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AN INVITATION TO HIGHER TEICHMÜLLER THEORY

1. Introduction

Riemann surfaces are of fundamental importance in many areas of math-ematics and theoretical physics. The study of the moduli space of Riemannsurfaces of a fixed topological type is intimately related to the study of theTeichmüller space of that surface, together with the action of the mappingclass group. Classical Teichmüller theory has many facets and involves theinterplay of various methods from geometry, analysis, dynamics and alge-braic geometry. In recent years, higher Teichmüller theory emerged as anew field in mathematics. It builds as well on a combination of methodsfrom different areas of mathematics. The goal of this article is to invite thereader to get to know and to get involved into higher Teichmüller theory bydescribing some of its many facets. Along the way we point to open ques-tions, and formulate some conjectures and task for the future. We will notbe able to discuss every aspect of higher Teichmüller theory, and will be verybrief on most of them. In particular we will not touch upon universal orinfinite higher Teichmüller spaces [76, 68], or algebraic structures developedin [74, 106, 105].

Higher Teichmüller theory is concerned with the study of representationsof fundamental groups of oriented surface S of negative Euler characteristicinto simple real Lie groups G of higher rank. The diversity of the methodsinvolved is due partly to the one-to-one correspondence between represen-tations, flat bundles, and Higgs bundles given by nonabelian Hodge theory[102], which was established in work of Donaldson, Hitchin, Corlette andSimpson, [38, 37, 69, 35, 103].

We will introduce higher Teichmüller spaces below as special subsets of therepresentation variety Hom(π1(S), G))/G, namely as connected componentsconsisting entirely of discrete and faithful representations. This is howevera definition which only arose a posteriori. The first family of higher Te-ichmüller spaces, the Hitchin components, has been introduced by Hitchin[70] using the theory of Higgs bundles. That they are higher Teichmüllerspaces in the sense of our definition was in general only proven ten yearslater by Labourie [75] and independently Fock and Goncharov [42] throughthe study of the space of positive decorated local systems or positive rep-resentations. The second family of higher Teichmüller spaces, the spaces ofmaximal representations, was defined completely independently as the levelset of a characteristic number on the representation variety, and its prop-erty of being a higher Teichmüller space in the sense of our definition wasshown by Burger, Iozzi, and Wienhard in [28], motivated by previous work

The author acknowledges support by the National Science Foundation under agreementDMS-1536017, by the Sloan Foundation, by the Deutsche Forschungsgemeinschaft, by theEuropean Research Council under ERC-Consolidator grant 614733, and by the Klaus-Tschira-Foundation.

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2 AN INVITATION TO HIGHER TEICHMÜLLER THEORY

of Goldman [53] in the context of classical Teichmüller space. The results of[70], [75], [42] and [28] arose completely independently, from different pointsof view and using very different methods. Only when comparing them it be-come apparent that the three spaces, Hitchin components, spaces of positiverepresentations, and spaces of maximal representations, have many similar-ities and provide examples of a new phenomenon. Now we consider them astwo families of what we call higher Teichmüller spaces. As the reader will see,we are still exploring the similarities and differences of these two families.It is interesting to note that the interplay between geometric and dynam-ical methods for representations of finitely generated groups and the moreanalytic and algebro-geometric methods from the theory of Higgs bundlesare at the heart of several recent advances in our understanding of higherTeichmüller spaces.

Many questions in higher Teichmüller theory are motivated by the thingswe know about classical Teichmüller space, its properties and interestinggeometric and dynamical structures it carries. However, there are also severalnew features that only arise for higher Teichmüller spaces and are not presentin classical Teichmüller theory, see for example Section 4, Section 10 andSection 12. Higher Teichmüller theory is a very young and active field ofmathematics. It is shaped by young mathematicians. There are still manyopen questions and unchartered territory to explore. We therefore hopethat many young (and older) mathematicians will accept this invitation andcontribute to the field in the future.

2. Classical Teichmüller space

Let S be a closed connected oriented topological surface of negative Eulercharacteristic χ(S) = 2−2g < 0, where g is the genus of S. The Teichmüllerspace T (S) of S is the space of marked conformal classes of Riemannianmetrics on S. It has been well studied using the theory of quasi-conformalmaps as well as methods from hyperbolic geometry. By the uniformizationtheorem, there is a unique hyperbolic, i.e. constant curvature −1, metric ineach conformal class. This identifies T (S) with the moduli space of markedhyperbolic structures. A marked hyperbolic structure is a pair (X, fX),where X is a hyperbolic surface and fX : S → X is an orientation preservinghomeomorphism. Two marked hyperbolic structures (X, fX) and (Y, fY ) areequivalent if there exists an isometry g : X → Y such that g fX is isotopicto fY . The mapping class group of S acts naturally on T (S) by changing themarking. This action is properly discontinuous, and the quotient of T (S) bythis action is the moduli space M(S) of Riemann surfaces of topological typegiven by S. Teichmüller space is homeomorphic to R

6g−6 and the universalcover of M(S).

