CLUSTER ALGEBRAS AND TRIANGULATED SURFACES PART II: …fomin/Papers/ft.pdf · PART II: LAMBDA...

CLUSTER ALGEBRAS AND TRIANGULATED SURFACES

PART II: LAMBDA LENGTHS

SERGEY FOMIN AND DYLAN THURSTON

Abstract. We construct geometric models for cluster algebras associated withbordered surfaces with marked points, for any choice of coefficients of geometrictype, using generalized decorated Teichmuller spaces. In this context, the clustervariables are interpreted as suitably renormalized lambda lengths.

Contents

Introduction 2Overview 31. Non-normalized cluster algebras 52. Rescaling and normalization 73. Cluster algebras of geometric type and their positive realizations 94. Bordered surfaces, arc complexes, and tagged arcs 115. Structural results 186. Lambda lengths on bordered surfaces with punctures 197. Lambda lengths of tagged arcs 238. Opened surfaces 279. Lambda lengths on opened surfaces 2910. Non-normalized exchange patterns from surfaces 3611. Laminations and shear coordinates 3912. Shear coordinates with respect to tagged triangulations 4213. Tropical lambda lengths 4414. Laminated Teichmuller spaces 4815. Topological realizations of some coordinate rings 5216. Principal and universal coefficients 5817. Tropical degeneration and relative lambda lengths 59Acknowledgments 62References 63

Date: May 1, 2008.2000 Mathematics Subject Classification. Primary 30F60, Secondary 16S99, 55U10.Key words and phrases. Cluster algebra, lambda length, decorated Teichmuller space, cluster

complex, arc complex, triangulated surface, hyperbolic surface, Ptolemy relations.Partially supported by NSF grant DMS-0555880 (S. F.) and a Sloan Fellowship (D. T.).

1

2 SERGEY FOMIN AND DYLAN THURSTON

Introduction

This paper continues the study of cluster algebras associated with marked Riemannsurfaces with holes and punctures. The emphasis of the first paper [8] was on thecombinatorial construction of the cluster complex, which we showed to be closelyrelated to the tagged arc complex of the surface, a particular extension of the classicalsimplicial complex of arcs connecting marked points. The focus of the current paperis on geometry, specifically on providing an explicit geometric interpretation for thecluster variables in any cluster algebra (of geometric type) whose exchange matrixcan be associated with a triangulated surface. More concretely, we demonstrate thateach cluster variable can be viewed as a properly normalized lambda length of thecorresponding (tagged) arc. Similarly to their precursors introduced by R. Penner[22, 23, 24], these lambda lengths serve as coordinates on appropriately definedgeneralizations of decorated Teichmuller spaces.

The main underlying idea of this line of inquiry—already present in the pioneeringworks of V. Fock and A. Goncharov [4, 5], and of M. Gekhtman, M. Shapiro, andA. Vainshtein [18]—is to interpret such a Teichmuller space as the real positive partof an algebraic variety. The coordinate ring A of this variety is generated by thevariables corresponding to the lambda lengths; these variables satisfy certain knownalgebraic relations among lambda lengths (the generalized Ptolemy relations). Therings A arising from various versions of this construction possess, in a very naturalway, a cluster algebra structure: the lambda lengths become cluster variables, andclusters correspond to tagged triangulations.

The main results obtained in this paper can be succinctly summarized as follows.

(1) We investigate a broad class of cluster algebras that includes any cluster al-gebra of geometric type whose (skew-symmetric) exchange matrix is a signedadjacency matrix of a triangulated bordered surface. Each such cluster al-gebra is naturally associated with a collection of integral laminations on thesurface. Every exchange relation is then readily written in terms of shearcoordinates of these laminations with respect to a given tagged triangulation.

(2) We describe the cluster variables in these cluster algebras as generalizedlambda lengths. To this end, we (a) extend the lambda length construc-tion to tagged arcs, (b) define Teichmuller spaces of opened surfaces, andthe associated lambda lengths, (c) define laminated Teichmuller spaces andtropical lambda lengths, and (d) combine all these elements together to realizecluster variables as certain rescalings of lambda lengths of tagged arcs.

(3) The underlying combinatorics of a cluster algebra is governed by its clustercomplex. In our first paper [8], written in collaboration with Michael Shapiro,we described this complex in terms of tagged arcs, and investigated its basicstructural properties. For technical reasons, the treatment there excluded thecase of closed surfaces with exactly two punctures. In this paper, we removethis restriction, extending the main results of the previous papre to the mostgeneral case of bordered surfaces with marked points.

CLUSTER ALGEBRAS AND TRIANGULATED SURFACES II 3

Overview

Sections 1–3 are devoted to preliminaries on cluster algebras and exchange pat-terns. It turns out that the proper algebraic framework for our main constructionis provided by the axiomatic setting of non-normalized cluster algebras. This wasin fact the original setting of cluster algebras [11], although it was largely overshad-owed by subsequent developments in cluster algebra theory, which dealt, almost ex-clusively, with the normalized case. Section 1 contains a review of (non-normalized)cluster algebras and the mutation rules used to define them. Section 2 discussesrescaling of cluster variables and the technicalities involved in constructing a nor-malized pattern by rescaling a non-normalized one. Section 3 recalls the notion ofcluster algebras of geometric type, and discusses realizations of these algebras inwhich both cluster and coefficient variables are represented by positive functions ona topological space, in anticipation of Teichmuller-theoretic applications.

In Section 4, we review the main constructions and some of the main results ofthe prequel [8] to this paper: ordinary and tagged arcs on a bordered surface withmarked points; triangulations, flips, and arc complexes (both ordinary and tagged);associated exchange graphs; and signed adjacency matrices and their mutations.

In Section 5, we formulate the first batch of our results, which describe the struc-ture of cluster algebras whose exchange matrices come from triangulated surfaces.(The proofs come much later.) In brief, we extend all main structural theorems of [8]to arbitrary surfaces, including closed surfaces with two marked points, which werenot covered in [8] because of the exceptional nature of the fundamental groups oftheir exchange graphs.

In Sections 6–10, we develop the hyperbolic geometry tools required for our mainconstruction. Section 6 presents Penner’s concept of lambda length, and its basicalgebraic properties. In Section 7, we adapt this concept to the tagged setting, andwrite down the appropriate versions of Ptolemy relations. This enables us to describe,in Theorem 7.5, the main structural features of a particular class of exchange patternswhose coefficient variables come from boundary segments. (This class of clusteralgebras already appeared in [4, 5, 18].) To handle most general coefficient systems,we need another geometric idea, introduced in Section 8: the concept of an openedsurface obtained by replacing interior marked points by geodesic circular boundarycomponents. Section 9 presents the corresponding version of lambda lengths (forthe lifted arcs on the opened surface), and an appropriate variation of the decoratedTeichmuller space. In Section 10, we show that suitably rescaled lambda lengthsof lifted arcs form a non-normalized exchange pattern, providing a crucial buildingblock for our main construction (to be completed in Section 14).

Sections 11–13 are devoted to the combinatorics of general coefficient systemsof geometric type (equivalently, extended exchange matrices). As noted by Fockand Goncharov [6], W. Thurston’s shear coordinates transform under ordinary flipsaccording to the rules of matrix mutations. In Section 11, we review this beautifultheory; in Section 12, we extend its main results, namely Thurston’s coordinatizationtheorem for integral laminations, and the matrix mutation rule, to the tagged setting.Section 13 discusses the notion of tropical lambda length, a discrete analogue of


Penner’s concept, in which hyperbolic lengths are replaced by a combination oftransverse measures with respect to a family of laminations. Tropical lambda lengthssatisfy the tropical version of exchange relations, which makes them ideally suitedfor the role of rescaling factors in our main construction.

This construction is presented in Section 14. Here we introduce the notion of alaminated Teichmuller space whose defining data include, in addition to the surface, afixed multi-lamination on it. This space can be coordinatized by yet another versionof lambda lengths, obtained by dividing the ordinary lambda lengths of lifted arcsby the tropical ones. (It turns out that the ratio does not depend on the choice of alift of the arc. And even though it does depend on the choice of lifts of laminations,this choice does not affect the resulting cluster algebra structure, up to a canonicalisomorphism.) The key property of these “laminated lambda lengths” is that theyform an exchange pattern (thus generate a cluster algebra) of the required kind. Inother words, they satisfy the exchange relations of Ptolemy type whose coefficientsare encoded by the shear coordinates of the chosen multi-lamination with respect tothe current tagged triangulation. This geometric realization allows us to prove allour claims, made in earlier sections, pertaining to the structural properties of thecluster algebras under consideration.

The next two sections are devoted to applications and examples. In Section 15,we provide topological models for cluster-algebraic structures in coordinate ringsof various algebraic varieties, which include certain Grassmannians and affine basespaces. In each case, the relevant cluster structure is encoded, in a uniform way,by a particular choice of a bordered surface with a collection of integral laminationson it. These examples provide a fascinating link between representation theory(canonical bases, rings of invariants) and combinatorial topology, a link awaitingfurther exploration. Section 16 treats in concrete detail two important classes ofcoefficient systems introduced in [13]: the principal coefficients and the universalcoefficients (the latter in finite types An and Dn).

Section 17 contains an informal discussion of how our main geometric constructionof renormalized lambda lengths on opened surfaces can be obtained by means oftropical degeneration from a somewhat more conventional setting in which, insteadof fixing a multi-lamination, one picks in advance a family of decorated hyperbolicstructures on an opened surface. As we let those geometries approach the Thurstonboundary of the appropriate Teichmuller space, certain ratios of lambda lengths oflifted tagged arcs (with respect to the different hyperbolic structures at hand) yield,in the “tropical limit,” the corresponding cluster variables.

Throughout this paper, we use notation and terminology consistent with the pre-vious paper in the series [8].

We plan to devote our next paper to the construction of “canonical” bases incluster algebras associated with bordered surfaces.

Several figures in this paper are best viewed in color.


1. Non-normalized cluster algebras

The original definition and basic properties of (non-normalized) cluster algebraswere given in [11]; see [2] for further developments. In this section, we recall thebasic notions of this theory following the aforementioned sources (cf. especially [2,Sections 1.1–1.2]). Note that our previous paper [8] used a more restrictive normal-ized setup.

The construction of a (non-normalized, skew-symmetrizable) cluster algebra beginswith a coefficient group P, an abelian group without torsion, written multiplicatively.Take the integer group ring ZP, and let F be (isomorphic to) a field of rationalfunctions in n independent variables with coefficients in (the field of fractions of) ZP.Later, we are going to call the positive integer n the rank of our yet-to-be-definedcluster algebra, and F its ambient field. The following definitions are central for thecluster algebra theory.

Definition 1.1 (Seeds). A seed in F is a triple Σ = (x,p, B) consisting of:

• the cluster x ⊂ F , a set of n algebraically independent elements (calledcluster variables) which generate F over the field of fractions of ZP;• the coefficient tuple p = (p±x )x∈x, a 2n-tuple of elements of P;• the exchange matrix B=(bxy)x,y∈x, a skew-symmetrizable n×n integer matrix.

That is, B can be made skew-symmetric by rescaling its columns by appropriatelychosen positive scalars. (In all our applications, B will in fact be skew-symmetric.)

Definition 1.2 (Seed mutations). Let Σ = (x,p, B) be a seed in F , as above. Picka cluster variable z ∈ x. We say that another seed Σ = (x,p, B) is related to Σ bya seed mutation in direction z if

• the cluster x is given by x = x − z ∪ z, where the new cluster variablez ∈ F is determined by the exchange relation

(1.1) z z = p+z

∏

x∈xbxz>0

xbxz + p−z∏

x∈xbxz<0

x−bxz ;

• the coefficient tuple p = (p±x )x∈x satisfies

p±z = p∓z ;(1.2)

p+x /p

−x =

(p+

z )bzx p+x /p

−x if bzx ≥ 0

(p−z )bzx p+x /p

−x if bzx ≤ 0

(for x 6= z)(1.3)

• the exchange matrix B = (bxy) is obtained from B by the matrix mutationrule

(1.4) bxy =

0 if x = y = z;

−bxz if y = z 6= x;

−bzy if x = z 6= y;

bxy if x 6= z, y 6= z, and bxzbzy ≤ 0;

bxy + |bxz|bzy if x 6= z, y 6= z, and bxzbzy ≥ 0.


It is easy to check that the mutation relation is symmetric: that is, Σ is in turnrelated to Σ by a mutation in direction z.

Remark 1.3. For the reader’s benefit, we spell out the crucial (and only) differencebetween Definition 1.2 and its counterpart [8, Definition 5.1] used in our previouspaper: here, we do not require P to be endowed with a semifield structure. Thus, wedo not need the “auxiliary addition” ⊕ in P, and drop the normalization conditionp+

z ⊕ p−z = 1. As in the original cluster algebra setup [11], there is a price to be paidfor getting rid of the semifield structure: in the absence of normalization, the seedΣ in Definition 1.2 is not uniquely determined by Σ and a choice of z. Indeed, then− 1 monomial formulas (1.3) prescribe the ratios p+

x /p−x only, leaving us with n− 1

degrees of freedom in choosing specific coefficients p±x .

Definition 1.4 (Exchange pattern). Let E be a connected (unoriented, possiblyinfinite) n-regular graph. That is, each vertex t in E is connected to exactly nother vertices; the corresponding n-element set of edges is denoted by star(t). Anattachment of a seed Σ = (x,p, B) at t is a bijective labeling of the n clustervariables in x by the n edges in star(t); cf. [12, Section 2.2]. With such a seedattachment, we typically use the notation x = (xe(t))e∈star(t). An exchange patternon E is, informally speaking, a collection of seeds attached at the vertices of E andrelated to each other by the corresponding mutations. Let us now be precise. Anexchange patternM = (Σt) is a collection of seeds Σt = (x(t),p(t), B(t)) labeled bythe vertices t in E and satisfying the following condition. Consider an edge e in E

connecting two vertices t and t. Let x(t) = (xe(t))e∈star(t) be the cluster at t, labeledusing the seed attachment. Then the definition requires that the seed Σt is relatedto Σt by a mutation in the direction of xe(t).

An exchange pattern on a regular graph E can be canonically lifted to an exchangepattern on its universal cover, an n-regular tree Tn, recovering the original definitionin [11].

Definition 1.5 (Cluster algebra). To define a (non-normalized) cluster algebra A,one needs two pieces of data:

• an exchange patternM = (Σt) as in Definition 1.4;• a ground ring R, a subring with unit in ZP that contains all coefficient

tuples p(t), for all seeds Σt = (x(t),p(t), B(t)).

The cluster algebra A is then defined as the R-subalgebra of the ambient field Fgenerated by the union of all clusters x(t).

Remark 1.6. Once an exchange pattern M as in Definition 1.4 has been fixed,we can start using the vertices and edges of the underlying graph E to label thecomponents of coefficient tuples p(t) = (p±e (t)) and the entries of the exchangematrices B(t) = (bfe(t)) (here e, f ∈ star(t)). With this notation, the exchangerelation (1.1) associated with an edge e between t and t takes the form

(1.5) xe(t)xe(t) = p+e (t)

∏

f∈star(t)bfe(t)>0

xf (t)bfe(t) + p−e (t)

∏

f∈star(t)bfe(t)<0

xf (t)−bfe(t) .


Furthermore, the adjacent clusters x(t) = (xf (t))f∈star(t) and x(t) = (xg(t))g∈star(t)

coincide (as unlabeled sets) except for the replacement of xe(t) by xe(t), resulting ina bijection

star(t) \ e → star(t) \ ebetween the corresponding sets of labels. These bijections form a discrete connection(see, e.g., [19]) on E that keeps track of the relabelings of each cluster variable.1

Suppose that the edges f ∈ star(t) \ e and f ∈ star(t) \ e are mapped toeach other under this discrete connection; in other words, xf (t) = xf (t). Then thecorresponding equations (1.2)–(1.3) become

p±e (t) = p∓e (t) ;(1.6)

p+

f(t)/p−

f(t) =

(p+

e (t))bef (t) p+f (t)/p−f (t) if bef (t) ≥ 0

(p−e (t))bef (t) p+f (t)/p−f (t) if bef (t) ≤ 0

(for f 6= e).(1.7)

2. Rescaling and normalization

In this section, we make a couple of observations related to rescaling of clustervariables; these observations will play an important role in the sequel.

The first algebraic observation (see Proposition 2.1 below) is that rescaling anexchange pattern gives again an exchange pattern. That is, if we replace each clustervariable by a new one that differs by a constant factor (these constant factors canbe chosen completely arbitrarily for different cluster variables), and then rewriteeach exchange relation in the obvious way in terms of the new variables, then thecoefficients in these new exchange relations satisfy the monomial relations for anexchange pattern.

To formulate the above statement precisely, we will need the following naturalnotion. We say that a collection (ce(t)) labeled by all pairs (t, e) with e ∈ star(t))is compatible with the discrete connection defined by an exchange pattern M (cf.Remark 1.6) if, for any edge e between t and t, we have cf (t) = cg(t) wheneverxf (t) = xg(t).

