ) Isome(H, dH)

CF

CF

→ F is an isometry of Riemannian Lie groups (G , ρ) and (H,F∗ρ),so an isomorphism.

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Corollary

If two Lie groups can be made isometric, they can be madeisometric using Riemannian distances. In particular, they arequasi-isometric.

Conjecture (folklore)

If two nilpotent simply connected Lie groups are quasi-isometric,they are isomorphic.

→ Inverse question: Which of the pairs of Lie groups that arequasi-isometric, can indeed be made isometric?

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Lie groups tend to have ’good representatives’ in terms ofisometry-relation:

Theorem (Breuillard)

Any connected Lie group of polynomial growth can be madeisometric to a nilpotent group, namely its nilshadow.

Theorem (Alekseevski)

Any Heintze group can be made isometric to a purely real Heintzegroup.

Definition

A Lie group of the form N oA R is a Heintze group, if N isnilpotent simply connected Lie group and the action (in the Liealgebra level) is given by A ∈ der(n) so that any root of thecharacteristic polynomial of A has strictly positive real part. AHeintze group is purely real if those roots are real numbers.

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G1CMI∼ G2 and both are

{nilpotent

purely real Heintze⇒ G1 ∼= G2

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Theorem (Gordon–Wilson + K.–Le Donne)

If two completely solvable Lie groups can be made isometric, theyare isomorphic.

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Theorem (CKLNO)

Let g be a solvable Lie algebra. Choose a vector subspace a ⊆ gsatisfying n⊕ a = g and ads(a)(a) = {0}. Define

ϕ : g→ der(g) ϕ(X ) = −adsi(πa(X ))

Then

i) Gr(ϕ) := {(X , ϕ(X ) | X ∈ g)} is a completely solvable idealof go der(g),

ii) When the vector space g is equipped with the operation

[X ,Y ]R := [X ,Y ] + ϕ(X )(Y )− ϕ(Y )(X )

then the map τ(X ) = (X , ϕ(X )) is a Lie algebra isomorphismfrom (g, [·, ·]R) to Gr(ϕ), the real shadow of g.

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n⊕ a = g ads(a)(a) = {0}ϕ : g→ der(g) ϕ(X ) = −adsi(πa(X ))

[X ,Y ]rrad := [X ,Y ] + ϕ(X )(Y )− ϕ(Y )(X )

Observations:

If g is completely solvable (e.g. nilpotent), then ϕ ≡ 0 and thereal shadow is itself.

If g has polynomial growth, then adsi(X ) = ads(X ) for allX ∈ g, so the real shadow equals to what is called nilshadow.

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Example: S̃E(2). Basis {X ,Y ,Θ} with[X ,Y ] = 0 [Θ,X ] = Y [Θ,Y ] = −X

→ solvable and polynomial growth.

n = span(X ,Y ) so a := span(Θ) →

{g = n⊕ aads(a)(a) = {0}

Now

ad(Θ) =

0 −1 01 0 00 0 0

→ adsi(Θ) = ad(Θ)so [Θ,X ]R = [Θ,X ]

ϕ(Θ)(X )︷︸︸︷−adsi(πa(Θ))(X ) +adsi(πa(X ))(Θ)

= Y − ad(Θ)(X ) + 0 = Y − Y = 0[Θ,Y ]R = 0 and [X ,Y ]R = [X ,Y ] = 0

So the real shadow is R3.16 / 18



Theorem (Breuillard + CKLNO, in the thesis of SebastianoNicolussi Golo)

Let G and N be solvable simply connected Lie groups, and assumeN is nilpotent. Then the following are equivalent:

G can be made isometric to N,

G has polynomial growth and N is the nilshadow of G .

Left open the following questions:

If groups G1 and G2 of polynomial growth can be madeisometric, do they have the same nilshadow?

If groups G1 and G2 of polynomial growth have the samenilshadow, can they be made isometric to themselves?

What about other solvable groups?

→ Attempt to make ’can be made isometric’ to purely algebraiclycheckable relation (at least for solvable groups)!

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Left open the following questions:

If groups G1 and G2 of polynomial growth can be madeisometric, do they have the same nilshadow?

If groups G1 and G2 of polynomial growth have the samenilshadow, can they be made isometric to themselves?

What about other solvable groups?

Solution:

Conjecture (CKLNO)

Let G1 and G2 be simply connected solvable Lie groups. Then thefollowing are equivalent:

G1 can be made isometric to G2,

G1 and G2 have isomorphic realshadows.

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IntroductionExistence of Lie group structuresFinding all Lie group structures: CMIReal shadow

On Lie group structures of homogeneous metric spacesusers.jyu.fi/~vikakivi/prujut/2020-03-30.pdf ·...

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