Higher Teichmüller theory builds on an algebraic realization of Teichmüllerspace. The universal cover X of the hyperbolic surface X naturally identifieswith the hyperbolic plane H

2, and the fundamental group π1(X) acts as

group of deck transformations by isometries on X ∼= H2. Thus, upon fixing

a base point, the marking induces a group homomorphism (fX)∗ : π1(S) →π1(X) < Isom+(H2) ∼= PSL(2,R), which is called the holonomy. Associatingto a marked hyperbolic structure its holonomy gives a well defined injective

AN INVITATION TO HIGHER TEICHMÜLLER THEORY 3

map

hol : T (S) → Hom(π1(S),PSL(2,R))/PSL(2,R).

The representation variety Hom(π1(S),PSL(2,R))/PSL(2,R) is the spaceof all group homomorphisms of π1(S) into PSL(2,R), up to conjugationby PSL(2,R). It carries a natural topology (induced from the topologyof PSL(2,R)). Teichmüller spce T (S) is a connected component of therepresentation variety Hom(π1(S),PSL(2,R))/PSL(2,R). It is one of thetwo connected components, which consist entirely of discrete and faithfulrepresentations of π1(S) into PSL(2,R). The other such component is T (S),where S is the surface with the opposite orientation.

Remark 1. From this point of, as set of discrete and faithful representation,T (S) was first studied by Fricke. Historically it would thus be more appro-priate to call it Fricke space and the generalizations higher Fricke spaces,but it seems hard to change a name that is well established.

Classical Teichmüller space has many interesting properties and carries ad-ditional structure. It is a Kähler manifold which admits several Riemannianand non-Riemannian metrics, has nice explicit parametrizations, and carriesinteresting flows and dynamical systems. We will not be able to recall mostof these interesting properties, but will come back to a few of them in thesequel.

3. What is higher Teichmüller theory?

We might interpret what higher Teichmüller theory is in a narrow or abroader sense. In a very broad sense it is the study of classes of repre-sentations of finitely generated groups into Lie groups of higher rank withparticularly nice geometric and dynamical behaviour. In the narrow senseone could characterize it as the study of higher Teichmüller spaces as wedefine them below. In this article we restrict most of our discussion to thisnarrow interpretation. In the broad sense it is touched upon also in thecontributions of Kassel [73] and Potrie [95].

Teichmüller space is a connected component of the representation varietyHom(π1(S),PSL(2,R))/PSL(2,R) - this is where higher Teichmüller theorytakes it starting point. Instead of focussing on group homomorphisms ofπ1(S) into PSL(2,R), we replace PSL(2,R) by a simple Lie group G ofhigher rank (this is what the higher refers to), such as PSL(n,R), n ≥ 3 orSp(2n,R), n ≥ 2, and consider the representation variety Hom(π1(S), G)/G.We make the following definition:

Definition 2. A higher Teichmüller space is a subset of Hom(π1(S), G)/G,which is a union of connected components that consist entirely of discreteand faithful representations.

Note that as soon as G is not locally isomorphic to PSL(2,R), the groupπ1(S) is not isomorphic to a lattice in G. Therefore the set of discrete andfaithful representations is only a closed subset of Hom(π1(S), G)/G. It is thusnot clear that higher Teichmüller spaces exist at all, and in fact they will onlyexist for special families of Lie groups G. In particular, when G is a simplyconnected complex Lie group, the representation variety Hom(π1(S), G)/G


is irreducible as an algebraic variety, and hence connected, and there can-not be any connected component consisting entirely of discrete and faithfulrepresentations. There are two known families of higher Teichmüller spaces,Hitchin components and spaces of maximal representations. They have beendiscovered from very different points of view and by very different meth-ods. It then became clear that they share many properties, in particularthe property requested in Definition 2. We describe a common underlyingcharacterization, which also suggests the existence of two further families ofhigher Teichmüller spaces in Section 7.