Proposition 2.1. Let M = (Σt) be an exchange pattern on an n-regular graph E,as in Definition 1.4 and Remark 1.6. Let (ce(t)) be a collection of scalars in P that iscompatible with the discrete connection associated with M. Then the following con-struction yields an exchange pattern M′ = (Σ′

t) on E, with Σ′t = (x′(t),p′(t), B(t)):

• the attached cluster x′(t) = (x′e(t))e∈star(t) is given by x′e(t) = xe(t)/ce(t);• the coefficient tuple p′(t) = (p′±e (t)) is defined by

(2.1) p′±e (t) =p±e (t)

ce(t)ce(t)

∏

±bfe(t)>0

cf (t)±bfe(t),

where t denotes the endpoint of e different from t;• the exchange matrices B(t) do not change.

1This connection has trivial monodromy since the cluster variables in each cluster are distinct.


Proof. It is straightforward to check that substituting xe(t) = x′e(t)ce(t) into (1.5) re-sults into the requisite exchange relation inM′. Together with the compatibility con-dition, this ensures that adjacent (attached) clusters are related by the correspondingmutations. It remains to demonstrate that the rescaled coefficient tuples p′(t) satisfythe requirements in Definition 1.2. The condition p′±e (t) = p′∓e (t) (cf. (1.6)) is easilyverified. Finally, in order to check the equation (1.7) for the coefficients p′±e (t), wesubstitute the expressions (2.1) into it, and factor out (1.7) for the original pattern.The resulting equation

∏

g∈star(t)

cg(t)bgf

(t) = (ce(t)ce(t))−bef (t)

∏

g : bge(t) bef (t)>0

cg(t)bge(t)|bef (t)|

∏

g∈star(t)

cg(t)bgf (t)

is easily seen to follow from the matrix mutation rules (1.4). ¤

Remark 2.2. The reader may want to see a more intuitive explanation of whythe axioms of an exchange pattern survive rescaling. Such an explanation can beprovided by checking this property against the alternative version of the mutationrules (1.2)–(1.4) given in [11, (2.7)] (cf. also the “Caterpillar Lemma” in [15]).

We next turn to the issue of normalization, that is, using the rescaling of clustervariables (as in Proposition 2.1) to obtain a “normalized” exchange pattern. Thelatter concept requires endowing the coefficient group P with a semifield structure(cf. Remark 1.3).

Definition 2.3 (Normalized exchange pattern). Suppose that (P,⊕, ·) is a (commu-tative) semifield, i.e., (P, ·) is an abelian multiplicative group, (P,⊕) is a commu-tative semigroup, and the auxiliary addition ⊕ is distributive with respect to themultiplication. The multiplicative group of any such semifield P is torsion-free [11,Section 5]. An exchange pattern M = (Σt) as in Definition 1.4/Remark 1.6 (or thecorresponding cluster algebra) is called normalized if the coefficients p±e (t) satisfythe normalization condition

(2.2) p+e (t)⊕ p−e (t) = 1 .

Our next algebraic observation is that rescaling of cluster variables in a non-normalized exchange pattern produces a normalized pattern if the rescaling factorsce(t) themselves satisfy the auxiliary-addition version of the same exchange relations.

Proposition 2.4. Continuing with the assumptions and constructions of Proposi-tion 2.1, let us furthermore suppose that the coefficient group P is endowed with an ad-ditive operation ⊕ making (P,⊕, ·) a semifield. Then the rescaled patternM′ = (Σ′

t)is normalized if and only if the scalars ce(t) satisfy the relations

(2.3) ce(t) ce(t) = p+e (t)

∏

f∈star(t)bfe(t)>0

cf (t)bfe(t) ⊕ p−e (t)

∏

f∈star(t)bfe(t)<0

cf (t)−bfe(t) .

Proof. Equation (2.3) is simply a rewriting of the normalization condition p′+e (t) ⊕p′−e (t) = 1 for the rescaled coefficients p′±e (t) given by (2.1). ¤


3. Cluster algebras of geometric type and their positive

realizations

The most important example of normalized exchange patterns (resp., normalizedcluster algebras) are the patterns (resp., cluster algebras) of geometric type.

Definition 3.1 (Tropical semifield, cluster algebra of geometric type [11, Exam-ple 5.6, Definition 5.7]). Let I be a finite indexing set, and let

(3.1) P = Trop(qi : i ∈ I)be the multiplicative group of Laurent monomials in the formal variables qi : i ∈ I,which we call the coefficient variables. Define the auxiliary addition ⊕ by

(3.2)∏

i

qai

i ⊕∏

i

qbi

i =∏

i

qmin(ai,bi)i .

The semifield (P,⊕, ·) is called a tropical semifield (cf. [1, Example 2.1.2]) A clusteralgebra (or the corresponding exchange pattern) is said to be of geometric type if it isdefined by a normalized exchange pattern with coefficients in some tropical semifieldP = Trop(qi : i ∈ I), over the ground ring R = Z[q±i : i ∈ I] or R = Z[qi : i ∈ I].

This definition differs slightly ffrom the one used in [2, 11, 12, 14], where only theformer choice of R was allowed. See loc. cit. for numerous examples.

Definition 3.2 (Extended exchange matrix ). For an exchange pattern of geometrictype, the coefficients p±e (t) are simply monomials in the variables qi. It is convenientand customary to encode these coefficients, along with the exchange matrix B(t),in a rectangular extended exchange matrix B(t) = (bef (t)) defined as follows. The

columns of B(t) are, as before, labeled by star(t). The top n rows of B(t) arealso labeled by star(t), while the subsequent rows are labeled by the elements of I.The top n× n submatrix of B(t) is B(t) (so our notation for the matrix elements isconsistent) while the entries of the bottom |I|×n submatrix are uniquely determinedby the formula

p+e (t)

p−e (t)=∏

i∈I

qbie(t)i .

As observed in [11], the mutation rules (1.6)–(1.7) can be restated, in the case ofgeometric type, as saying that the matrices B(t) undergo a matrix mutation givenby the same formulas (1.4) as before—now with a different set of row labels.

A pair (x(t)), B(t)) consisting of a cluster and the corresponding extended ex-change matrix will be referred to as a seed (of geometric type).

Example 3.3 (The ring C[SL2]). This was the first example given on the first pageof the first paper about cluster algebras [11]. The coordinate ring

A = C[SL2] = C[z11, z12, z21, z22]/〈z11z22 − z12z21 − 1〉carries a structure of a cluster algebra of geometric type, or rank n = 1, with thecoefficient semifield Trop(z12, z21), the cluster variables z11 and z22, and the soleexchange relation

z11z22 = z12z21 + 1.


The clusters are x(t1) = z11 and x(t2) = z22, and the extended exchange matricesare

B(t1) =

011

, B(t2) =

0−1−1

.

Cf. Example 15.4.

The following concept is rooted in the original motivations of cluster algebras,designed in part to study totally positive parts of algebraic varieties of Lie-theoreticorigin, in the sense of G. Lusztig [20] (see also [9, 14, 21] and references therein). Inthe context of cluster structures arising in Teichmuller theory, similar notions werefirst considered in [4, 18].

Definition 3.4 (Positive realizations). A positive realization of a cluster algebra Aof geometric type is, informally speaking, a faithful representation of A in the spaceof positive real functions on a topological space T of appropriate real dimension.An accurate definition follows.

Let A be a cluster algebra of geometric type over the semifield Trop(qi : i ∈ I).As before, let n denote the rank of A, let E be the underlying n-regular graph,and let x(t) = (xe(t))e∈star(t), for t ∈ E, be the clusters. A positive realization of Ais a topological space T together with a collection of functions xe(t) : T → R>0

(resp., qi : T → R>0) representing each cluster variable xe(t) (resp., each coefficientvariable qi) so that

• these functions satisfy all appropriate exchange relations, and• for each t ∈ E, the map

(3.3)∏

e∈star(t)

xe(t)×∏

i∈I

qi : T → Rn+|I|>0

is a homeomorphism.

To construct such a T , one starts with an arbitary homeomorphism of the form (3.3),and then determines the rest of the maps xe(t) : T → R>0 using exchange relations.The key feature of exchange patterns that makes this construction work is that there-parametrization maps relating adjacent clusters are birational and subtraction-free(hence positivity preserving).

The following simple observation will be useful in the sequel.

Proposition 3.5. A collection of extended exchange matrices B(t) labeled by thevertices of an n-regular graph E and related to each other by the corresponding matrixmutations (cf. Definition 3.2) defines an exchange pattern (or cluster algebra) ofgeometric type if and only if such a pattern can be given a positive realization, i.e., ifand only if there exists a topological space T and a collection of positive real functionson it which satisfy the conditions in Definition 3.4.

Proof. The “only if” direction is trivial (cf. the concluding remarks in Definition 3.4).For the “if” part, use the fact that two rational functions in m variables are equal ifand only if they coincide pointwise as functions on Rm

>0. ¤


4. Bordered surfaces, arc complexes, and tagged arcs

This section offers a swift review of the main constructions in [8]. For a detailedexposition with lots of examples and pictures, see [8, Sections 2–5, 7].

Definition 4.1 (Bordered surface with marked points). Let S be a connected oriented2-dimensional Riemann surface with (possibly empty) boundary ∂S. Fix a non-empty finite set M of marked points in the closure of S, so that there is at least onemarked point on each connected component of ∂S. Marked points in the interiorof S are called punctures. We will want S to have at least one triangulation by anon-empty set of arcs with endpoints at M. Consequently, we do not allow S to be asphere with one or two punctures; nor an unpunctured or once-punctured monogon;nor an unpunctured digon or triangle. Such a pair (S,M) is called a bordered surfacewith marked points. An example is shown in Figure 1.

Figure 1. A bordered surface with marked points. In this example,S is a torus with a hole; the boundary ∂S has a single componentwith 3 marked points on it; the set M consists of those 3 points plus2 punctures in the interior of S.

Definition 4.2 (Ordinary arcs). An arc γ in (S,M) is a curve in S, considered upto isotopy, such that

• the endpoints of γ are marked points in M;• γ does not intersect itself, except that its endpoints may coincide;• except for the endpoints, γ is disjoint from M and from ∂S;• γ does not cut out an unpunctured monogon or an unpunctured digon.

We denote by A(S,M) the set of of all arcs in (S,M). See Figure 2.

An arc whose endpoints coincide is called a loop.

Definition 4.3 (Compatibility of ordinary arcs). Two arcs are compatible if they(more precisely, some of their isotopic deformations) do not intersect in the interiorof S. For example, the three arcs shown in Figure 2 on the right are pairwisecompatible.


Figure 2. The three curves on the left do not represent arcs; thethree curves on the right do.

Definition 4.4 (Ideal triangulations). A maximal collection of distinct pairwisecompatible arcs forms an (ordinary) ideal triangulation. The arcs of a triangulationcut S into ideal triangles ; note that we do allow self-folded triangles. Each idealtriangulation consists of

(4.1) n = 6g + 3b+ 3p+ c− 6

arcs, where g is the genus of S, b is the number of boundary components, p is thenumber of punctures, and c is the number of marked points on the boundary ∂S.

Figure 3 shows two triangulations of a sphere with 4 punctures. Each triangulationhas 4 ideal triangles, 2 of which are self-folded.

Figure 3. Two triangulations of a 4-punctured sphere. Each trian-gulation has n = 6 arcs. Cf. (4.1), with g = 0, b = 0, p = 4, c = 0.

The assumptions made above ensure that (S,M) possesses a triangulation withoutself-folded triangles (see [8, Lemma 2.13]).

Definition 4.5 (Ordinary flips). Ideal triangulations are connected with each otherby sequences of flips. Each flip replaces a single arc γ in a triangulation T by a(unique) arc γ ′ 6= γ that, together with the remaining arcs in T , forms a new idealtriangulation. See Figure 4; also, the two triangulations in Figure 3 are related by aflip.

It is important to note that this operation cannot be applied to an arc γ that liesinside a self-folded triangle in T .

Definition 4.6 (Arc complex and its dual graph). The arc complex ∆(S,M) is the(possibly infinite) simplicial complex on the ground set A(S,M) defined as the cliquecomplex for the compatibility relation. In other words, the vertices of ∆(S,M) are


γ−→ γ′

Figure 4. A flip inside a quadrilateral

the arcs, and the maximal simplices are the ideal triangulations. The dual graph of∆(S,M) is denoted by E(S,M); its vertices are the triangulations, and its edgescorrespond to the flips. See Figure 5.

∆(S,M) E(S,M)

Figure 5. The arc complex and its dual graph for a once-punctured digon

In general, the arc complex ∆(S,M) has nonempty boundary since his dual graphis not n-regular: not every arc can be flipped. In [8], we suggested a natural wayto extend the arc complex beyond its boundary, obtaining an n-regular graph whichcan be used to build the desired exchange patterns. This requires an introduction ofthe concept of a tagged arc.

Definition 4.7 (Tagged arcs). A tagged arc is obtained by taking an arc that doesnot cut out a once-punctured monogon and marking (“tagging”) each of its ends inone of the two ways, plain or notched, so that the following conditions are satisfied:

• an endpoint lying on the boundary of S must be tagged plain, and• both ends of a loop must be tagged in the same way.

See Figure 6The set of all tagged arcs in (S,M) is denoted by A./(S,M).

Definition 4.8 (Representing ordinary arcs by tagged arcs). Ordinary arcs can beviewed as a special case of tagged arcs, via the following dictionary. Let us canonicallyrepresent any ordinary (untagged) arc β by a tagged arc τ(β) defined as follows. Ifβ does not cut out a once-punctured monogon, then τ(β) is simply β with both ends


./ ./ ./

./

./

./

Figure 6. Different types of tagged arcs

tagged plain. Otherwise, β is a loop based at some marked point a and cutting outa punctured monogon with the sole puncture b inside it. Let α be the unique arcconnecting a and b and compatible with β. Then τ(β) is obtained by tagging α plainat a and notched at b. See Figure 7.

β

a

b./

τ(β)

a

b

Figure 7. Representing an arc bounding a punctured monogon by a tagged arc

Definition 4.9 (Compatibility of tagged arcs). This is an extension of the corre-sponding notion for ordinary arcs. Tagged arcs α and β are compatible if and only if

• their untagged versions α and β are compatible;• if α and β share an endpoint a, then the ends of α and β connecting to a

must be tagged in the same way—unless α = β , in which case at least oneend of α must be tagged in the same way as the corresponding end of β.

It is easy to see that the map γ 7→ τ(γ) described in Definition 4.8 preservescompatibility.

Definition 4.10 (Tagged triangulations). A maximal (by inclusion) collection ofpairwise compatible tagged arcs is called a tagged triangulation.

Each ideal triangulation T can be represented by a tagged triangulation τ(T ) viathe dictionary τ described above. Figure 8 shows two tagged triangulations obtainedby applying τ to the triangulations in Figure 3.

./ ./ ./ ./

Figure 8. Two tagged triangulations of a 4-punctured sphere.

By [8, Theorem 7.9], all tagged triangulations have the same cardinality n givenby (4.1).


Definition 4.11 (Tagged arc complex ). The tagged arc complex ∆./(S,M) is thesimplicial complex whose vertices are tagged arcs and whose simplices are collectionsof pairwise compatible tagged arcs. See Figure 9 on the left.

The ordinary arc complex ∆(S,M) can be viewed a subcomplex of ∆./(S,M) (viathe map τ); cf. Figures 5 and 9.

./

./

∆./(S,M)

./

./

./

./

E./(S,M)

Figure 9. Tagged arc complex and its dual graph for a once-punctured digon

The maximal simplices of ∆./(S,M) are the tagged triangulations, so ∆./(S,M) ispure of dimension n−1. Furthermore, ∆./(S,M) is a pseudomanifold, i.e., each sim-plex of codimension 1 is contained in precisely two maximal simplices. To rephrase,every tagged arc in an arbitrary tagged triangulation can be flipped (replaced by adifferent tagged arc) in a unique way to produce another tagged triangulation.

Definition 4.12 (Dual graph of the tagged arc complex ). The dual graph E./(S,M)of the pseudomanifold ∆./(S,M) has tagged triangulations as its vertices. Two suchvertices are connected by an edge if these tagged triangulations are related by a“tagged flip.” Thus, E./(S,M) is a (possibly infinite) n-regular graph. An exampleis shown in Figure 9 on the right.

Remark 4.13 (Nomenclature of tagged triangulations and tagged flips). It is easyto see (cf. [8, Section 9.2]) that compatibility of tagged arcs is invariant with respectto a simultaneous change of all tags at a given puncture. Let us take any taggedtriangulation T ′ and perform such change at every puncture where all ends of T ′ arenotched. It is not hard to verify that the resulting tagged triangulation T ′′ representssome ideal triangulation T (possibly containing self-folded triangles): T ′′ = τ(T ). Inother words, each tagged triangulation can be obtained from an ordinary one byapplying τ and then placing notches around some punctures.

This observation can be used to give a concrete description of all possible “taggedflips.” Let T ′ be a tagged triangulation, and let T ′′ and T be as above. Then eachof the n “tagged flips” out of T ′ is of one of the two kinds:


(4.D) a flip performed inside a once-punctured digon, as represented by one of the4 edges of the graph E./(S,M) in Figure 9. The tagging at the vertices ofthe digon does not change; or

(4.Q) an ordinary flip inside a quadrilateral in T (cf. Definition 4.5). As before,the sides of the quadrilateral do not have to be distinct. Moreover, thosesides (stripped of their tagging) should be arcs of T but not necessarily of T ′:specifically, such a side can be a loop in T enclosing a once-punctured mono-gon. The tagging at each vertex of the quadrilateral remains the same.