Hitchin components TH(S,G) are defined when G is a split real simpleLie group, the space of maximal representations Tmax(S,G) is defined whenG is a non-compact simple Lie group of Hermitian type. In the case whenG = PSL(2,R), the Hitchin component and the space of maximal represen-tations agree and coincide with Teichmüller space T (S). For other groupsG not locally isomorphic to PSL(2,R), which are at the same time split andof Hermitian type, i.e. Sp(2n,R) or SO(2, 3), there is a proper inclusionTH(S,G) ⊂ Tmax(S,G).

Remark 3. We assume that S is a closed surface.There is a related theory forsurfaces with punctures or boundary components. However, in this case thecorresponding subset of the representation variety is not a union of connectedcomponents. We comment on the situation for surfaces with punctures inSection 7.

We shortly review the definitions of Hitchin components and maximalrepresentations. For more details and further properties we refer the readerto the survey [29].

3.1. Hitchin components. Hitchin components are defined when G is asplit real simple Lie group. Any split real simple Lie group G contains athree-dimensional principal subgroup, i.e. an embedding ιp : SL(2,R) → G,which is unique up to conjugation. For the classical Lie groups SL(n,R),Sp(2n,R), and SO(n, n + 1) this is just the irreducible representation ofSL(2,R) in the appropriate dimension. Precomposing ιp with a discreteembedding of π1(S) into SL(2,R) we obtain a representation ρp : π1(S) → G,which we call a principal Fuchsian representation.

Definition 4. The Hitchin component TH(S,G) is the connected compo-nent of Hom(π1(S), G)/G containing a principal Fuchsian representationρp : π1(S) → G.

Remark 5. Note that we are a bit sloppy in our terminology, e.g. when G =PSL(3,R) there are 2 connected components in Hom(π1(S),PSL(3,R))/PSL(3,R)which consists of discrete and faithful representation which preserve the ori-entation. We refer to each of them as the Hitchin component.

Hitchin showed, using methods from the theory of Higgs bundle thatthe Hitchin component is homeomorphic to a vector space of dimensiondim(G)(2g − 2). He conjectured that these components are geometricallysignificant and parametrize geometric structures. This was supported by oneexample. Goldman [54] had investigated the spaces of convex real projectivestructures on S and shown that it is isomorphic to R

16g−16 and that the


holonomy of a convex real projective structure is in the Hitchin component.Soon afterwards Choi and Goldman [31] proved that in fact TH(S,PSL(3,R))parametrizes the space of convex real projective structures on S. It tookanother ten years before further progress was made, when Labourie [75] in-troduced methods from dynamical systems to the study of representationsin the Hitchin component and showed that for G = PSL(n,R),PSp(2n,R),and PO(n, n+1) representations in the Hitchin component are discrete andfaithful. That representations in any Hitchin component are discrete andfaithful follows from work of Fock and Goncharov [42]. They investigatedthe space of positive representations in Hom(π1(S), G)/G, when G is a splitreal simple Lie group, and showed that it coincides with the Hitchin compo-nent (see Section 7).

3.2. Maximal representations. Maximal representations are defined whenthe simple Lie group G is of Hermitian type. They are singled out by a char-acteristic number, the Toledo number, which for G = PSL(2,R) is just theEuler number. In [53] Goldman showed that the Euler number distinguishesthe connected components of Hom(π1(S),PSL(2,R))/PSL(2,R), and thatTeichmüller space corresponds to the connected component formed by rep-resentation of Euler number 2g−2, which is the maximal value it can attain.In general, the Toledo number is bounded in terms of the Euler characteris-tic of S and the real rank of G, and constant on connected components ofHom(π1(S), G)/G. The space of maximal representations Tmax(S,G) is theset of all representations for which the Toledo number assumes it maximalpossible value. It is a union of connected components. Using methods frombounded cohomology, it was proven in [28] that any maximal representationis faithful with discrete image.

Remark 6. There are two types of Hermitian Lie groups, those of tube typeand those not of tube-type. Maximal representations into Lie groups thatare not of tube type satisfy a rigidity theorem [111, 67, 28, 21, 19]: Theimage of a maximal representation is always contained in the stabilizer in Gof a maximal subsymmetric space of tube type. This reduces the study ofmaximal representation essentially to the case when G is of tube type.

3.3. Anosov representations. Anosov representation are homomorphismsof finitely generated hyperbolic groups Γ into arbitrary reductive Lie groupsG with special dynamical properties. They have been introduced by Labourie[75] to investigate representations in the Hitchin component, and extendedto hyperbolic groups in [64]. The set of Anosov representations is an opensubset of Hom(Γ, G)/G, but in general not a union of connected componentsof Hom(Γ, G)/G. Representations in the Hitchin component and maximalrepresentations were the first examples of Anosov representations [75, 25,64, 27]. We refer to Kassel’s contribution [73] for the definition, more detailsand more references on Anosov representations.

The following key properties of Hitchin representations and maximal rep-resentations follow from them being Anosov representations (with respect tocertain parabolic subgroups).

(1) Every representation in the Hitchin component and every maximalrepresentation is discrete and faithful.


(2) Let ρ : π1(S) → G be a Hitchin representation, then there existsa ρ-equivariant continuous boundary map ξ : S1 → G/B into thegeneralized flag variety G/B, where B is the Borel subgroup of G.The map sends distinct points in S1 to transverse points in G/B.

(3) Let ρ : π1(S) → G be a maximal representation, then there exists aρ-equivariant continuous boundary map ξ : S1 → G/S into the gen-eralized flag variety G/S, where S is a maximal parabolic subgroupof G which fixes a point in the Shilov boundary of the symmetricspace X = G/K. The map sends distinct points in S1 to transversepoints in G/S.

4. Topology of the representation variety

For a connected Lie group G the obstruction to lifting a representationπ1(S) → G to the universal cover of G defines a characteristic invariant inH2(S, π1(G)) ∼= π1(G). For compact simple Lie groups [7] and complex sim-ple Lie groups [53, 86] the connected components of Hom(π1(S), G)/G are inone to one correspondence with elements in π1(G). This does not hold any-more for real simple Lie groups in general. Of course, characteristic invari-ants in H2(S, π1(G)) still distinguish some of the connected components, butthey are not sufficient to distinguish all of them. In [70] Hitchin determinedthe number of connected components of Hom(π1(S),PSL(n,R))/PSL(n,R),and showed that Hitchin components have the same characteristic invariantsas other components. The space of maximal representations, which is de-fined using characteristic invariants, in fact decomposes itself into severalconnected components, which hence cannot be distinguished by any charac-teristic invariant [56, 19].

A precise count of the number of connected components for several clas-sical groups, and in particular for the connected components of the space ofmaximal representations has been given using the Morse theoretic methodson the moduli space of Higgs bundles Hitchin introduced [92, 48, 20, 56, 19,47, 46]. The situation is particularly interesting for G = Sp(4,R) (and simi-larly for the locally isomorphic group SO(2, 3)) as there are 2g−4 connectedcomponents in which all representations are Zariski dense [63, 22]. Maximalrepresentations in these components cannot be obtained by deforming anappropriate Fuchsian representation ρ : π1(S) → SL(2,R) → Sp(4,R). Anexplicit construction of representations in these exceptional components isgiven in [63]. Any maximal representations into Sp(2n,R) with n ≥ 3 on theother hand can be deformed either to a principal Fuchsian representation (ifit is in a Hitchin component) or to a (twisted) diagonal Fuchsian representa-tion ρ : π1(S) → SL(2,R) × O(n) < Sp(2n,R). In order to distinguish theconnected components in the space of maximal representations, additionalinvariants are necessary. Such additional invariants have been defined on theone hand using methods from the theory of Higgs bundles [56, 19, 32, 9, 6]and on the other hand using the Anosov property of representations [63].

We shortly describe the additional invariants arising from the Anosovproperty. If a representations is Anosov with respect to a parabolic subgroupP < G, then the pull-back of the associated flat G-bundle to T 1S admits areduction of the structure group to L, where L is the Levi subgroup of P .


The characteristic invariants of this L-bundle provide additional invariantsof the representation. These additional invariants can be used in particularto further distinguish connected components consisting entirely of Anosovrepresentation. Note that representations can be Anosov with respect todifferent parabolic subgroups - each such parabolic subgroup gives rise toadditional invariants. For the case of maximal representations into Sp(2n,R)it is shown in [63] that these additional invariants in fact distinguish allconnected components.

Conjecture 7. Let G be a simple Lie group of higher rank. The connectedcomponents of Hom(π1(S), G)/G can be distinguished by characteristic in-variants and by additional invariants associated to unions of connected com-ponents consisting entirely of Anosov representations.

Note that, since Anosov representations are discrete and injective, anyconnected component consisting entirely of Anosov representations also pro-vides an example of a higher Teichmüller space. Thus Conjecture 7 impliesin particular the following

Conjecture 8. There are connected components of Hom(π1(S), G)/G whichare not distinguished by characteristic invariants, if and only if there existhigher Teichmüller spaces in Hom(π1(S), G)/G

Combining Conjecture 8 with Conjecture 19 gives a precise list of groups(see Theorem 17) for which we expect additional connected components toexist. A particularly interesting case is SO(p, q), p 6= q. Here additionalcomponents and additional invariants have recently been found via Higgsbundle methods [32, 9, 6].