To illustrate, the two tagged triangulations in Figure 8 are related by a tagged flipof type (4.Q).

By [8, Proposition 7.10], the dual graph E./(S,M) (hence the complex ∆./(S,M))is connected—i.e., any two tagged triangulations can be connected by a sequenceof flips—unless (S,M) is a closed surface with a single puncture, in which caseE./(S,M) (resp., ∆./(S,M)) consists of two isomorphic connected components, onein which the ends of all tagged arcs are plain, and another in which they are allnotched.

Definition 4.14 (Exchange graph of tagged triangulations). We denote by E(S,M)a connected component of E./(S,M). More precisely, E(S,M) = E./(S,M) unless(S,M) is closed with a single puncture; in the latter case, E(S,M) is the connectedcomponent of E./(S,M) in which all arcs are plain.

As shown in [8, Theorem 7.11], there is a natural class of normalized exchangepatterns (equivalently, cluster algebras) whose underlying graph is E = E(S,M).(Strictly speaking, this result was obtained under the assumption thet (S,M) is nota closed surface with two punctures. We are not going to make this assumptionherein.) In such a pattern, the cluster variables are labeled by the tagged arcs whileclusters correspond to tagged triangulations. We next proceed to reviewing thisconstruction.

Definition 4.15 (Signed adjacency matrix ). The key ingredient in building an ex-change pattern on E(S,M) is a rule that associates with each tagged triangulation Ta skew-symmetrizable exchange matrix B(T ) called a signed adjacency matrix of T .The rows and columns of B(T ) are labeled by the arcs in T . The direct definition ofB(T ) is fairly technical, and we refer the reader to [8, Definitions 4.1, 9.18] for those(unfortunately unavoidable) technicalities. For the immediate purposes of this re-view, we make the following shortcut. Let us start by defining B(T ) for an ordinaryideal triangulation T without self-folded triangles; this was first done in [4, 5, 18].Under that assumption, one sets

(4.2) B(T ) =∑

∆

B∆ ,


the sum over all ideal triangles ∆ in T of the n× n matrices B∆ = (b∆ij) given by

(4.3) b∆ij =

1 if ∆ has sides i and j, with j following i in the clockwise order;

−1 if the same holds, with the counterclockwise order;

0 otherwise.

See Figure 10 for an example.

23

4 1

65

B(T ) =

0 −1 0 0 1 −1

1 0 −1 0 0 0

0 1 0 −1 0 0

0 0 1 0 −1 0

−1 0 0 1 0 1

1 0 0 0 −1 0

Figure 10. The signed adjacency matrix for a triangulation of a once-punctured hexagon

One can then extend the definition of B(T ) to arbitrary tagged triangulations byrequiring that

• whenever T and T are related by a flip of a tagged arc k, the associatedsigned adjacency matrices are related by the corresponding mutation:

(4.4) B(T ) = µk(B(T ));

• if T is a triangulation without self-folded triangles, and T ′ is obtained fromT by putting a notch at each end of each arc in T adjacent to a puncture,then B(T ′) = B(T ).

It follows from [8, Proposition 7.10, Definition 9.6, Lemma 9.7] that this definition isconsistent, that is, there exists a unique collection of matrices B(T ) satisfying (4.4).

Each matrix B(T ) is skew-symmetric, with entries equal to 0, ±1, or ±2.It turns out that replacing ideal triangulations by (more general) tagged triangula-

tions does not extend the class of the associated exchange matrices B(T ) [8, Propo-sition 12.3]: each matrix B(T ) corresponding to a tagged triangulation is identical,up to simultaneous permutations of rows and columns, to a matrix corresponding toan ordinary ideal triangulation.

Remark 4.16. There are various ways to extend the matrices B(T ) to rectangularmatrices B(T ) (cf. Definition 3.2), creating coefficient systems of geometric type.A very general construction of this kind will be discussed in Section 11. Here webriefly discuss an easy special case that was already pointed out in [4, 18].


Let B(S,M) denote the set of boundary segments between adjacent marked pointson ∂S. Consider the tropical coefficient semifield

P = Trop(qβ : β ∈ B(S,M))

generated by the variables qβ labeled by such boundary segments. For an idealtriangulation T without self-folded triangles, define the (|B(S,M)|+n)×n matricesB(T ) by the same equations (4.2)–(4.3) as before, but with the understanding thatthe row index i can now be either an arc or a boundary segment. The matrices B(T )still satisfy the mutation rule B(T ) = µk(B(T )). This can be deduced from (4.4)by gluing a triangle on the other side of each boundary segment (thus making itinto a legitimate arc), then “freezing” all these arcs (i.e., not allowing to flip them).The same argument allows us to extend the definition of B(t) to arbitrary taggedtriangulations, as was done for B(t)’s, resulting in a well defined tropical coefficientsystem.

5. Structural results

In this section, we formulate those of our results whose statements do not requireany references to Teichmuller theory or hyperbolic geometry—even though theirproofs will rely on geometric arguments. These results concern structural propertiesof exchange patterns (or cluster algebras) whose exchange matrices can be describedas signed adjacency matrices of triangulations of a bordered surface. Most crucially,we show (see Theorem 5.1) that, for any choice of coefficients, there is an exchangepattern on the n-regular graph E = E(S,M) (see Definition 4.14) whose exchangematrices are the signed adjacency matrices B(T ).

Theorem 5.1. Let T be a tagged triangulation consisting of n tagged arcs in (S,M).Let Σ=(x(T),p(T), B(T)) be a seed as in Definition 1.1, that is:

• x(T) is an n-tuple of formal variables labeled by the arcs in T;• p(T) is a 2n-tuple of elements of an abelian torsion-free group P;• B(T) is the signed adjacency matrix of T.

Then there is a unique exchange pattern (ΣT ) on E(S,M) such that ΣT= Σ.

More precisely, let F be the field of rational functions in the variables x(T) withcoefficients in ZP. Then there exist unique elements xγ(T ) ∈ F and p±γ (T ) ∈ P

labeled by the tagged triangulations T ∈ E(S,M) and the tagged arcs γ ∈ T suchthat

• every triple ΣT = (x(T ),p(T ), B(T )) is a seed, where x(T ) = (xγ(T ))γ∈T ,p(T ) = (p±γ (T ))γ∈T , and B(T ) is the signed adjacency matrix of T ;• each cluster variable xγ = xγ(T ) does not depend on T ;• if a tagged triangulation T ′ is obtained from T by flipping a tagged arc γ ∈ T ,

then ΣT is obtained from ΣT ′ by the seed mutation replacing xγ by xγ′.

Furthermore, all cluster variables xγ (hence all seeds ΣT ) are distinct.

In the terminology of [11, Section 7], the last statement means that E(S,M) isthe exchange graph of the exchange pattern (ΣT ).


Theorem 5.1 implies that the structural results obtained in [8, Theorem 5.6] holdin full generality, for arbitrary bordered surfaces with marked points:

Corollary 5.2. Let A be a cluster algebra whose exchange matrices arise from tri-angulations of a surface (S,M). Then each seed in A is uniquely determined by itscluster; the cluster complex (see [8, Definition 5.4]) and the exchange graph E of Ado not depend on the choice of coefficients in A; the seeds containing a given clustervariable form a connected subgraph of E; and several cluster variables appear togetherin the same cluster if and only if every pair among them does. The cluster complexis the complex of tagged arcs, as in [8, Theorem 7.11]; it is the clique complex for its1-skeleton, and is a connected pseudomanifold.

Theorem 5.1 and Corollary 5.2 are proved in Section 14, using results and con-structions from the intervening sections. The proof is based on interpreting thecluster variables xγ as generalized lambda lengths, which are particular elements of(the field of fractions of) the appropriately defined extension of the Teichmuller spaceof (S,M).

6. Lambda lengths on bordered surfaces with punctures

The machinery of lambda lengths was introduced and developed by R. Penner [22,24] in his study of decorated Teichmuller spaces. In this section, we adapt Penner’sdefinitions to the case at hand, and give a couple of useful geometric lemmas.

Throughout the paper, (S,M) is a bordered surface with marked points as de-scribed at the beginning of Section 4. The Teichmuller space T (S,M) consists ofall complete finite-area hyperbolic structures with constant curvature −1 on S \M,with geodesic boundary at ∂S \M. (Thus there is a cusp at each point of M.) Ourassumptions on S guarantee that T (S,M) is non-empty and is more than a singlepoint.

For a given hyperbolic structure in T (S,M), each arc can be represented by aunique geodesic. Since there are cusps at the marked points, such a geodesic segmentis of infinite length. So if we want to measure the “length” of a geodesic arc betweentwo marked points, we need to renormalize. This is done as follows.

Definition 6.1 (Decorated Teichmuller space [22, 23, 24]). A point in a decorated

Teichmuller space T (S,M) is a hyperbolic structure as above together with a col-lection of horocycles, one around each marked point. (Appropriately interpreted,a horocycle around a cusp c is the set of points at an equal distance from c: al-though the cusp is infinitely far away from any point in the surface, there is still awell-defined way to compare distances from two different points.)

Remark 6.2. We note that our definition is a common generalization of those explic-itly given by Penner in loc. cit., as we simultaneously decorate both the punctures andthe marked points on the boundary. The possibility of extending his theory to thisgenerality was already mentioned by Penner [24, comments following Theorem 5.10].


Recall that B(S,M) denotes the set of segments of the boundary ∂S between twoadjacent marked points. The cardinality of B(S,M) is thus equal to c, the numberof marked points on ∂S.

Definition 6.3 (Lambda lengths [22, 23, 24]). Fix σ ∈ T (S,M). Let γ be an arcin A(S,M), or a boundary segment in B(S,M). We will use the notation γσ forthe geodesic representative of γ (relative to σ).

Let l(γ) = lσ(γ) be the signed distance along γσ between the horocycles at eitherend of γ (positive if the two horocycles do not intersect, negative if they do intersect).The lambda length λ(γ) = λσ(γ) of γ is defined by2

(6.1) λ(γ) = exp(l(γ)/2).

Definition 6.3 may at first seem somewhat arbitrary. It can be interpreted (see [22];we will not need this interpretation here) as a certain dot product between two nullvectors corresponding to the two endpoints.

For a given γ ∈ A(S,M) ∪B(S,M), one can view the lambda length

λ(γ) : σ 7→ λσ(γ)

as a function on the decorated Teichmuller space T (S,M). Penner shows that such

lambda lengths can be used to coordinatize T (S,M), as follows.

Theorem 6.4 (see [24, Theorems 5.8, 5.10]). For any triangulation T of (S,M)without self-folded triangles, the map

∏

γ∈T∪B(S,M)

λ(γ) : T (S,M)→ Rn+c>0

is a homeomorphism.

(Recall that n is the total number of arcs in T , and c is the number of markedpoints on the boundary.)

For our purposes, the crucial property of lambda lengths is the “Ptolemy relation,”the basic prototype of an exchange relation in cluster algebras.

Proposition 6.5 (Ptolemy relations [22, Proposition 2.6(a)]). Let

α, β, γ, δ ∈ A(S,M) ∪B(S,M)

be arcs or boundary segments (not necessarily distinct) that cut out a quadrilateral;we assume that the sides of the quadrilateral, listed in cyclic order, are α, β, γ, δ.Let η and θ be the two diagonals of this quadrilateral; see Figure 11. Then thecorresponding lambda lengths satisfy the Ptolemy relation

(6.2) λ(η)λ(θ) = λ(α)λ(γ) + λ(β)λ(δ).

There is a Ptolemy relation (6.2) associated to each ordinary flip in an ideal trian-gulation (cf. Definition 4.5). Note that some sides of the relevant quadrilateral maybe glued to each other, changing the appearance of the relation. See for exampleFigure 12.

2This definition coincides with the one in [24, Section 4] (or [18]), and differs by a factor of√

2from the definition in [22]. The choice made here makes Lemma 6.8 work with no factors.


η

θ

αβ

γ

δ

Figure 11. Sides and diagonals in a quadrilateral

α

β

γθη

λ(η)λ(θ) = λ(α)λ(γ) + λ(β)2

Figure 12. Ptolemy relation on a 4-punctured sphere. Cf. Figure 3.

By Theorem 6.4, each triangulation provides a set of coordinates on T (S,M), whileProposition 6.5 allows us to relate the coordinatizations corresponding to differenttriangulations. In the absence of punctures, this leads to an exchange pattern (witha special choice of coefficients) in which the lambda lengths play the role of clustervariables; cf. [6, 18]. For a punctured surface, the situation is much more delicate,for reasons both geometric and combinatorial: as we know, not every arc can beflipped without leaving the realm of ordinary triangulations. There is also a (related)algebraic reason, provided by following lemma, a special case of Proposition 6.5.

Lemma 6.6. Let α, β, γ, η, θ ∈ A(S,M)∪B(S,M) be as shown in Figure 13, that is:α and β bound a digon with a sole puncture p inside it; θ and γ connect p to thevertices of the digon; η is the loop enclosing γ. Then

(6.3) λ(η)λ(θ) = λ(α)λ(γ) + λ(β)λ(γ).

Lemma 6.6 shows that in the punctured case, complexities associated with settingup a cluster algebra structure already arise for ordinary flips, namely those thatcreate self-folded triangles. Indeed, the Ptolemy relation (6.3) cannot be an instanceof a cluster exchange (1.1) since the two terms on the right-hand side of (6.3) havea common factor λ(γ).

This issue can be resolved by introducing tagged arcs and their lambda lengths,and by extending the Ptolemy relations to the case of tagged flips. In the case ofa tagged arc with a notched end, our definition of a lambda length will require thenotion of (the hyperbolic distance from) a conjugate horocycle.


α

β

θ

η

γp

Figure 13. Arcs in a punctured digon.

Definition 6.7. For an horocycle h around a puncture in the interior of S, we denoteby L(h) the hyperbolic length of h. Two horocycles h and h around the same interiormarked point are called conjugate if L(h)L(h) = 1.

Lemma 6.8 ([24, Lemma 4.4]). Fix a decorated hyperbolic structure in T (S,M).Consider a triangle in S\M with vertices p, q, r∈M whose sides have lambda lengthsλpq, λpr, and λqr. Then the length hr of the horocyclic segment cut out by the triangleat vertex r is given by

hr =λpq

λprλqr

.

Proof. First let us see how the side lengths lij, the lambda lengths λij, and the horo-cyclic lengths hi change when we change the choice of horocycle. For convenience,we work in the upper-half-plane model for the hyperbolic plane, with the metric

ds2 =dx2 + dy2

y2,

and put the three vertices at 0, 1, and ∞, as in Figure 14 on the left. Let us movethe horocycle around ∞ from an initial Euclidean height of y to a height of y ′. Byelementary integration, the new lengths are h′

∞ = h∞(y/y′), l′i∞ = li∞+ln(y′/y), and

λ′i∞ = λi∞

√y′/y, for i ∈ 0, 1. The other lambda lengths and horocyclic lengths

are unchanged. Similar statements are true with 0, 1,∞ permuted.

Up to scale,λ01

λ0∞λ1∞

is the unique expression in the λij that is covariant in the

same way as h∞ with respect to the R3>0-action associated with moving the three

horocycles. To fix the scale, consider the case where all three horocycles just touch,as in Figure 14 on the right. In this case, h∞ = 1, as in the statement of thelemma. ¤

We note that a version of Lemma 6.8 appears as [22, Corollary 3.4], but thestatement there is off by a factor of

√2.

Lemma 6.9. Consider a punctured monogon with the vertex q ∈ ∂S and a solepuncture p in the interior. Choose a horocycle around q, and a horocycle h around p.Let λqq and λpq be the corresponding lambda lengths for the boundary of the monogonand the arc γpq connecting p and q inside it, respectively. Let h be the horocycle


l0 1 l1 1

h 1

0 1

1

−→l00 1 l01 1

h 01

0 1

1

−→

0 1

1

Figure 14. Moving a horocycle in the proof of Lemma 6.8. At thelast step, all the hyperbolic lengths lxy become 0.

around p conjugate to h, and let λpq be the corresponding lambda length of γpq; seeFigure 6. Then λqq = λpqλpq.

qp q q

pq

pq

h

h

Figure 15. The punctured monogon in Lemma 6.9. The geodesicsegments whose lengths are measured are shown in red.

Proof. By Lemma 6.8, we have L(h) =λqq

λ2pq

and L(h) =λqq

λ2pq

. Since h and h are

conjugate, we obtain 1 = L(h)L(h) =λ2

qq

λ2pqλ

2pq

, and the claim follows. ¤

7. Lambda lengths of tagged arcs

Lemma 6.9 can be used to define a cluster algebra structure associated with the

decorated Teichmuller space T (S,M) of a general bordered surface with punctures,extending the construction in [4, 5, 18]. As mentioned earlier, the key idea is to inter-pret a notched end of a tagged arc as an indication that in defining the correspondinglambda length, we should take the distance to the conjugate horocycle.

Definition 7.1 (Lambda lengths of tagged arcs). Fix a decorated hyperbolic struc-

ture σ ∈ T (S,M). The lambda length λ(γ) = λσ(γ) of a tagged arc γ ∈ A./(S,M) isdefined as follows. If both ends of γ are tagged plain, then the definition of λ(γ) given


in Definition 6.3 stands. Otherwise, the definition should be adjusted by replacingeach horocycle h at a notched end of γ by the corresponding conjugate horocycle h.