5. Geometric Structures

Classical Teichmüller space T (S) is not just a space of representations,but in fact a space of geometric structures: every representation is the ho-lonomy of a hyperbolic structure on S. For higher Teichmüller spaces, sucha geometric interpretation is less obvious. The quotient of the symmetricspace Y associated to G by ρ(π1(S)) is of infinite volume. In order to findgeometric structures on compact manifolds associated, other constructionsare needed.

For any representation ρ : π1(S) → G in the Hitchin component or in thespace of maximal representation, there is t a domain of discontinuity in ageneralized flag variety X = G/Q, on which ρ acts cocompactly. The quo-tient is a compact manifold M with a locally homogeneous (G,X)-structure.This relies on the the construction of domains of discontinuity for Anosovrepresentations given by Guichard and Wienhard in [64] and generalizedby Kapovich, Leeb, and Porti in [72]. We do not describe this construc-tion here in detail, but refer the reader to [73], where locally homogeneous(G,X)-structures, Anosov representations and the construction of domainsof discontinuity are discussed in more detail.

The construction of the domains of discontinuity, together with some topo-logical considerations, allows one to deduce the general statement


Theorem 9. [64] For every split real simple Lie group G there exists ageneralized flag variety X and a compact manifold M such that TH(S,G)parametrizes a connected component of the deformation space of (G,X)-structures on M . For every Lie group of Hermitian type G there exists ageneralized flag variety X and a compact manifold M such that for everyconnected component C of Tmax(S,G) the following holds: A Galois cover ofC parametrizes a connected component of the deformation space of (G,X)-structures on M .

In particular, any Hitchin representation or maximal representation is es-sentially the holonomy of a (G,X)-structures on a compact manifold. It ishowever quite hard to get an explicit description of the deformation space of(G,X)-structures they parametrize. First it is nontrivial to determine thetopology of the quotient manifold, and second it is rather difficult to givea synthetic description of the geometric properties which ensure that theholonomy representation of a (G,X) structure lies in the Hitchin componentor in the space of maximal representations. In three cases we such a syn-thetic description: the Hitchin component for TH(S,PSL(3,R)), the Hitchincomponent TH(S,PSL(4,R)) and TH(S,PSp(4,R)), the space of maximalrepresentations Tmax(S,SO

(2, n)).

Theorem 10. [31] A representation ρ : π1(S) → PSL(3,R) is in Hitchincomponent TH(S,PSL(3,R)) if and only if it is the holonomy representationof a convex real projective structures on S

A convex real projective structure on S is a realization of S as the quotientof a convex domain Ω ⊂ RP

2 by a group Γ < PSL(3,R) of projectivelinear transformation preserving Ω. One aspect which makes this case veryspecial is that the group PSL(3,R) acts as transformation group on the two-dimensional homogeneous space RP

2. The subgroup ρ(π1(S)) preserves aconvex domain Ω and acts cocompactly on it. The quotient is a surfacehomeomorphic to S. For more general simple Lie groups of higher rank,there is no two-dimensional generalized flag variety on which they act, andso the quotient manifold M is higher dimensional.

Theorem 11. [62] The Hitchin component TH(S,PSL(4,R)) parametrizesthe space of properly convex foliated projective structures on the unit tan-gent bundle of S. The Hitchin component TH(S,PSp(4,R)) parametrizesthe space of properly convex foliated projective contact structures on the unittangent bundle of S.

Theorem 12. [33] The space of maximal representations Tmax(S,SO(2, n))

parametrizes the space of fibered photon structures on O(n)/O(n−2) bundlesover S.

Conjecture 13 (Guichard-Wienhard). Let ρ : π1(S) → G be a representa-tion in a higher Teichmüller space, then there exists a generalized flag varietyX and compact fiber bundle M → S, such that ρ : π1(M) → π1(S) → G,where π1(M) → π1(S) is induced by the bundle map and π1(S) → G is givenby ρ, is the holonomy of a locally homogeneous (G,X)-structure on M .

In fact, we expect, that for any cocompact domain of discontintuity whichis constructed through a balanced thickening in the sense of [72], the quotient


manifold is homeomorphic to a compact fiber bundle M over S. A relatedconjecture has been made by Dumas and Sanders [40, Conjecture 1.1] fordeformations of Hitchin representations in the complexification of G. Theyproved the conjecture in the case of PSL(3,C). Guichard and Wienhard[59] determine the topology of the quotient manifold for maximal represen-tations and Hitchin representations into the symplectic group, Alessandriniand Li [4] prove the conjecture for Hitchin representations into PSL(n,R),and their deformations into PSL(n,C), and Alessandrini, Maloni, and Wien-hard [5] analyze the topology of quotient manifold for complex deformationsof symplectic Hitchin representations.

It is very interesting to note that the recent advantages [4, 33] on un-derstanding the topology of the quotient manifolds rely on a finer analysisand description of the Higgs bundle associated to special representations inthe Hitchin component or in the space of maximal representations. Thatthe explicit description of the Higgs bundles can be used to endow the do-main of discontinuity naturally with the structure of a fiber bundle was firstdescribed by Baraglia [8] for TH(S,PSL(4,R)), where he recovered the pro-jective structures on the unit tangent bundle from Theorem 11.

6. Relation to the moduli space of Riemann surfaces

The outer automorphism group fo π1(S) is isomorphic to the mappingclass group of S. It acts naturally on Hom(π1(S), G)/G. This action isproperly discontinuous on higher Teichmüller spaces - in fact more gener-ally on the set of Anosov representations [78, 114, 64]. It is natural to askabout the relation between the quotient of higher Teichmüller spaces bythis action and the moduli space of Riemann surfaces M(S). For Hitchincomponents Labourie made a very precise conjecture, based on Hitchin’sparametrization of the Hitchin component. We state the parametrizationand Labourie’s conjecture for G = PSL(n,R) to simplify notation. Hitchinintroduced the Hitchin component in [70] using methods from the theory ofHiggs bundles. This requires the choice of a conformal structure on S. HeHitchin [70] showed, using methods from the theory of Higgs bundles, thatthe Hitchin component is homeomorphic to a vector space. Namely, it ishomeomorphic to the space of holomorphic differentials on S with respect toa chosen conformal structure, i.e. TH(S,PSL(n,R)) ∼=

∑ni=2 H

0(S,Ki) Thisparametrization depends on the choice of a conformal structure and is notinvariant under the mapping class group.

Conjecture 14. [78] The quotient of TH(S,PSL(n,R)) by the mappingclass group is a holomorphic vector bundle over M(S), with fiber equal to∑n

i=3H0(S,Ki).

This conjecture has been proven by Labourie [77] and Loftin [88] for G =PSL(3,R) and by Labourie [79], using Higgs bundle methods, for all splitreal Lie groups of rank 2. It is open in all other cases.

For maximal representations we expect similarly to get a mapping classgroup invariant projection from Tmax(S,G) to T (S), see [2, Conjecture 10].For G = SO(2, n) such a projection is constructed in [33]. For G of rank 2Alessandini and Collier [2] construct a mapping class group invariant com-plex structure on Tmax(S,G). For G = Sp(4,R) they show that the quotient


of Tmax(S,G) by the mapping class group is a holomorphic vector bundleover M(S), and describe in detail the fiber over a point, which is rather com-plicated since the space of maximal representations has nontrivial topologyand singular points.

7. Positivity

Hitchin components and maximal representation were introduced andstudied by very different methods. It turns out that they do not onlyshare many properties, but also admit a common characterization in termsof positive structures on flag varieties. Only the flag varieties and notionsof positivity in question are different for Hitchin components and maximalrepresentations.

For Hitchin components we consider full flag varieties and Lusztig’s totalpositivity [90]. For maximal representations the flag variety in question is theShilov boundary of the symmetric space of G and positivity is given by theMaslov cocycle. In order to keep the description simple, we illustrate bothnotions in examples. We consider G = SL(n,R) for Hitchin components,and G = Sp(2n,R) for maximal representations.

The relevant flag variety for TH(S,SL(n,R)) is the full flag variety

F(Rn) := F = (F1, F2, · · · , Fn−1) |Fi ⊂ Rn, dim(Fi) = i, Fi ⊂ Fi+1.

Two flags F,F ′ are said to be transverse if Fi ∩ F ′n−i = 0. We fix the

standard basis (e1, · · · , en) of Rn. Let F ∈ F be the flag with Fi =

span(e1, · · · , ei), and E ∈ F the flag with Ei = span(en, · · · , en−i+1).Any flag T transverse to F , is the image of E under a unique unipotent

matrix uT . The triple of flags (E,T, F ) is said to be positive if and onlyif uT is a totally positive unipotent matrix. Note that a unipotent (herelower triangular) matrix is totally positive if and only of every minor ispositive, except those that have to be zero by the condition that the matrixis unipotent. Any two transverse flags (F1, F2) can be mapped to (E,F ) byan element of SL(n,R) and we can extend the notion of positivity to anytriple of pairwise transverse flags.