In order to write relations among these lambda lengths, we will need the followinglemma.

Lemma 7.2. Let γ and γ ′ be two tagged arcs connecting marked points p, q ∈ M.Assume that the untagged versions of γ and γ ′ coincide. Also assume that γ and γ ′

have identical tags at q, and different tags at p. (See Figure 16.) Let η be the loopbased at q wrapping around p, so that η encloses a monogon with a sole puncture pinside it. Then λ(η) = λ(γ)λ(γ ′).

./

η

q

p

γ γ′

Figure 16. Arcs γ, γ ′ and the enclosing loop η in Lemma 7.2

Proof. In view of Definition 7.1, Lemma 7.2 is a restatement of Lemma 6.9. ¤

Remark 7.3. One delicate aspect associated with Lemma 7.2 is that η itself is nota legal tagged arc since it encloses a once-punctured monogon. Assume furthermorethat both ends of η are tagged plain. Then η is an arc in A(S,M); as such, it isrepresented by τ(η) = γ ′ (cf. Figure 7). However, λ(η) is not the same as λ(γ ′).

Lemma 7.2 allows us to write the exchange relations associated with the taggedflips of types (4.D) and (4.Q) described in Remark 4.13.

Definition 7.4 (Ptolemy relations for tagged arcs). If two tagged triangulations T1

and T2 are related by a flip of type (4.Q), then the corresponding lambda lengths arerelated by an appropriate specialization of the equation (6.2). We will continue torefer to such relations among lambda lengths of tagged arcs as (generalized) Ptolemyrelations. Note that these relations can be more complicated than their counterpartsfor the ordinary arcs: some of the arcs α, β, γ, δ appearing in (6.2) may bound aonce-punctured monogon and so might not be present in T1 and T2. In such a case,Lemma 7.2 suggests that we should replace the lambda length of each such loop bythe product of lambda lengths of the two tagged arcs in T1 (equivalently, T2) that itencloses. See Figure 17.

We now take care of the tagged flips of type (4.D). Consider a punctured digonas in Lemma 6.6 and Figure 13. Making use of Lemma 7.2, we can rewrite therelation (6.3) in the form

(7.1) λ(γ ′)λ(θ) = λ(α) + λ(β),


./ ./

α′α′′

β

γ′γ′′

θη

λ(η)λ(θ) = λ(α′)λ(α′′)λ(γ′)λ(γ′′) + λ(β)2

Figure 17. Ptolemy relation for a tagged flip in a 4-punctured sphere.Cf. Figures 8 and 12.

where γ′ denotes γ with a notch at p, as in Figure 16.Our next goal is to show that lambda lengths of tagged arcs on a given bordered

surface with marked points naturally form a normalized exchange pattern whose ex-change relations are the generalized Ptolemy relations of Definition 7.4 together withequations (7.1). Making these statements precise will require a bit of preparation.

First, the coefficient semifield P is going to be the tropical semifield (see Def-inition 3.1) generated by the lambda lengths of the boundary segments (cf. Re-mark 4.16):

(7.2) P = Trop(λ(γ) : γ ∈ B(S,M)).

(If S is closed, then P = 1 is the trivial one-element semifield.) We note thatformula (7.2) makes sense in view of Theorem 6.4, which enables us to treat theselambda lengths as independent variables.

Second, let us describe the clusters. As in the case of ordinary arcs, each lambdalength of a tagged arc γ ∈ A./(S,M) can be viewed as a function σ 7→ λσ(γ) on the

decorated Teichmuller space T (S,M). For a tagged triangulation T of (S,M), let

(7.3) x(T ) = λ(γ) : γ ∈ Tdenote the collection of lambda lengths of the tagged arcs in T .

Third, the ambient field. Let us pick an ordinary triangulation T without self-folded triangles. It follows from Theorem 6.4 that the lambda lengths in x(T) arealgebraically independent over the field of fractions of P. Let F = F(T) be the fieldgenerated (say over R) by these lambda lengths. That is, F consists of all functions

defined on a dense subset of T (S,M) which can be written as a rational expression(say with real coefficients) in the lambda lengths of the arcs in T.

Theorem 7.5. There exists a unique normalized exchange pattern (ΣT ), here iden-tified with its positive realization (see Definition 3.4 and Proposition 3.5) with thefollowing properties:

• the coefficient semifield P is the tropical semifield generated by the lambdalengths of boundary segments, as in (7.2);• the ambient field F = F(T) is generated over P by the lambda lengths of a

given triangulation T with no self-folded triangles;


• the underlying n-regular graph E = E(S,M) is the exchange graph of taggedtriangulations (see Definition 4.14), and the seeds ΣT = (x(T ),p(T ), B(T ))are labeled by the vertices of E(S,M), as in Theorem 5.1;• each cluster x(T ) consists of the lambda lengths of the tagged arcs in T , as

in (7.3);• each exchange matrix B(T ) is the signed adjacency matrix of T ;• the exchange relations out of each seed ΣT are the Ptolemy relations (see

Definition 7.4) and the equations (7.1) associated with the two respective typesof tagged flips from T (cf. Remark 4.13).

Neither the ambient field F(T) nor the entire exchange pattern (ΣT ) depend on thechoice of the initial triangulation T.

Proof. We know that the signed adjacency matrices B(T ) associated with taggedtriangulations T satisfy the mutation rule (4.4), as required in the definition of anexchange pattern. We also know from Theorem 6.4 that the lambda lengths formingthe initial cluster x(T) are algebraically independent. It remains to verify that

(i) the relations (6.2) and (7.1) associated with arbitrary tagged flips can beviewed as exchange relations (1.5) for the signed adjacency matrices B(T ) oftagged triangulations T ,

(ii) the coefficients appearing in these relations satisfy the mutation rules (1.6)–(1.7), and

(iii) the normalization condition (2.2) holds in the tropical semifield P.

Straightforward albeit somewhat tedious details of these verifications are omitted. Ithelps to note that a statement essentially equivalent to claim (ii) has been alreadychecked in Remark 4.16. ¤

Remark 7.6. It is tempting to try to deduce Theorem 5.1 from Theorem 7.5 byexpressing cluster variables for any exchange pattern with exchange matrices B(T )as lambda lengths of tagged arcs, perhaps rescaled to get different coefficients. Itturns out however that this simplistic approach does not produce the most generalcoefficient patterns, as required for Theorem 5.1. Instead, we will need to develop, insubsequent sections, a much more complicated concept of generalized lambda lengthsfor laminated Teichmuller spaces associated with opened surfaces.

Remark 7.7. We note that Theorem 7.5 implies that lambda lengths of taggedarcs in any cluster (i.e., tagged triangulation) parametrize the decorated Teichmuller

space T (S,M), extending Theorem 6.4 verbatim to the case of tagged triangulations.

Example 7.8. Let (S,M) be a once-punctured digon, with notation as in Figure 13.The coefficient semifield is P = Trop(λ(α), λ(β)). The four cluster variables λ(γ),λ(θ), λ(γ ′), λ(θ′), are labeled by the tagged arcs in (S,M). The four clusters cor-respond to the four tagged triangulations, cf. Figure 9. The two exchange relationshave the form (7.1). The resulting exchange pattern has finite type A1 × A1, in thenomenclature of [12].

In the unpunctured case, theere is no tagging, and Theorem 7.5 specializes toits counterparts given by V. Fock and A. Goncharov [4, 5] and by M. Gekhtman,


M. Shapiro, and A. Vainshtein [18]. The case of an unpunctured disk discussed inExample 7.9 below was already treated in [12, Section 12.2].

Example 7.9. Let (S,M) be an unpunctured (n+3)-gon with vertices v1, . . . , vn+3,labeled counterclockwise. For 1≤ i < j ≤ n+3, let γij denote the arc or boundarysegment connecting vi and vj, that is, a diagonal or a side of the (n+3)-gon. Denoteλij = λ(γij). Applying the construction in Theorem 7.5 to this special case, we getthe coefficient semifield

P = Trop(λ12, λ23, . . . , λn+3,1)

generated by the lambda lengths of the sides of the (n+3)-gon; the cluster variablesare the lambda lengths of diagonals. The corresponding cluster algebra (of type An)can be interpreted (see [12, Proposition 12.7]) as a homogeneous coordinate ring ofthe Grassmannian Gr2,n+3 of 2-dimensional subspaces in Cn+3. See Example 15.1 fora more detailed treatment.

In the case of a once-punctured disk, we recover a particular cluster algebra oftype Dn that has been described (from a different perspective) in [12, Section 12.4].

Example 7.10. Let (S,M) be an n-gon (n ≥ 3) with a single puncture p inside it.Let the vertices of the n-gon be v1, . . . , vn, labeled counterclockwise. For 1 ≤ i <j ≤ n, there are two arcs or boundary segments connecting vi and vj, depending onwhich side of the curve the puncture p is on. Let γij (resp., γij = γji) denote thecurve that has p on the left as we move from vi to vj. There are also plain arcs γii

connecting vi to p, and tagged arcs γii that have a notched end at p. Replace γ’swith λ’s to denote the corresponding lambda lengths. Then λ12, . . . , λn,1 generate thetropical semifield of coefficients; the remaining λ’s are cluster variables. As always,clusters correspond to tagged triangulations. The resulting cluster algebra coincideswith the cluster algebra A described in [12, Example 12.15], and identified in [12,Proposition 12.16] with the coordinate ring of the affine cone over the Schubertdivisor in Gr2,n+2.

8. Opened surfaces

As shown in Section 7, the lambda lengths of tagged arcs form an exchange pat-tern. It is important to note that the coefficients in such an exchange pattern areof a very special kind: they are monomials in the lambda lengths of the boundarysegments. (For example, in the case of a closed surface with punctures, the coeffi-cients are trivial.) In order to construct exchange patterns with general coefficients(as in Theorem 5.1), we will need to modify our geometric setting, extending theTeichmuller space from surfaces with cusps at marked points to opened surfaces.

It is not unusual in Teichmuller theory to allow both cusped surfaces and surfaceswith geodesic boundary in the same moduli and Teichmuller spaces. One standardmodel is the space of all complex structures on the complement of the marked points.A complex structure on the neighborhood of a singularity can have two possiblebehaviours: it can have a removable singularity at the marked point, correspondingto a cusp in the hyperbolic structure after uniformization; or it can be equivalent


to the complex plane minus a closed disk, corresponding to a non-finite volume endafter uniformization. It will be more convenient for us to use a different model:instead of complex structures, we will work with hyperbolic metrics, truncated sothat they have geodesic boundary and finite volume.

On the combinatorial/topological level, our construction will be based on the fol-lowing concept.

Definition 8.1 (Opening of a surface). Let M = M \ ∂S denote the set of markedpoints in the interior of S. For a subset P ⊂M, the corresponding opened surface SP

is obtained from S by removing a small open disk around each point in P . For p ∈ P ,let Cp be the boundary component of SP created in this way. We then introduce anew marked point Mp on each Cp, and set

MP = (M \ P ) ∪ Mpp∈P ,

creating a new bordered surface with marked points (SP ,MP ), with no punctures.The sets of marked points MP and M can be identified with each other in a naturalway. See Figure 18.

p qa b

(S,M) Mp Mq

Cp Cq

a b

(SP ,MP )

Figure 18. Opening of a surface. Here (S,M) is a twice-punctureddigon, M = a, b, p, q, P = M = p, q, and MP = a, b,Mp,Mq.

There is a natural “projection” map

(8.1) κP : A(SP ,MP )→ A(S,M)

(surjective but not injective) that corresponds to collapsing the new boundary com-ponents Cp. We will refer to any γ ∈ A(SP ,MP ) that projects onto a given arcγ ∈ A(S,M) as a lift of γ. To describe these lifts, we introduce, for every p ∈ P ,the map

(8.2) ψp : A(SP ,MP )→ A(SP ,MP )

that takes each arc ending at Mp and twists it once clockwise around Cp. Then, forexample, an arc γ ∈ A(S,M) connecting two distinct points p, q ∈ P has the lifts

κ−1P (γ) = (ψp)

n(ψq)mγn,m∈Z,

where γ is some particular lift of γ. See Figure 19. If γ ∈ A(S,M) goes from apoint p ∈ P back to itself, then κ−1

P (γ) consists of two orbits under the action of ψp .Opening all the punctures in M results in the “largest” opened surface

(8.3) (S,M) = (SM,M

M).


p qγ

Mp

Mq

γ

ψp(γ)

ψq(γ)

Figure 19. An arc connecting two punctures, and three of its lifts

Its arc complex

(8.4) A(S,M)def= A(S

M,M

M)

naturally projects onto all the other arc complexes A(SP ,MP ); that is, the mapκ

Mfactors through every other map κP , for P ⊂M.

Definition 8.2 (Lifts of tagged arcs). To lift a tagged arc γ ∈ A./(S,M) to anopened surface (SP ,MP ) (in particular, to (S,M)), we simply lift the untaggedversion of γ to A(SP ,MP ) (resp., A(S,M)), and then affix the same tags as theones used at the corresponding ends of γ. See Figure 20. Thus, the lifted tagged arcγ may have a notched end at an unopened point p /∈ P , or at a marked point Mp

(p ∈ P ). We denote by A./(SP ,MP ) (resp., A./(S,M)) the set of all such taggedarcs γ on (S,M).

./

γ

q

p

./γ

q

Mp

Figure 20. Lift of a tagged arc

9. Lambda lengths on opened surfaces

We are now prepared to describe our main Teichmuller-theoretic construction.This will be done in two steps, Definitions 9.1 and 9.4.

Definition 9.1. For a subset P ⊂ M, the partial Teichmuller space TP (SP ,MP )is the space of all finite-volume, complete hyperbolic metrics on SP \ (M \ P )with geodesic boundary, modulo isotopy. The decorated partial Teichmuller space

TP (SP ,MP ) is the same set of metrics as in TP (SP ,MP ), modulo isotopy relativeto Cpp∈P , and with a choice of horocycle around each point in M \ P .

That is, a hyperbolic structure in TP (SP ,MP ) has a cusp at each point in M \ P(i.e., at each original marked point on ∂S and at each interior marked point notin P ), and a new circular geodesic boundary component Cp arising from each pointp ∈ P . The boundary is otherwise geodesic. Note in particular that there is not


a cusp at the points Mp, which are introduced merely to help parametrize the newboundary.

The decorated Teichmuller space TP (SP ,MP ) is a fibration over TP (SP ,MP ) withfibers RM. The decorations look different depending on the marked point: for pointsnot in P , the decoration is a choice of horocycle as in the previous sections, whileat the new geodesic boundary the decoration comes from restricting the isotopies tothose that leave the boundary components Cp (p ∈ P ) fixed. (We could equivalentlyrestrict to isotopies that leave the points Mp fixed.)

Suppose we have chosen P ⊂ M, an orientation on Cp for each p ∈ P , and adecorated geometric structure σ ∈ TP (SP ,MP ). To each arc γ ∈ A(S,M), we canassociate a unique infinite, non-selfintersecting geodesic γσ on SP (relative to σ): atendpoints of γ that are not opened, the geodesic γσ runs out to the cusp, while atendpoints that are in P , it spirals around Cp in the chosen direction. See Figure 21.

−→

Figure 21. Representing arcs by geodesics on an opened surface.Shown on the left is a portion of the original surface; on the right is aparticular opening of the surface, endowed with a hyperbolic structure.The lower right marked point, the only one not in P , has been left asan interior cusp; the remaining three have been opened into circulargeodesic boundary components. The lower left component is orientedcounterclockwise and the other two are oriented clockwise.

For an arc γ ∈ A(SP ,MP ) on an opened surface, we set γσ = γσ, where γ =κP (γ) is obtained from γ by the collapsing map κP of (8.1). Thus the geodesicrepresentative γσ does not depend on how γ winds around its opened ends.

To represent a tagged arc γ ∈ A./(S,M) (or any of its lifts to (SP ,MP ), seeDefinition 8.2) by a geodesic on the opened surface (SP ,MP ), we apply the sameconstruction as above, except that the spiralling at the notched ends Mp (p ∈ P )should go against the direction that has been chosen on Cp.


We now coordinatize the partial Teichmuller spaces TP (SP ,MP ) by introducingappropriate generalizations of Penner’s lambda lengths. This will require the follow-ing notion.

Definition 9.2. For each p ∈ P , there is a perpendicular horocyclic segment hp

near Cp: we take a (short) segment of the horocycle from Mp ∈ Cp which is perpen-dicular to Cp and to all geodesics γσ that spiral to Cp in the direction given by thechosen orientation. In Figure 21, the segments hp are drawn as dashed blue curves,as is the horocycle decorating the marked point not in P .

To deal with tagged arcs, we will also need the conjugate perpendicular horocyclicsegment hp which satisfies the same requirements as hp with respect to the geodesicsspiralling in the opposite direction.

Definition 9.3 (Lambda lengths on an opened surface). We next define lambdalengths λ(γ) for the arcs γ ∈ A(SP ,MP ), or more generally for the tagged arcsγ ∈ A./(SP ,MP ). As in Definition 6.3, we set

(9.1) λ(γ) = λσ(γ) = el(γ)/2,

where l(γ) = lσ(γ) is the distance between appropriate intersections of the geo-desic γσ with the horocycles at its two ends. At ends of γσ that spiral around oneof the openings Cp, there will be many intersections between γσ and the horocyclicsegment hp (or the conjugate horocyclic segment hp if γ is notched at Mp), and weneed to pick one of them. Assume that γ connects two ends Mp and Mq, with bothp and q in P (this is the most complicated case). Suppose furthermore that γ twistssufficiently far in the direction prescribed by the tagging at each end. Then thereare unique intersections between γσ and hp, hq (or hp or hq, respectively, as needed)such that the path that runs

• along hp (or hp) from Mp to one intersection, then• along γσ to the other intersection, then• along hq (or hq) to the other endpoint Mq

is homotopic to the original arc γ, as shown in Figure 22. If one or both of theends of γ are not in P , we leave γσ unmodified at that end and pick the uniqueintersection between γσ and the corresponding horocycle. In either case, l(γ) is the(signed) distance between the chosen intersections with the two horocycles.

−→

Figure 22. Finding the correct intersection with the perpendicularhorocycles.


In order to extend the definition to all tagged arcs γ ∈ A./(SP ,MP ), not justthose that twist sufficiently much, we postulate how l(γ) and λ(γ) change when wetwist γ around the boundary. Specifically, we mandate, in the case of plain tagging,that if γ has an endpoint Mp (p ∈ P ), then

(9.2) l(ψpγ) = l(p) + l(γ),

where ψp is the clockwise twist defined by (8.2), and we use the notation

(9.3) l(p) =

−length of Cp p ∈ P, if Cp is oriented counterclockwise;

0 if p 6∈ P ;

length of Cp p ∈ P, if Cp is oriented clockwise.

Accordingly (cf. (9.1)), in the case of plain tagging, we have

(9.4) λ(ψpγ) = λ(p)λ(γ),

where

(9.5) λ(p) = λσ(p) = el(p)/2.

In order for l(γ) and λ(γ) to be well defined, we need of course to check that therequirements (9.2)–(9.4) are consistent with the earlier definitions given in the casewhere γ twists sufficiently much. Indeed they are, since the distance along thegeodesic γσ between successive intersections with the horococyle hp is always equalto |l(p)|, as shown in Figure 23.

The cases where some of the ends are notched and/or both endpoints of γ areat the same puncture p are handled in a similar way. In particular, if γ is notchedat Mp then (9.4) becomes

(9.6) λ(ψpγ) = λ(p)−1 λ(γ),

where ψp is the clockwise twist around Cp as before.

j l(p ) j ~ ~C p

~h p

−→

Figure 23. Shown on the right is a geodesic γ spiralling to a bound-ary component Cp, and the corresponding horocycle hp. On the left,

their respective universal covers γ, Cp, and hp. The distance betweenhorocycles is the length of Cp.


Definition 9.4. The complete decorated Teichmuller space T (S,M) is a topologicalspace defined as follows. As a set, T (S,M) is the disjoint union over all subsets

P ⊂ M of 2|P | copies of TP (SP ,MP ), one for each choice of orientation on each

boundary circle Cp . (Thus T (S,M) consists of 3|M| strata TP (SP ,MP ).)It remains to describe the topology on T (S,M). For an arc γ ∈ A./(S,M) (see

Definition 8.2), define the lambda length

λ(γ) : T (S,M)→ R

on each stratum of T (S,M) by projecting γ to the appropriate set A./(SP ,MP ) andusing the construction above. The topology on T (S,M) (making it into a connectedspace) is the weakest in which all of these λ(γ) are continuous.

Proposition 9.5. Let T be an ideal triangulation of (S,M). For each γ ∈ T , fix anarc γ ∈ A(S,M) that projects to γ. Then the map

Φ =

(∏

p∈M

λ(p)

)×(

∏

β∈B(S,M)

λ(β)

)×(∏

γ∈T

λ(γ)

): T (S,M)→ R

n+|M|>0

is a homeomorphism.

Proof. For any vector Λ in Rn+|M|>0 , we can construct Φ−1(Λ), the unique geometric

structure in T (S,M) with the corresponding set of lambda lengths, as follows. Firstnote that if the arc γ has endpoints p and q in M, then for any alternate lift γ ′ of γthere are n,m ∈ Z so that

γ′ = ψnpψ

mq (γ).

Then by equation (9.4), for any hyperbolic structure with these lambda coordinates,we have

λ(γ′) = λ(p)nλ(q)mλ(γ).

This equation can then be used to compute lambda lengths of all lifts of the arcsin T .

Now for each ideal triangle in T with sides γ1, γ2, γ3, pick lifts γ ′1, γ′2, γ

′3 that

form a triangle in (S,M). (Note that if the triangle is self-folded, say with γ1 = γ2,then we will end up with two different lifts of this arc γ1.) Then take a decoratedideal hyperbolic triangle (i.e., a triangle with choice of horocycles around each cusp)so that the lambda length of the sides of the triangle match with the λ(γ ′

i), as inPenner’s proof of Theorem 6.4.

We next need to glue these triangles together to form a hyperbolic surface. Foreach arc γ ∈ T , we have two different lifts γ ′, γ′′ of γ coming from the two differenttriangles that have this arc as a side (or the two different sides of the same triangle,in case γ is the repeated edge of a self-folded triangle). Suppose that γ ′′ = ψn

pψmq (γ′),

where p and q are the endpoints of γ as before. Then glue the two hyperbolic trianglesso that the the horocycles around the vertex corresponding to p are offset by n · l(p)and the horocycles around the vertex q are offset by n · l(q). It is then elementaryto verify that the resulting glued surface has the desired lambda lengths.


By definition of the topology on T (S,M), the map Φ is continuous. It remains toshow that Φ−1, as constructed above, is continuous. To do this, we must show thatfor an arbitrary arc α ∈ A(S,M), the lambda length λ(α) is a continuous functionof the given coordinates. Let α = κ(α) be the arc in A(S,M) corresponding to α.We can move from T to a triangulation that contains α by a series of diagonal flipsin quadrilaterals. For each diagonal flip, an exchange move lets us write the lambdalength of one lift of the new diagonal in terms of lambda lengths of lifts of the oldlambda lengths. (By the remarks above, if we can write one lift of a given arc interms of the given lambda coordinates, we can write all lifts in terms of these lambdacoordinates by multiplying by appropriate powers of the λ(p) for p ∈ M.) We endup inductively writing λ(α) as an algebraic function with non-zero denominator interms of the original coordinates. ¤

Proposition 9.5 can be extended without any changes to the case of tagged trian-gulations, as in Remark 7.7. In order to do that—and, more importantly, in order tocomplete our construction of exchange patterns associated with opened surfaces—wewill need exchange relations involving

(i) the lambda lengths λ(γ), for γ ∈ A./(S,M),(ii) the lambda lengths λ(β), for β ∈ B(S,M), and(iii) the lambda lengths λ(p), for p ∈M.

Some of these relations are easy to obtain. We see right away that the lambdalengths of types (i) and (ii) still obey the Ptolemy relation (Proposition 6.5) foreach quadrilateral on the opened surface whose sides do not involve the circularboundaries Cp—provided the tags of the three arcs meeting at each vertex of thequadrilateral agree with each other. Indeed, the geometry is identical to what wehad before, once a lift to the universal cover is made.

To relate different lifts of the same arc to each other, we will use the monomialrelation (9.4) which involves the lambda lengths of types (i) and (iii).

We will also need a relation associated with a tagged flip inside an opened mono-gon, an analogue of Lemma 6.9.

Lemma 9.6. Inside an opened surface, consider a monogon with a marked vertex qand a single boundary component Cp in the interior, oriented counterclockwise. Letδ and % be two tagged arcs connecting q and Mp as shown in Figure 24 on the left,and let η be the outer boundary of the monogon. Then

(9.7) λ(δ)λ(%) =λ(η)

1− λ+(p)−2,

where λ+(p) denotes the lambda length of the hole at p (see (9.3) and (9.5)) withrespect to the clockwise orientation of Cp, so that

(9.8) λ+(p) = exp((length of Cp)/2).

Proof. Strange as it may seem at first glance, the equation (9.7) is yet anotherinstance of the same Ptolemy relation. To see that, consider Figure 25, in whichthe arc δ′ has been obtained from δ by a counterclockwise twist around Cp, and


6Mp q

./

δ

%

η qC p

%

h

h

Figure 24. An opened monogon

the tagged arc %′ (notched at Mp) has been obtained from % by a clockwise twistaround Cp. We have

λ(δ′) = λ(δ)λ(p)−1,(9.9)

λ(%′) = λ(%)λ(p)−1,(9.10)

λ+(p) = λ(p)−1(9.11)

(since Cp is oriented counterclockwise). Lifting this configuration to the universalcover, we identify a quadrilateral whose sides, in cyclic order, have lambda lengths(i.e., exponentiated hyperbolic lengths) λ(δ), λ(η), λ(%), and λ+(p)2 (the latter corre-sponds to travelling twice along Cp). The diagonals of this quadrilateral have lambdalengths λ(δ′) and λ(%′). Combining (9.9)–(9.11) with the Ptolemy relation

λ(δ′)λ(%′) = λ(δ)λ(%) + λ(η)λ+(p)2,

we obtain (9.7). ¤

6Mp qδ

%δ′

%′

η

./

Figure 25. Proof of Lemma 9.6


10. Non-normalized exchange patterns from surfaces

In this section, we describe a construction of a non-normalized exchange pattern onthe exchange graph E(S,M) of tagged triangulations of the original surface (S,M).This construction is different from the one given in Section 7: although it is morecomplicated, it is eventually going to provide—after proper rescaling—a more generalclass of coefficients. Here is the basic idea: rather than designating the lambda lengthof a tagged arc γ as the corresponding cluster variable, we take the lambda lengthof an arbitrary lift of γ to the opened surface (S,M) (see (8.3) and Figure 19),rescaled in a particular way. It turns out that we do not have to coordinate theselifts: the corresponding lambda lengths will always form an exchange pattern. Incontrast to the simpler construction in Section 7, this new exchange pattern will notbe normalized.

We begin by setting up the coefficient group P = P(S,M) as the (free) abelianmultiplicative group generated by the set

(10.1) λ(p) : p ∈M ∪ λ(β) : β ∈ B(S,M)of lambda lengths of boundary components β and circular components Cp. By Propo-sition 9.5, we can view (and treat) these lambda lengths either as functions on thecomplete decorated Teichmuller space T (S,M) or as formal variables (=coordinatefunctions).

We next introduce what will soon become the cluster variables. Their definitioninvolves the rescaling factors ν(a), defined for the marked points a ∈ M

Mon the

opened surface (S,M) by

(10.2) ν(a) =

√1− λ+(p)−2 if a = Mp, with p ∈M;

1 if a ∈M ∩ ∂S.

For a tagged arc γ ∈ A./(S,M) with endpoints a and b, we then set

(10.3) x(γ) = λ(γ)ν(a)ν(b).

Thus x(γ) is a function on the Teichmuller space T (S,M). For example,

x(γ) = λ(γ)√

1− λ+(p)−2√

1− λ+(q)−2

if γ connects Mp and Mq, with p, q ∈M, and x(γ) = λ(γ) if γ connects two markedpoints on ∂S.

Rewriting (6.2) in terms of the rescaled lambda lengths x(γ), we conclude thatthe latter satisfy the same Ptolemy relations

(10.4) x(η)x(θ) = x(α)x(γ) + x(β)x(δ),

for each quadrilateral with sides α, β, γ, δ and diagonals η and θ. Here we assume thatthe arcs contained in α, β, γ, δ, η, θ (remember that some of these can be boundarysegments) are tagged so that any two of them are compatible, with the exception ofthe pair (η, θ).

We will also need the relations associated with the tagged flips of type (4.D).


Lemma 10.1. Consider an opened digon with vertices r and q and a counterclockwiseoriented opening Cp with a marked point Mp. Let α, β, %, and θ be the tagged arcsshown in Figure 26. (The tags at r and q have been suppressed in the picture.) Weassume that the arcs in α, β, %, θ are tagged so that any two of them are compatible,with the exception of the pair (%, θ). Then

(10.5) x(%)x(θ) = λ(p)x(α) + x(β).

Alternatively, if Cp is oriented clockwise, θ is notched at Mp, and % is tagged plainat Mp, then

(10.6) x(%)x(θ) = λ(p)−1 x(α) + x(β).

Mpr q./

α

β

θ

%

6

Figure 26. An exchange relation in an opened digon.

Proof. Let us introduce the arcs γ, δ, and η as in Figure 27. The lambda lengths ofthe six arcs in Figure 27 satisfy the Ptolemy relation (6.2), and therefore the identity(10.4) holds. (Even though η /∈ A./(S,M), the notation x(η) has obvious meaning.)We also have

(10.7) x(γ) = x(δ)λ(p)

(by (9.4)) and

(10.8) x(η) = x(δ)x(%)

(by Lemma 9.6). Putting everything together, we obtain (10.5). The second partof the lemma (see equation (10.6)) is completely analogous—and is in fact nothingmore than a restatement of the first part. ¤

For each tagged arc γ ∈ A./(S,M), let us fix an arbitrary lift γ ∈ A./(S,M)(see Definition 8.2), and set x(γ) = x(γ) (cf. (10.2)–(10.3)). Then, for each taggedtriangulation T ∈ E(S,M), define

(10.9) x(T ) = x(γ) : γ ∈ T.In view of Proposition 9.5, the rescaled lambda lengths in x(T ) can be treated asformal variables algebraically independent over the field of fractions of P(S,M).


Mpr q

α

β

θ

ηγ

δ

6

Figure 27. Proof of Lemma 10.1.

We are now ready to state our next theorem: the rescaled lambda lengths of liftsof tagged arcs form a non-normalized exchange pattern.

Theorem 10.2. For an arbitrary choice of lifts γ of the tagged arcs γ ∈ A./(S,M),there exists a (unique) non-normalized exchange pattern on E(S,M) with the follow-ing properties:

• the coefficient group is P = P(S,M) (see (10.1));• the cluster variables are the rescaled lambda lengths x(γ) defined by (10.3);• the cluster x(T ) at a vertex T ∈ E(S,M) is given by (10.9);• the ambient field is generated over P by some (equivalently, any) cluster x(T );• the exchange matrices are the signed adjacency matrices B(T );• the exchange relations out of each seed are the relations (10.4) and (10.5)–

(10.6) associated with the corresponding tagged flips, properly rescaled us-ing (9.4) to reflect the choices of lifts.

To be more accurate, the description of cluster variables above should refer toa “positive realization” of the exchange pattern in question, in the spirit of Defi-nition 3.4. Even though this pattern is not of geometric type, the correspondingnotions still have clear meaning, and the analogue of Proposition 3.5 holds.

Proof. The proof is similar to the proof of Theorem 7.5. As before, the real issue iscoefficients: we need to demonstrate that they satisfy the requisite mutation rules(1.2)–(1.2). It is straightforward to check that these rules hold for each triple offlips/mutations

T1µx←→ T2

µz←→ T3µx←→ T4 ,

if the lifts of the arcs involved are chosen in a coordinated way (this is essentiallythe same verification as before)—and therefore this rule would hold for any lifts, byProposition 2.1. ¤


11. Laminations and shear coordinates

In this section, we briefly review a small fragment—as this is all we need—ofW. Thurston’s theory of measured laminations [25, 28], and its relationship withmatrix mutations. Our exposition is an abridged adaptation of the one given byV. Fock and A. Goncharov [6, Section 3]. An interested reader is refered to the citedsources for further details.

In the next section, we will extend these constructions to the tagged setting.

Definition 11.1. An integral unbounded measured lamination—in this paper, fre-quently just a lamination—on a marked surface (S,M) is a finite collection of non-selfintersecting and pairwise non-intersecting curves in S, modulo isotopy relativeto M, subject to the restrictions specified below. Each curve must be one of thefollowing:

• a closed curve;• a curve connecting two unmarked points on the boundary of S;• a curve starting at an unmarked point on the boundary and, at its other end,

spiraling into a puncture (either clockwise or counterclockwise);• a curve whose both ends spiral into punctures (not necessarily distinct).

Also, the following types of curves are not allowed:

• a curve that bounds an unpunctured or once-punctured disk;• a curve with two endpoints on the boundary of S which is isotopic to a piece

of boundary containing no marked points, or a single marked point.

See Figures 28 and 29.

Figure 28. A lamination in a twice-punctured annulus.

Laminations on a marked surface (S,M) can be coordinatized using W.Thurston’snotion of shear coordinates.

Definition 11.2 (Shear coordinates). Let L be an integral unbounded measuredlamination. Let T be a triangulation without self-folded triangles. For each arc γ


Figure 29. Curves that are not allowed in a lamination.

in T , the corresponding shear coordinate of L with respect to the triangulation T ,denoted by bγ(T, L), is defined as a sum of contributions from all intersections ofcurves in L with the arc γ. Specifically, such an intersection contributes +1 (resp,−1)to bγ(T, L) if the corresponding segment of a curve in L cuts through the quadrilateralsurrounding γ as shown in Figure 30 on the left (resp., on the right). Note that eventhough a spiralling curve can intersect an arc infinitely many times, the number ofintersections that contribute to the computation of bγ(T, L) is always finite.

An example is shown in Figure 31.

γ+1 −1

γ

Figure 30. Defining the shear coordinate bγ(T, L). The curve on theleft contributes +1, the one on the right contributes −1.