Theorem 15. [42, 75, 57] Let ρ : π1(S) → SL(n,R) be a representa-tion. Then ρ ∈ TH(S,SL(n,R)) if and only if there exists a continuousρ-equivariant map ξ : S1 → F(Rn) which sends positive triples in S1 topositive triples in F(Rn).

To describe the analogous characterization of maximal representationsinto Sp(2n,R) we consider R2n with the standard symplectic form ω and lete1, · · · , en, f1, · · · fn be a symplectic basis. Then

L(R2n) := L ⊂ R2n | dimL = n, ω|L×L = 0

is the space of Lagrangian subspaces, and two Lagrangians L and L′ aretransverse if L ∩ L′ = 0.

Fix LE = span(e1, · · · , en) and LF = span(f1, · · · , fn). Any LagrangianLT ∈ L transverse to LF is the image of LE under an element vT =(

Idn 0MT Idn

)

∈ V , where MT is a symmetric matrix.


The triple of Lagrangians (LE , LT , LF ) is said to be positive if and onlyif MT ∈ Pos(n,R) ⊂ Sym(n,R) is positive definite. This is equivalent tothe Maslov cocycle of (LE , LT , LF ) being n, which is the maximal value itcan attain. The symplectic group Sp(2n,R) acts transitively on the space ofpairs of transverse Lagrangians and we can extend the notion of positivityto any triple of pairwise transverse Lagrangian.

Theorem 16. [28] Let ρ : π1(S) → Sp(2n,R be a representation. Thenρ ∈ Tmax(S,Sp(2n,R)) if and only if there exists a continuous ρ-equivariantmap ξ : S1 → L(R2n) which sends positive triples in S1 to positive triples inL(R2n.

In [61, 60] we introduce the notion of Θ-positivity. It generalizes Lusztig’stotal positivity, which is only defined for split real Lie groups, to arbitrarysimple Lie groups. There are four families of Lie groups admitting a Θ-positive structure:

Theorem 17. [61, Theorem 4.3.] A simple Lie group G admits a Θ-positivestructure if and only if:

(1) G is a split real form.(2) G is of Hermitian type of tube type.(3) G is locally isomorphic to SO(p, q), p 6= q,.(4) G is a real form of F4, E6, E7, E8, whose restricted root system is of

type F4.

Θ refers to a subset of the simple roots. If G admits a Θ-positive structure,then there is positive semigroup U>

Θ ⊂ UΘ < PΘ with which we can define thenotion of positivity for a triple of pairwise transverse points in the generalizedflag variety G/PΘ as above. Here PΘ is the parabolic group associated tothe subset of simple roots Θ, and UΘ is its unipotent radical.

Definition 18. A representation ρ : π1(S) → G is said to be Θ-positiveif there exists a continuous ρ-equivariant map ξ : S1 → G/PΘ which sendspositive triples in S1 to positive triples in G/PΘ.

Conjecture 19 (Guichard-Labourie-Wienhard). The set of Θ-positive rep-resentations ρ : π1(Σg) → G is open and closed in Hom(π1(S), G)/G. Inparticular, Θ-positive representations form higher Teichmüller spaces.

For more details on Θ-positivity and Θ-positive representations we referthe reader to [61] and the upcoming papers [60, 58], in which Conjecture 13will be partly addressed. In particular we prove that Θ-positive representa-tions are PΘ-Anosov and form an open subset of Hom(π1(S), G)/G, and aclosed set, at least in the subset of irreducible representations.

The existence of a Θ-positive structure provides a satisfying answer onwhen and why higher Teichmüller spaces exist, and we expect that the fam-ilies of Lie groups listed in Theorem 17 are the only simple Lie groups forwhich higher Teichmüller spaces in Hom(π1(S), G)/G exist. A particularinteresting case is the family of Θ-positive representations for G = SO(p, q).Here the connected components have recently been determined with Higgsbundle methods, and several of them contain Θ-positive representations[32, 6].