0−2

−2 2

−31

Figure 31. Shear coordinates of a lamination L with respect to anordinary triangulation T .


Theorem 11.3 (W. Thurston). For a fixed triangulation T without self-folded tri-angles, the map

L 7→ (bγ(T, L))γ∈T

is a bijection between integral unbounded measured laminations and Zn.

Example 11.4. Figure 32 shows six “elementary” laminations L1, . . . , L6 of a once-punctured digon (each lamination Li consists of a single curve), and their shearcoordinates with respect to a particular triangulation T . It is easy to see that anyintegral lamination of the digon consists of several (possibly none) curves homotopicto some Li, together with several (possibly none) curves homotopic to L(i+1) mod 6.It is also easy to see that, in agreement with Theorem 11.3, each vector in Z2 canbe uniquely written as a nonnegative integer linear combination

pyi + qy(i+1) mod 6 (p, q ∈ Z≥0),

where the six vectors

y1 = [−1, 0], y2 = [−1, 1], y3 = [0, 1], y4 = [1, 0], y5 = [1,−1], y6 = [0,−1]

represent the shear coordinates of L1, . . . , L6, respectively.

L1

0−1

L2

1−1

L3

10

L6

−10

L5

−11

L4

01

-

6I

¾

? R

y1

y2

y3

y4

y5y6

Figure 32. Six “elementary” laminations of a once-punctured digon,and the corresponding vectors of shear coordinates.

Definition 11.5 (Multi-laminations and associated extended exchange matrices).A multi-lamination is simply a finite family of laminations. Let us fix a multi-lamination

L = (Ln+1, . . . , Lm)

of size m−n; this choice of indexing will be convenient in the sequel. For a triangu-lation T of (S,M) without self-folded triangles, define an m× n matrix

B = B(T,L) = (bij)


(cf. Definition 3.2) as follows. The top n × n part of B is the signed adjacencymatrix B(T ) = (bij)1≤i,j≤n, whereas the bottom m− n rows are formed by the shearcoordinates of the laminations Li with respect to the triangulation T :

(11.1) bij = bj(T, Li) if n < i ≤ m.

B =

0 −1 0 0 1 −1

1 0 −1 0 0 0

0 1 0 −1 0 0

0 0 1 0 −1 0

−1 0 0 1 0 1

1 0 0 0 −1 0

2 0 −2 −2 1 −3

Figure 33. The matrix B = B(T,L) for the example in Figure 31.The multi-lamination L = (L) consists of a single lamination L. Weuse the labeling of arcs in T shown in Figure 10.

The key observation is that, under ordinary flips, the matrices B(T ) transformaccording to the mutation rules.

Theorem 11.6 (W. Thurston–V. Fock–A. Goncharov). Fix a multi-lamination L.If triangulations T and T ′ without self-folded triangles are related by a flip of an arc k,then the corresponding matrices B(T,L) and B(T ′,L) are related by a mutation indirection k.

Although the reader will not find the above version of Theorem 11.6 in the citedsources, it can be seen to be a restatement of results contained therein. More specif-ically, applying the definition of a matrix mutation to the case under considerationresults in identities for the shear coordinates (with respect to T and T ′) that appear,for example, at the end of Section 3.1 in [6].

12. Shear coordinates with respect to tagged triangulations

In this section, we define shear coordinates for triangulations with self-folded trian-gles, and more generally, for tagged triangulations. We then obtain the appropriateanalogues of Theorems 11.3 and 11.6.

Definition 12.1 (Shear coordinates with respect to a tagged triangulation). We ex-tend Definition 11.2 by defining the shear coordinates bγ(T, L) of an integral un-bounded measured lamination L with respect to an arbitrary tagged triangulation T .(Here γ is a tagged arc in T .) These coordinates are uniquely defined by the followingproperties:


(i) Suppose that tagged triangulations T1 and T2 coincide with each other exceptthat at a particular puncture p, the tags of the arcs in T1 are different fromthe tags of the arcs in T2. Suppose that laminations L1 and L2 coincide witheach other except that each curve in L1 that spirals into p has been replacedin L2 by a curve that spirals in the opposite direction. Then bγ1

(T1, L1) =bγ2

(T2, L2) for each tagged arc γ1 ∈ T1 and its counterpart γ2 ∈ T2.(ii) By performing tag-changing transformations described above, we can convert

any tagged triangulation into a tagged triangulation T that does not containany notches except possibly inside once-punctured digons. Let T denote theideal triangulation that is represented by T ; that is, T = τ(T ) in the notationof Definitions 4.8 and 4.10. Let γ be an arc in T that is not contained insidea self-folded triangle, and let γ = τ(γ). Then, for a lamination L, we definebγ(T, L) by applying the rule in Definition 11.2 to the ordinary arc γ viewedinside the triangulation T .

Note that if γ is contained inside a self-folded triangle in T enveloping a puncture p,then we can first apply the tag-changing transformation (i) at p, and then use therule (ii) to determine the shear coordinate in question.

Example 12.2. Let T be the tagged triangulation of a punctured digon shown inFigure 34 (cf. also Figure 32), and let T1, T2, and T12 be the tagged triangulationsobtained from T as follows:

• T1 is obtained from T by a tagged flip replacing the arc γap by a taggedarc γbp;• T2 is obtained from T by a tagged flip replacing the arc γbp by a tagged

arc γap;• T12 is obtained from T by performing both of these (commuting) tagged flips.

a p bγap γbp

Figure 34. A triangulation T of a once-punctured digon.

Let L be a lamination in this once-punctured digon, with shear coordinates b1(T, L)and b2(T, L) corresponding to the arcs γap and γbp of T , respectively. We similarlydefine b1(Ts, L) and b2(Ts, L) for each subscript s ∈ 1, 2, 12. Note that for exampleb1(T12, L) and b2(T12, L) refer to the shear coordinates associated with γbp and γap,respectively, since these tagged arcs replace γap and γbp, in this order. Then the rules(i)–(ii) of Definition 12.1 yield:

bj(Ts, L) =

−bj(T, L) if j appears in s;

bj(T, L) if j does not appear in s.


For example, the shear coordinates of the laminations L1 and L3 (in the notation ofFigure 32) with respect to the tagged triangulations T, T1, T2, T12 are:

b1(T, L1) = −1 b2(T, L1) = 0 b1(T, L3) = 0 b2(T, L3) = 1

b1(T1, L1) = 1 b2(T1, L1) = 0 b1(T1, L3) = 0 b2(T1, L3) = 1

b1(T2, L1) = −1 b2(T2, L1) = 0 b1(T2, L3) = 0 b2(T2, L3) = −1

b1(T12, L1) = 1 b2(T12, L1) = 0 b1(T12, L3) = 0 b2(T12, L3) = −1

We now extend Definition 11.5 to the tagged case.

Definition 12.3. An extended exchange matrix B(T,L) of a multi-lamination L

with respect to a tagged triangulation T is defined in exactly the same way as inDefinition 11.5, this time using the shear coordinates from Definition 12.1.

We are now prepared to provide the promised generalizations of Theorems 11.3and 11.6.

Theorem 12.4. Fix a multi-lamination L. If tagged triangulations T and T ′ arerelated by a flip of a tagged arc k, then the corresponding matrices B(T,L) andB(T ′,L) are related by a mutation in direction k.

Proof. The proof is a straightforward verification based on Theorem 11.6, Defini-tion 12.1, and Remark 4.13. ¤

Theorem 12.5. For a fixed tagged triangulation T , the map

L 7→ (bγ(T, L))γ∈T

is a bijection between integral unbounded measured laminations and Zn.

Proof. This statement follows from Theorem 11.3, Theorem 12.4, and the invertibilityof matrix mutations. Proceed by induction on the number of flips required to obtainT from a triangulation without self-folded triangles. ¤

13. Tropical lambda lengths

A naıve definition of a tropical lambda length of an arc γ with respect to a(multi-)lamination L is based on the notion of a transverse measure of γ with respectto L, i.e., the corresponding intersection number. The latter notion is however illdefined as a spiralling curve in a lamination intersects an arc infinitely many times.(This issue does not come up in the absence of punctures.) We bypass this problemby passing to the opened surface where γ is replaced by a family of lifts γ. This setsthe stage for Section 14, where the tropical lambda lengths of those lifts are used asrescaling factors which allow us to construct the requisite normalized patterns witharbitrary coefficients of geometric type.

Remark 13.1. One can alternatively define tropical lambda lengths via a limit-ing procedure that degenerates a hyperbolic structure on (S,M) into a discrete, ortropical, version thereof, in the spirit of W. Thurston’s approach to compactifyingTeichmuller spaces. Further hints are provided in Section 17. In this paper, we donot systematically pursue this course of action.


Definition 13.2 (Lifts of a lamination). In order to lift a lamination L to a (non-unique) lamination L on an opened surface (S,M), or more generally on (SP ,MP )(see (8.3) and Definition 8.1, respectively), we perform the following procedure, foreach p ∈ P . Recall that in SP , the puncture p is opened into a circular boundarycomponent Cp with a marked point Mp (cf. Figure 18). Let us replace each curvein the lamination L that spirals into p by a new curve in L that runs into a pointon Cp, making sure that different curves do not intersect, and that their endpointslying on Cp are pairwise distinct and different from Mp. The (ends of) curves in Lthat do not spiral into punctures are lifted to the opened surface SP in the obvious“tautological” way.

Mp

-

Mp

-

Figure 35. Different lifts of lamination L1 from Figure 32.

This notion straightforwardly generalizes to multi-laminations: a lifted multi-lami-nation L consists of (uncoordinated) lifts of the individual laminations that makeup L.

Definition 13.3 (Transverse measures). Let L be a lift of a lamination L to theopened surface (S,M). Let γ be a tagged arc in A(S,M), or a boundary segmentin B(S,M). The transverse measure of γ with respect to L is an integer denoted bylL(γ) and defined as follows. If γ does not have ends at boundary components Cp (p ∈M), then lL(γ) is simply the (nonnegative) number of intersection points betweenγ and the curves in L. (More precisely, we should take curves in the correspondinghomotopy classes that minimize this number, for example by introducing a hyperbolicstructure and taking the geodesic representatives.) If γ has one or two such ends, andtwists sufficiently far around the corresponding opening(s) in the direction consistentwith their orientation (respectively, in the opposite direction if the end is notched),then again, lL(γ) is equal to the number of intersections between γ and L. (We notethat the notion of “sufficiently far” will depend on the choice of lift L.)

We then extend the definition to all arcs using the approach utilized earlier todefine l(γ) in Definition 9.3. That is, we require that adding an extra twist to γalways changes the value of lL(γ) by a fixed amount. Making this precise will requirea little preparation.

Let us denote by lL(Cp) the transverse measure of Cp with respect to L, i.e., thenumber of ends of curves in L that lie on Cp. By analogy with (9.2) and (9.3), we


define the function lL : A(S,M)→ Z so that it satifies, in the case of plain taggingat Mp, the equation

(13.1) lL(ψpγ) = lL(p) + lL(γ),

where ψp is the clockwise twist defined by (8.2), and

(13.2) lL(p) =

−lL(Cp) p ∈ P, if Cp is oriented counterclockwise;

0 if p 6∈ P ;

lL(Cp) p ∈ P, if Cp is oriented clockwise.

If γ is notched at Mp, then (13.1) is replaced by

(13.3) lL(ψpγ) = −lL(p) + lL(γ).

Note that with this definition, the numbers lL(γ) are no longer nonnegative. Alsonote that lL(p) does not depend on the choice of lift (so we could use the notationlL(p) instead). See Figure 36.

0

1Mp

- −1

0Mp

-

Figure 36. Transverse measures of arcs with respect to a liftedlamination. Here lL(Cp) = 1, so lL(p) = −1, and (13.1) becomeslL(ψpγ) = lL(γ)− 1.

Definition 13.4 (Tropical semifield associated with a multi-lamination). Let L =(Li)i∈I be a multi-lamination in (S,M); here I is a finite indexing set. We introducea formal variable qi for each lamination Li, and set (see Definition 3.1)

(13.4) PL = Trop(qi : i ∈ I).Definition 13.5 (Tropical lambda lengths). We continue in close analogy with Defi-nition 9.3. Let L = (Li)i∈I be a lift of a multi-lamination L. Instead of exponentiat-ing the distances to get the lambda lengths (cf. (9.1)), we define the tropical lambdalength of γ ∈ A(S,M) ∪B(S,M) with respect to L by

(13.5) cL(γ) =

∏

i∈I

q−l

Li(γ)/2

i ∈ PL.

In the case of plain tagging, the tropical lambda lengths satisfy

(13.6) cL(ψpγ) = c

L(p) c

L(γ),


where

(13.7) cL(p) = cL(p) =

∏

i∈I

q−l

Li(p)/2

i ∈ PL.

(cf. (9.4) and (9.5)). On the other hand, if γ is notched at Mp then

(13.8) cL(ψpγ) = c

L(p)−1 c

L(γ)

(cf. (9.6)).

Remark 13.6. Our definition of a tropical lambda length of a (lifted) tagged arcdepends on the choice of a lift L of the multi-lamination L. Different choices willresult in different notions of a tropical lambda length. However, they will all differfrom each other by gauge transformations which simultaneously rescale the lambdalengths of all arcs incident to a given puncture, and do not affect the underlyingcluster algebra structure.

On the other hand, the tropical (or ordinary) lambda lengths of boundary seg-ments, holes, or arcs which are not incident to punctures do not depend on the choiceof a lift L. Consequently, we can use notation cL(β) = c

L(β) for β ∈ B(S,M), or

cL(p) = cL(p) for p ∈M.

Remark 13.7. We can also define, in a similar way, the quantities lL(γ) and λ

L(γ)

when γ is a loop based at a marked point and enclosing a sole puncture/opening.

It is perhaps worth emphasizing that all our lambda lengths, whether tropical orordinary, do not depend on a (tagged) triangulation containing the (tagged) arc inquestion.

The main property of the tropical lambda lengths is that they satisfy the tropicalversion of the exchange relations (10.4) and (10.5)–(10.6).

Lemma 13.8. On the opened surface (S,M), consider a quadrilateral with sidesα, β, γ, δ and diagonals η and θ, cf. Figure 11. Assume that the tagging of the arcsin α, β, γ, δ, η, θ is consistent at each marked point. Then

(13.9) cL(η)c

L(θ) = c

L(α)c

L(γ)⊕ c

L(β)c

L(δ),

where ⊕ denotes the tropical addition in PL.

Proof. It is immediate from the definition of the tropical semifield (see Definition 3.1)that it suffices to prove (13.9) in the case when L consists of a single lamination L.In that case, (13.9) becomes

(13.10) lL(θ) + lL(η) = max(lL(α) + lL(γ), lL(β) + lL(δ)),

a well known (and easy to check) relation for intersection numbers (see, e.g., [6,Section 3]). ¤

Lemma 13.9. Consider an opened digon shown in Figure 26. Under the assumptionsof (the first part of) Lemma 10.1, we have

(13.11) cL(%) c

L(θ) = c

L(p) c

L(α)⊕ c

L(β).


Alternatively, if the opening Cp is oriented clockwise, θ is notched at Mp, and % istagged plain at Mp, then

(13.12) cL(%) c

L(θ) = c

L(p)−1 c

L(α)⊕ c

L(β).

Proof. These statements can be either verified directly, or obtained by tropical de-generation from their ordinary counterparts mentioned above. We note in particularthat in the language of transverse measures (cf. (13.10)), the relation (13.11) corre-sponds to the identity

¤(13.13) lL(%) + lL(θ) = max(lL(α) + lL(γ), lL(β) + lL(δ)).

14. Laminated Teichmuller spaces

In this section, we use the notions developed in previous sections—specifically, or-dinary and tropical lambda lengths of tagged arcs on an opened surface—to presentour main construction, a geometric realization of cluster algebras associated withsurfaces. The main idea is rather natural. As the lifts γ of an arc γ vary, the cor-responding tropical lambda lengths, just like the ordinary ones, form a geometricprogression (cf. (9.4) and (13.6)). Making sure that the ratios of the two progres-sions (ordinary and tropical) coincide, we proceed by dividing an ordinary (rescaled)lambda length of γ by a tropical one, thus obtaining an invariant of λ that plays therole of a cluster variable.

Definition 14.1 (Laminated Teichmuller space). Let L = (Li)i∈I be a multi-lamina-tion in (S,M). The laminated Teichmuller space T (S,M,L) is defined as follows. Apoint (σ, q) ∈ T (S,M,L) is a decorated hyperbolic structure σ ∈ T (S,M) togetherwith a collection of positive real weights q = (qi)i∈I which are chosen so that thefollowing boundary conditions hold:

• for each boundary segment β ∈ B(S,M), we have λσ(β) = cL(β);• for each hole Cp, with p ∈M), we have λσ(p) = cL(p).

In these equations, the quantities cL(β) and cL(p) (cf. Remark 13.6) are given by theformulas (13.5) and (13.7), with each qi specialized to a given positive real value. Inmore concrete terms, we require that

(14.1) lσ(β) = −∑

i∈I

lLi(β) ln(qi),

and similarly with β replaced by p. Informally, our boundary conditions (14.1) askthat for each boundary segment or hole, the total weighted sum of its transversemeasures with respect to the laminations in L is equal to the ordinary hyperboliclength, with respect to σ, of the segment between the appropriate horocycles.

We next coordinatize the laminated Teichmuller space T (S,M,L). A system ofcoordinates will include the weights of laminations plus the lambda lengths of thearcs in a (tagged) triangulation.


Proposition 14.2. Let L = (Li)i∈I be a multi-lamination in (S,M). The lami-nated Teichmuller space T (S,M,L) can be coordinatized as follows. Fix an ideal (ortagged) triangulation T of (S,M). Choose a lift L = (Li) of L, and choose a lift ofeach of the n arcs γ ∈ T to an arc γ ∈ A(S,M). Then the map

Ψ : T (S,M,L)→ Rn+|I|>0

defined byΨ(σ, q) = (λσ(γ))γ∈T × q

is a homeomorphism. The same is true with the lambda lengths λ(γ) replaced bytheir rescaled versions x(γ) defined by (10.2)–(10.3).

Proof. This is a direct consequence of Proposition 9.5 and Definition 14.1. ¤

Proposition 14.2 enables us to view the coordinate functions qi and λ(γ) as “vari-ables,” similarly to our treatment of the earlier versions of the Teichmuller space.

We next define the quantities that will play the role of cluster variables in ourmain construction.

Definition 14.3 (Laminated lambda lengths). Fix a lift L of a multi-lamination L.For a tagged arc γ ∈ A./(S,M), the laminated lambda length x

L(γ) is a function on

the laminated Teichmuller space T (S,M,L) defined by

(14.2) xL(γ) = x(γ)/c

L(γ),

where γ is an arbitrary lift of γ, the rescaled lambda length x(γ) is defined by (10.2)–(10.3), and c

L(γ) is the tropical lambda length of Definition 13.5. The value of x

L(γ)

does not depend on the choice of the lift γ since x(γ) and cL(γ) rescale by the same

factor cL(p) = λσ(p) (see Definition 14.1) as γ twists around the opening Cp (cf. (9.4)and (13.6)). On the other hand, x

L(γ) does depend on the choice of the lifted multi-

lamination L. This will not create problems as the resulting cluster structure willbe unique up to gauge transformations (see Remark 13.6), hence up to a canonicalisomorphism.

Remark 14.4. The laminated lambda lengths xL(γ) can be intuitively interpreted

as follows. Suppose that the lift γ is such that it twists sufficiently far around eachof its ends lying on opened components. (Since the latter notion depends on a liftedlamination Li, it may be impossible to satisfy this condition for all i simultaneously;still, let us assume that we can.) Then we combine the definition (14.2) with (9.1),(10.3), and (13.5) to get

(14.3) xL(γ) = ν(a)ν(b) el(γ)/2

∏

i∈I

qlLi

(γ)/2

i

(here a and b are the endpoints of γ, and ν is the function defined by (10.2)). Thusx

L(γ) is obtained by exponentiating a sum of three kinds of contributions:

• the hyperbolic length of the lifted arc γ between appropriate horocycles (“thecost of fuel while traveling along γ”);• a fixed contribution, depending on a lifted lamination Li, associated with

each crossing of Li by γ (“the tolls”);


• a contribution associated with each endpoint of γ.

Corollary 14.5. For a tagged triangulation T , the map

T (S,M,L) → Rn+|I|>0

(σ, q) 7→ (xL(γ))γ∈T × q

is a homeomorphism. (As before, L is a lift of a multi-lamination L.)

Proof. This is a corollary of Proposition 14.2 and Definition 14.3. ¤

We are finally prepared to present our main construction.

Theorem 14.6. For a given multi-lamination L = (Li)i∈I , there exists a uniquenormalized exchange pattern (ΣT ) of geometric type with the following properties:

• the coefficient semifield is the tropical semifield PL = Trop(qi : i ∈ I);• the cluster variables xL(γ) are labelled by the tagged arcs γ ∈ A./(S,M);• the seeds ΣT = (x(T ), B(T,L)) are labeled by the tagged triangulations T ;• the exchange graph is the graph E(S,M) of tagged flips (see Definition 4.14);• each cluster x(T ) consists of the cluster variables xL(γ), for γ ∈ T ;• the ambient field F is generated over PL by any given cluster x(T );• the extended exchange matrix B(T,L) is described in Definition 12.3;

This exchange pattern has a positive realization (see Definition 3.4) by functions onthe laminated Teichmuller space T (S,M,L). To obtain this realization, choose alift L of the multi-lamination L; then represent each cluster variable xL(γ) by thecorresponding laminated lambda length x

L(γ), and each coefficient variable qi by the

corresponding function on T (S,M,L).

Proof. We already know (see Theorem 12.4) that the matrices B(T,L) are related toeach other by mutations associated with the tagged flips. In view of Propositions 3.5and 14.2, all we need to prove is that the laminated lambda lengths x

L(γ) satisfy

the exchange relations encoded by the matrices B(T,L).Fix arbitrary lifts γ of all tagged arcs γ ∈ A./(S,M). By Theorem 10.2, the

lambda lengths of lifted arcs x(γ) form a (non-normalized) exchange pattern onE(S,M). This statement remains true after we pass from T (S,M) to T (S,M,L),even as we replace the coefficient group P(S,M) (see (10.1)) by Trop(qi). Indeed,the monomial mutation rules (1.2)–(1.3) are satisfied by the coefficients in exchangerelations for x(γ)’s viewed as functions on T (S,M), and therefore they are satisfiedafter the lambda lengths of boundary segments and holes are replaced by monomialsin the lamination weights.

By Lemmas 13.8 and 13.9, their tropical counterparts cL(γ) satisfy the tropical

versions of the same exchange relations. By Proposition 2.4, this is exactly whatis needed in order for the rescaled lambda lengths x(γ)/c

L(γ) to form a normalized

exchange pattern. It remains to verify that the extended exchange matrices of thisexchange pattern are the matrices B(T,L). In fact, by Theorem 12.4, it suffices todo this for some triangulation, say an ordinary triangulation T without self-foldedtriangles. In this setting, we need to verify that the coefficients of the exchange


relations associated with the flips from T (recall that these coefficients are givenby (2.1)) coincide with the coefficients encoded by B(T,L).

These exchange relations are obtained by dividing the Ptolemy relations (10.4) bytheir tropical counterparts (13.9). If arcs α, β, γ, δ ∈ T form a quadrilateral withdiagonals η ∈ T and θ, as in Figure 11, then the corresponding exchange relation is

xL(η)x

L(θ) = p+

η xL(α)x

L(γ) + p−η xL

(β)xL(δ),

where

(14.4) p+η =

cL(α) c

L(γ)

cL(η) c

L(θ)

, p−η =cL(β) c

L(δ)

cL(η) c

L(θ)

,

and α, β, γ, δ, η, θ are lifts of the corresponding arcs in T , coordinated so that α, β, γ, δform a quadrilateral with diagonals η and θ. If, for example, α is a boundary segmentrather than an arc, then x(α) = λ(α) = cL(α) = c

L(α) (see Definition 14.1), and the

exchange relation becomes

xL(η)x

L(θ) = p+

η xL(γ) + p−η xL

(β)xL(δ),

with the coefficients p±η given by (14.4) as before. Since these coefficients satisfy thenormalization condition p+

η ⊕ p−η = 1 (by Proposition 2.4, or by (13.9)), the claimreduces to showing that their ratio coincides with the Laurent monomial encoded bythe appropriate column of the matrix B(T,L). That is, we need to check that

p+η

p−η=cL(α) c

L(γ)

cL(β) c

L(δ)

=∏

i∈I

qbη(T,Li)i ,

where, as in (11.1), bη(T, Li) denotes the shear coordinate of the arc η with respectto the lamination Li. In view of (13.5), this is equivalent to the formula

(−lLi(α) + lLi

(β)− lLi(γ) + lLi

(δ))/2 = bη(T, Li) (i ∈ I).The latter is a version of a well known formula (see, e.g., [29, p. 44], [3, Section 4.6])relating shear coordinates to transverse measures. ¤

Definition 14.7. Let L be a multi-lamination on (S,M), a bordered surface withmarked points. Then the cluster algebra of geometric type described in Theorem 14.6is denoted by A(S,M,L).

Remark 14.8. The exchange pattern in Theorem 7.5—with the coefficient vari-ables corresponding to the boundary segments in B(S,M)—can be obtained as aparticular case of the main construction of this section. The corresponding multi-lamination L = Lββ∈B(S,M) contains one lamination Lβ for each boundary segmentβ ∈ B(S,M). The lamination Lβ consists of a single curve (also denoted by Lβ)defined as follows. Let p, q ∈ M be the endpoints of β. If p = q (so that β lies ona boundary component with a single marked point), then Lβ is a closed curve in S

encircling β. Otherwise, let p′ and q′ be two points on ∂S \ β located very close top and q, respectively. Then Lβ connects p′ and q′ within a small neighborhood of β.

See Example 15.1 for a particular instance of this construction.


Proof of Theorem 5.1. Uniqueness of an exchange pattern described in the theoremis clear: the mutation rules determine all seeds uniquely from the initial one. To proveexistence, we need to show that a sequence of mutations that returns us back to thesame tagged triangulation must recover the original seed. To use the terminologyof [13, Definition 4.5], we want to show that the exchange graph of our pattern iscovered by (and in fact is identical to) the graph E(S,M) of tagged flips.

We first reduce this claim to the case of patterns of geometric type. By [13,Theorem 4.6], the exchange graph of any cluster algebra A is covered by the exchangegraph of the cluster algebra with the same exchange matrices as A that has principalcoefficients (see Definition 16.1). Since the latter is of geometric type, the desiredreduction follows.

For an exchange pattern of geometric type, we produce a positive realization (cf.Definition 3.4) using the construction of Theorem 14.6. The key role in the argumentis played by (our generalization of) Thurston’s coordinatization theorem (see Theo-rems 11.3 and 12.5), which guarantees that we can get any coefficients (of geometrictype) by making an appropriate (unique) choice of a multi-lamination L.

It remains to show that all laminated lambda lengths xL(γ), for γ ∈ A(S,M),

are distinct. There are several ways to prove this; here we give a sketch of onesuch proof. It is easy to see that if in some normalized cluster algebra, a particularsequence of mutations yields a seed containing a cluster variable equal to one ofthe cluster variables of the initial cluster, then the same phenomenon must holdin the cluster algebra defined by the same initial exchange matrix over the one-element semifield 1. It is therefore enough to check the claim in the case of trivialcoefficients. In our setting, this case can be viewed as a special instance of theconstruction presented in Theorem 7.5, with the length of each boundary segmentequal to 1. In that instance, the cluster variables x(γ) are the lambda lengths ofcertain (pairwise non-isotopic) geodesics between appropriate horocycles.

We can use the torus action associated with changing the horocycles to show thatwe only need to consider pairs of tagged arcs connecting the same horocycles (aroundthe same marked points). Now, consider a pair of pants bounded by two horocyclesand a curve surrounding them and one of the arcs, and take the length of this curveto 0. Then the length of the other arc goes to ∞ whereas the length of the first arcstays bounded. (We thank Y. Eliashberg for suggesting this last argument.) ¤

Proof of Corollary 5.2. By Theorem 5.1, the cluster variables in A are in bijectionwith tagged arcs, and the seeds with tagged triangulations. The claims in the Corol-lary then follow from the basic properties of the tagged arc complex (see [8, Sec-tion 7]). ¤

15. Topological realizations of some coordinate rings

The main construction of Section 14 can be used to produce topological realizationsof (well known) cluster structures in coordinate rings of various algebraic varieties.Several such examples are presented in this section.


In each of these examples, the corresponding generalized decorated Teichmullerspace T (S,M) is naturally interpreted as the totally positive part of the correspond-ing cluster variety X, the spectrum of the associated cluster algebra. In other words,T (S,M) can be identified with the set of those points in X at which all cluster andcoefficient variables (equivalently, those belonging to a given extended cluster) takepositive values. In each case, one recovers the usual notion of total positivity ofmatrices or its well known extensions to Grassmannians and other G/P ’s (see, e.g.,[9, 10, 21] and references therein).

Example 15.1 (Grassmannians Gr2,n+3(C)). We already considered this clusteralgebra in Example 7.9. The homogeneous coordinate ring of the GrassmannianGr2,n+3(C) (with respect to its Plucker embedding) is generated by the Plucker co-ordinates ∆ij, for 1 ≤ i < j ≤ n+ 3, subject to the Grassmann-Plucker relations

∆ik ∆jl = ∆ij ∆kl + ∆il ∆jk (i < j < k < l).

These relations can be viewed as exchange relations in the cluster algebra An =C[Gr2,n+3] of cluster type An. This cluster algebra has n+ 3 coefficient variables

∆12, ∆23, . . . ,∆n+2,n+3, ∆1,n+3,

naturally corresponding to the sides of a convex (n + 3)-gon. The remaining n(n+3)2

Plucker coordinates ∆ij are the cluster variables; they are naturally labeled by thediagonals of the (n + 3)-gon. Since the exchange relations in An can be viewed asPtolemy relations between the lambda lengths of sides and diagonals of an (n+3)-gon(=unpunctured disk with n + 3 marked points on the boundary), we find ourselvesin a situation described in Remark 14.8, and can apply the construction discussedtherein. The case n = 3 is illustrated in Figure 37.

∆46

∆26

∆24

∆35

∆13

∆15

∆14

∆25

∆36

∆16

∆56

∆23

∆34

∆12

∆45

Figure 37. Representing the cluster structure on the GrassmannianGr2,6 by a multi-lamination of a hexagon.


Example 15.2 (Grassmannian Gr3,6; cf. [26]). The homogeneous coordinate ringA = C[Gr3,6] has a natural structure of a cluster algebra of (cluster) type D4, de-scribed in detail by J. Scott [26]. As a ring, A is generated by the Plucker coordinates∆ijk, for 1 ≤ i < j < k ≤ 6. As a cluster algebra, A has 16 cluster variables, whichinclude the cluster

(15.1) x = (x1, x2, x3, x4) = (∆245,∆256,∆125,∆235),

which we will use as an initial cluster. The cluster algebraA has 6 coefficient variables

(x5, . . . , x10) = (∆123,∆234,∆345,∆456,∆156,∆126),

which generate its ground ring. The exchange relations from the initial seed, and thecorresponding extended exchange matrix, with rows labeled by the cluster variables(rows 1–4) and coefficient variables (rows 5–10), are:

∆245 ∆356 = ∆345 ∆256 + ∆456 ∆235

∆256 ∆145 = ∆456 ∆125 + ∆156 ∆245

∆125 ∆236 = ∆126 ∆235 + ∆123 ∆256

∆235 ∆124 = ∆123 ∆245 + ∆234 ∆125

∆245

∆256

∆125

∆235

∆123

∆234

∆345

∆456

∆156

∆126

0 −1 0 1

1 0 −1 0

0 1 0 −1

−1 0 1 0

0 0 −1 1

0 0 0 −1

1 0 0 0

−1 1 0 0

0 −1 0 0

0 0 1 0

The topological realization of this cluster algebra, in which the marked surface (S,M)is a once-punctured quadrilateral, is shown in Figure 38. The initial cluster that wechose in (15.1) corresponds to the triangulation shown in Figure 39.

Example 15.3 (The ring C[Mat3,3]). The ring of polynomials in 9 variables zij

(i, j ∈ 1, 2, 3) viewed as matrix entries of a 3× 3 matrix

z =

z11 z12 z13

z21 z22 z23

z31 z32 z33

carries a natural cluster algebra structure of cluster type D4; see, e.g., [9, 14, 27].This cluster structure is very similar to the one discussed in Example 15.2, and isobtained as follows. The map

z 7→ rowspan

z11 z12 z13 0 0 1

z21 z22 z23 0 −1 0

z31 z32 z33 1 0 0

∈ Gr3,6


∆123 ∆234 ∆345

∆456 ∆156 ∆126

Figure 38. Representing the cluster structure on the GrassmannianGr3,6 by a multi-lamination L on a once-punctured quadrilateral. Eachof the 6 laminations in L consists of a single curve, and correspondsto a particular coefficient variable (a Plucker coordinate).

12

3 4

∆245∆256

∆125 ∆235

Figure 39. The triangulation representing the initial cluster (15.1)for the Grassmannian Gr3,6. The remaining 12 tagged arcs (not shownin the picture) correspond to the rest of the cluster variables.

induces a ring homomorphism

ϕ : C[Gr3,6]→ C[z11, . . . , z33].

For example, ϕ(∆145) = z11. (More generally, ϕ maps the 19 Plucker coordinates∆ijk—all but ∆456—into the 19 minors of z.) In fact, ϕ sends each of the 16 clus-ter variables in C[Gr3,6] into a cluster variable in C[z11, . . . , z33], and sends all butone coefficient variables in C[Gr3,6] into coefficient variables in C[z11, . . . , z33], thesole exception being ϕ(∆456) = 1. The exchange relations in the cluster algebraC[z11, . . . , z33] are obtained from those in C[Gr3,6] by applying ϕ. As a result, we geta cluster structure that can be realized by a multi-lamination on a once-puncturedquadrilateral that consists of 5 laminations: the ones shown in Figure 38 with theexception of the leftmost lamination in the bottom row.


Example 15.4 (The special linear group SL3; cf. [9]). This cluster algebra (also ofcluster type D4) is obtained from the one in Example 15.3 by further specializingthe coefficients, this time setting det(z) = 1. Equivalently, we send both coefficientvariables ∆123 and ∆456 in Example 15.2 to 1. This translates into removing theleftmost laminations in each of the two rows in Figure 38.

Example 15.5 (Affine base space SL4 /N). This example—with SL4 replaced byan arbitrary complex semisimple Lie group—was one of the main examples thatmotivated the introduction of cluster algebras [11]. Let N denote the subgroup ofunipotent upper-triangular matrices in SL4(C). The group N acts on SL4 by leftmultiplication; let

A = C[SL4 /N ] ⊂ C[z11, . . . , z44]/〈det(z)− 1〉be the ring of N -invariant polynomials in the matrix entries zij. The ring A isgenerated by the flag minors

∆I : z = (zij) 7→ det(zij|i ∈ I, j ≤ |I|),for I ( 1, 2, 3, 4, I 6= ∅. These flag minors satisfy well-known Plucker-type rela-tions.

The ringA carries a natural cluster structure of type A3, with 6 coefficient variables

∆1,∆12,∆123,∆4,∆34,∆234,

and 9 cluster variables

∆2,∆3,∆13,∆14,∆23,∆24,∆124,∆134,Ω = −∆1∆234 + ∆2∆134 .

Let the initial cluster be x = (∆2,∆3,∆23). The exchange relations from the ini-tial seed are:

∆2∆13 = ∆12∆3 + ∆1∆23

∆3∆24 = ∆4∆23 + ∆34∆2

∆23 Ω = ∆123∆34∆2 + ∆12∆234∆3

This algebra can be described using the multi-lamination L of a hexagon shownin Figure 40. The 9 cluster variables are in bijection with the 9 arcs (diagonals) inthe hexagon, while the 6 coefficient variables are labeled by the 6 laminations in L.The labeling is shown in Figure 41.

Remark 15.6. The cluster algebra of Example 15.5 illustrates a relatively rare phe-nomenon where the same algebra can be given two different topological realizations—because its exchange matrices can be obtained from two topologically different sur-faces, in this case an unpunctured hexagon and a once-punctured triangle. (Infor-mally speaking, types A3 and D3 coincide.) Consequently, one can represent thecluster structure in C[SL4 /N ] by a multi-lamination in a once-punctured triangle,consisting of 6 single-curve laminations. Cf. [8, Examples 4.3 and 4.5] and the dis-cussion at the beginning of [8, Section 14].


Figure 40. Representing the cluster structure on the affine base spaceSL4 /N by a multi-lamination of a hexagon. Each of the 6 curves is aseparate lamination, corresponding to a coefficient variable.

∆2

∆3

∆23

∆134

∆124

∆14

∆24

∆13

Ω

∆4

∆1

∆123

∆234

∆34

∆12

Figure 41. Affine base space SL4 /N : Dictionary between topologyand algebra

Example 15.7 (Unipotent subgroup N− ⊂ SL4). The coordinate ring of the subgroupN− of unipotent lower-triangular matrices has a cluster algebra structure that canbe obtained by specialization of coefficients from the cluster structure in C[SL4 /N ]discussed in Example 15.5: we set ∆1 = ∆12 = ∆123 = 1. (Cf. [2] and [17, Sec-tions 4.2.6–4.3].) This corresponds to removing the 3 laminations with these labelsfrom Figure 41.

Example 15.8 (Affine base space SL5 /N and unipotent subgroup N− ⊂ SL5). Inthese two cases, one gets cluster algebras of type D6, which can be represented by acollection of 8 (respectively, 4) single-curve laminations in a once-punctured hexagon.We omit the details for the sake of brevity.


Further examples of similar nature include various coordinate rings discussed in[8, Examples 6.7, 6.9, 6.10], among others.

16. Principal and universal coefficients

In this section, we work out two particular cases of our main construction, yieldingtwo distinguished choices of coefficients introduced in [13].

Definition 16.1 (Principal coefficients [13, Definition 3.1, Remark 3.2]). Let (Σt) =(x(t), B(t)) be an exchange pattern of geometric type. We say that this pattern (orthe corresponding cluster algebra A) has principal coefficients with respect to aninitial vertex t0 if the extended exchange matrix B(t0) has order 2n× n (as always,n is the rank of the pattern), and the bottom n × n part of B(t0) is the identitymatrix. We denote A = A•(B(t0)), where B(t0) is the initial n×n exchange matrix.

To rephrase, let x(t0) = (x1, . . . , xn) be the initial cluster, and let B(t0) = (bij).Then the algebra A = A•(B(t0)) is defined as follows. The ground semifield of A isTrop(q1, . . . , qn), and the exchange relations (1.5) out of the initial seed Σt0 are

xkx′k = qk

∏

1≤i≤nbik>0

xbik

i +∏

1≤i≤nbik<0

x−bik

i .

To construct a topological model for a cluster algebra with principal coefficients,we will need the following notion.

Definition 16.2 (Elementary lamination associated with a tagged arc). Let γ ∈A(S,M) be a tagged arc in (S,M). We denote by Lγ a lamination consisting of asingle curve (also denoted by Lγ, by an abuse of notation) defined as follows (up toisotopy). The curve Lγ runs along γ in a small neighborhood thereof; to complete itsdescription, we only need to specify what happens near the ends. Assume that γ hasan endpoint a on the boundary of S, more specifically on a circular component C.Then Lγ begins at a point a′ ∈ C located near a in the counterclockwise direction,and proceeds along γ as shown in Figure 42 on the left.

If γ has an endpoint a ∈M (a puncture), then Lγ spirals into a: counterclockwiseif γ is tagged plain at a, and clockwise if it is notched. See Figure 42 on the right.

a

γ

ba′ b′

Lγ

./

a

γLγ

Figure 42. Elementary laminations associated with tagged arcs

The following statement is straightforward to check.


Proposition 16.3. Let T be a tagged triangulation with a signed adjacency ma-trix B(T ). Then A•(B(T )) ∼= A(S,M,LT ) (cf. Definition 14.7), where LT =(Lγ)γ∈T is the multi-lamination consisting of elementary laminations associated withthe tagged arcs in T .

Example 16.4. Let T = (γap, γbp) be the triangulation of a once-punctured digonshown in Figure 34. Then LT = (L4, L3), in the notation of Figure 32.

We next turn to the discussion of cluster algebras with universal coefficients, whichwere constructed in [13, Section 12] for any finite Cartan-Killing (cluster) type. Moreexplicit versions of this construction for the types A and D have been given in theunpublished work [16]; the type A case was reproduced in [30, Proposition 7.2]. Herewe provide a restatement of the aforementioned results for the types A and D.

Proposition 16.5. Let (S,M) be a marked surface of finite cluster type, so thatthe set of tagged arcs A./(S,M) is finite. The corresponding cluster algebra withuniversal coefficients can be realized as A(S,M,L), where the multi-lamination L

consists of all elementary laminations (see Definition 16.2) associated with the taggedarcs γ ∈ A./(S,M).

The straightforward proof of this proposition is omitted.We note that among finite cluster types, only types A and D, and their direct

products, have topological realizations (see [8, Examples 6.6–6.7, Remark 13.5]).

Example 16.6. Let (S,M) be the once-punctured digon, so that the cluster typeis A1 × A1. Then L = (L1, L3, L4, L6), in the notation of Figure 32.

Example 16.7. Let (S,M) be an unpunctured (n + 3)-gon (type An). Then L

consists of all elementary (i.e., single-curve) laminations in (S,M). These are then(n+3)

2curves connecting non-adjacent midpoints of the sides of the polygon.

Example 16.8. Let (S,M) be a once-punctured n-gon (typeDn). Again, L consistsof all elementary laminations in (S,M). They include: (n − 3)n curves as in theprevious example (the number is doubled since we can go on either side of thepuncture); plus n curves beginning and ending near the same midpoint, and goingaround the puncture; plus 2n curves starting at one of the n midpoints and spirallinginto the puncture (either clockwise or counterclockwise); for the grand total of n2.

17. Tropical degeneration and relative lambda lengths

Here we sketch how our main construction of exchange patterns formed by gener-alized lambda lengths in “laminated Teichmuller spaces” can be obtained by tropicaldegeneration from a somewhat more traditional construction in which laminationsare replaced by hyperbolic structures of ordinary kind. This approach bears somesimilarity to the work of V. Fock and A. Goncharov on cluster varieties, and inparticular to their notion of symplectic double [7, Section 2.2].

First, a simple observation concerning (commutative) semifields; see Definition 2.3.

Lemma 17.1. A direct product of semifields is a semifield, under component-wiseoperations.


The tropical semifield of Definition 3.1 is one such example:

(17.1) Trop(q1, q2, . . . ) ∼= Trop(q1)× Trop(q2)× · · · .We next describe, somewhat informally, the general procedure of tropical degen-

eration. Fix a real number k, and define the binary operation ⊕k by

(17.2) y⊕k z = (yk + zk)1/k,

where y and z are positive reals, or positive-valued functions on some fixed set. It iseasy to see that the binary operation ⊕k is commutative, associative, and distribu-tive with respect to the ordinary multiplication. This makes (P, ⊕k , ·) a semifield,where P is any set of numbers or functions that is closed under ⊕k and undermultiplication. The simplest instance of this construction produces the semifield(R>0, ⊕k , ·) of positive real numbers, with ordinary multiplication and with addi-tion ⊕k defined by (17.2). In the “tropical limit,” as k → −∞, we get the semifield(R>0,⊕, ·) with the addition ⊕ given by

(17.3) y ⊕ z = limk→−∞

y⊕k z = min(y, z).

This is a close relative of the tropical addition (3.2), in the one-dimensional version,with the set I in (3.1) consisting of a single element. To obtain a multi-dimensionalversion, we apply the direct product construction of Lemma 17.1, either after takingthe limit (as in (17.1)), or before. Let us discuss the latter approach in more concretedetail. Start with a collection of real parameters k = (ki)i∈I , and consider thesemifield of tuples y = (yi)i∈I with the ordinary (pointwise) multiplication and withthe addition ⊕k defined by

(y ⊕k z)i = yi ⊕kizi = (yki

i + zki

i )1/ki .

Taking the limits ki → −∞ for all i, we obtain a close relative of the tropicalsemifield (3.1), with the addition defined by

(y ⊕ z)i = min(yi, zi) .

The geometric counterpart of the tropical degeneration procedure consists in ob-taining a lamination, viewed as a point on the Thurston boundary [25, 29] of theappropriate Teichmuller space, as a limit of ordinary hyperbolic structures. For sim-plicity, we discuss the case of surfaces with no marked points in the interior. Thegeneral case can be handled in exactly the same way as before, by lifting to the openedsurface. We still want to get the most general coefficients, so we cannot use the con-struction of Theorem 7.5 (coefficients coming from boundary segments); rather, weshall aim at obtaining the construction of Theorem 14.6 (coefficients coming from amulti-lamination) in the restricted generality of surfaces with no punctures.

Let σ ∈ T (S,M) be a decorated hyperbolic structure. (In the presence of punc-tures, we would need to consider σ ∈ T (S,M).) The lambda lengths λσ(γ) satisfygeneralized exchange/Ptolemy relations (6.2):

(17.4) λσ(η)λσ(θ) = λσ(α)λσ(γ) + λσ(β)λσ(δ).

Choose k ∈ R, and setλσ,k(γ) = (λσ(γ))1/k.


These quantities satisfy the ⊕k -version of (17.4):

(17.5) λσ(η)λσ(θ) = λσ(α)λσ(γ)⊕k λσ(β)λσ(δ).

Recalling Theorem 6.4, pick a triangulation T and a collection of positive reals(c(γ))γ∈T∪B(S,M). Then let σ and k vary so that k → −∞ while the coordinates(λσ,k(γ)) remain fixed:

λσ,k(γ) = c(γ), for γ ∈ T ∪B(S,M).

In other words, we set λσ(γ) = c(γ)k and let k → −∞. As a result, σ goes to a point

on the Thurston boundary of T (S,M) which can be identified with a real measuredlamination L whose tropical lambda lengths match the values we picked:

cL(γ) = c(γ), for γ ∈ T ∪B(S,M).

Meanwhile, the exchange relations (17.5) degenerate into

cL(η)cL(θ) = cL(α)cL(γ)⊕ cL(β)cL(δ),

where ⊕ denotes the minimum (the tropical addition). This fairly standard argumentshows how the tropical exchange relations for the tropical lambda lengths associatedwith a single lamination can be obtained by tropical degeneration from the ordinaryexchange relations for the lambda lengths with respect to a hyperbolic structure.

To extend this construction to the case of multiple laminations, we can use thedirect product construction of Lemma 17.1, yielding a solution of exchange relationsin the tropical semifield with several parameters, as in Lemmas 13.8 and 13.9. Thissolution can then be used to renormalize the ordinary lambda lengths as in (14.2),delivering the requisite normalized exchange pattern by virtue of Proposition 2.4, asin the proof of Theorem 14.6.

As we have seen, the process of creating a normalized exchange pattern formed byrenormalized lambda lengths consists of three main stages:

(1) producing a tropical solution of exchange relations from an ordinary one bymeans of “tropical degeneration;”

(2) applying a direct product construction to get a solution depending on severalreference geometries; and

(3) dividing the original non-normalized solution by the one produced at stage (2)to obtain a normalized pattern.

Alternatively, stage (1) (tropical degeneration) can be executed after stage (2) oreven after stage (3). We conclude this section by illustrating the latter approach inthe simple case of a single reference geometry; the resulting coefficients will lie ina tropical semifield with one variable. To compensate for this restriction, we willconsider a general case of a bordered surface with punctures.

Fix a reference geometry, a decorated hyperbolic structure ρ ∈ T (S,M). Now, forany other σ ∈ T (S,M) satisfying the boundary conditions

λσ(β) = λρ(β) for all β ∈ B(S,M),(17.6)

λσ(p) = λρ(p) for all p ∈M,(17.7)


the relative lambda length λσ/ρ(γ) of an arc γ ∈ A(S,M) is defined by

(17.8) λσ/ρ(γ) = λσ(γ)/λρ(γ),

where γ is an arbitrary lift of γ to the opened surface (S,M). Note that this ratiodoes not depend on the choice of a lift γ since the numerator and the denominatorrescale by the same factor under the twists ψp (see (9.4)).

By Theorem 10.2, the lambda lengths λρ(γ) (respectively, λσ(γ)) form a non-normalized exchange pattern with coefficients in the multiplicative group generatedby the boundary parameters (17.6)–(17.7). Consequently, by Proposition 2.4, the rel-ative lambda lengths λσ/ρ(γ) form a normalized exchange pattern over the semifieldgenerated by the lambda lengths for ρ, with ordinary addition and multiplication. Toillustrate what this amounts to, consider a quadrilateral in (S,M) with sides α, β, γ, δand diagonals η and θ, as in Proposition 6.5. Let α, β, γ, δ be lifts of α, β, γ, δ to theopened surface (S,M), coordinated so that α, β, γ, δ still form a quadrilateral. (Cf.the discussion surrounding (14.4).) Define the cross-ratio τ by

(17.9) τ =λρ(α)λρ(γ)

λρ(β)λρ(δ);

again, the value of τ does not depend on the choice of a quadruple of coordinatedlifts α, β, γ, δ. Then

(17.10) λσ/ρ(η)λσ/ρ(θ) =τ

1 + τλσ/ρ(α)λσ/ρ(γ) +

1

1 + τλσ/ρ(β)λσ/ρ(δ).

This exchange relation can of course be obtained by dividing the correspondingPtolemy relations for the two sets of lambda lengths (viz., λσ and λρ) by one an-other. Furthermore, the coefficients τ

1+τand 1

1+τ(or their ratio τ , in the “Y -pattern

version [13]) transform under flips according to the appropriate mutation rules, aspredicted by Propositions 2.1–2.4. (In the case of ordinary flips on (S,M), thisobservation was already made in [4, 5, 18].)

Degenerating the hyperbolic structure ρ to a lift L of a lamination L as explainedin the first part of this section, we obtain a normalized exchange pattern of geometrictype. More concretely, we get

τ

1 + τ=λρ(α)λρ(γ)

λρ(η)λρ(θ)→ cL(α)cL(γ)

cL(η)cL(θ)and

1

1 + τ=λρ(β)λρ(δ)

λρ(η)λρ(θ)→ cL(β)cL(δ)

cL(η)cL(θ),

recovering the coefficients (14.4). It is also easy to see directly that these coefficientsare obtained by exponentiating the shear coordinates of L, conforming to our generalrecipe.

Acknowledgments

The authors thank Leonid Chekhov, Yakov Eliashberg, Vladimir Fock, DavidKazhdan, Bernard Leclerc, Robert Penner, Alek Vainshtein, and Andrei Zelevin-sky for helpful advice and stimulating discussions. We are especially grateful toMichael Shapiro for the collaboration [8] that led to this paper, and for valuableinsights.


S. F. gratefully acknowledges the hospitality of IRMA (Strasbourg) in Spring 2006,and of MSRI (Berkeley) in Spring 2008.

A preliminary report on some of the main results contained in this paper waspresented at the LMS Workshop Teichmuller Theory and Cluster Algebras (Leicester,May 2006).

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

E-mail address: [email protected]

Department of Mathematics, Barnard College, Columbia University, New York, NY

10027, USA

E-mail address: [email protected]

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