8. Coordinates and Cluster structures

Teichmüller space carries several nice sets of coordinates. The best knownare Fenchel-Nielsen coordinates, which encode a hyperbolic structure by thelength of and the twist around a set of 3g − 3 disjoint simple closed non-homotopic curves which give a decomposition of S into a union of 2g − 2pair of pants. Goldman [54] introduced Fenchel-Nielsen type coordinates onthe Hitchin component TH(S,PSL(3,R). Here, there are two length andtwo twist coordinates associated to the curves of a pants decomposition,and in addition two coordinates which associated to each of the pairs ofpants. This a new feature arises because a convex real projective structureon a pair of pants is not uniquely determined by the holonomies around theboundary. For maximal representations, Fenchel-Nielsen type coordinateswere constructed in [104].

It is often easier to describe coordinates in the situation when the surfaceis not closed, but has at least one puncture. In this case one can considerdecorated flat bundles (or decorated representations), which is a flat bundleor a representation together with additional information around the punc-ture. Fixing an ideal triangulation, i.e. a triangulation where all the verticesare punctures, this additional information can be used to define coordinates.Examples of this are Thurston shear coordinates or Penner coordinates fordecorated Teichmüller space. In the context of higher Teichmüller spaces, fordecorated representations into split real Lie groups Fock and Goncharov [42]introduced two sets of coordinates, so called X - coordinates, which general-ize Thurston shear coordinates, and A-coordinates, which generalize Pennercoordinates. They show that when performing a flip of the triangulation(changing the diagonal in a quadrilateral formed by two adjacent triangles),the change of coordinates is given by a positive rational function. As aconsequence, the set of decorated representations where all coordinates arepositive, is independent of the triangulation. In fact, Fock and Goncharovprove that this set of positive representations is precisely the set of posi-tive representations in the sense of Section 7, where the notion of positivitystems from Lusztig’s positivity. In the case when G = PSL(n,R), the Fock-Goncharov coordinates admit a particularly nice geometric description basedon triple ratios and cross ratios. In particular there is a close relation betweenthe coordinates and cluster structures, which received a lot of attention. Thechange of coordinates associated to a flip of the triangulations is given bya sequence of cluster mutations. This has since been generalized to otherclassical groups in [83], see also [82, 55] for general split real Lie groups.Related coordinates have been defined by Gaiotto, Moore and Neitzke, us-ing the theory of spectral networks [44, 45]. For an interpretation of theWeil-Petersson form in terms of cluster algebras see [49].

Inspired by Fock-Goncharov coordinates for surfaces with punctures, Bona-hon and Dreyer defined coordinates on the Hitchin component TH(S,PSL(n,R))and showed that TH(S,PSL(n,R)) is real analytically homeomorphic to theinterior of a convex polygon of dimension (n2 − 1)(2g − 2) [16, 17]. Thesecoordinates are associated to a maximal lamination of the surface S andgeneralize Thurston’s shear coordinates of closed surfaces. A special case forsuch a maximal lamination is an ideal triangulation of S which is subordinate


to a pair of pants decomposition, i.e. the lamination consists of 3g−3 disjointsimple closed non-homotopic curves which give a pair of pants decomposi-tion, and three curves in each pair of pants, that cut the pair of pants into twoideal triangles. In this case, Zhang [118] provided a reparametrization of theBonahon-Dreyer coordinates, which give a genuine generalization of Fenchel-Nielsen type coordinates for the Hitchin component TH(S,PSL(n,R)), seealso [18] for a direct comparision with Goldman coordinates when n = 3.

In forthcoming work [3] we introduce X -type and A-type coordinates fordecorated maximal representations of the fundamental group of a puncturedsurface into the symplectic group Sp(2n,R). These coordinates have thefeature of behaving like the coordinates for G = PSL(2,R) but with valuesin the space of positive definite symmetric matrices Pos(n,R). In particular,even though they are noncommutative, they exhibit a cluster structure. Thisstructure is similar to the noncommutative cluster structure considered byBerenstein and Retakh [14], except for a difference in some signs.

It would be interesting to develop similar coordinates for Θ-positive rep-resentations, in particular for those into SO(p, q), and to investigate theirproperties. The properties of the Θ-positive structure suggests that in thiscase the cluster-like structure would combine noncommutative and commu-tative aspects.

Task 20. Develop X -type and A-type coordinates for decorated Θ-positiverepresentations into SO(p, q). Analyze their cluster-like structures.

9. Symplectic geometry and dynamics

For any reductive Lie group, the representation variety of a closed surfaceHom(π1(S), G)/G is a symplectic manifold [51]. On Teichmüller space thissymplectic structure interacts nicely with Fenchel-Nielsen coordinates. Thelength and twist coordinates give global Darboux coordinates: the lengthcoordinate associated to a simple closed curve in a pair of pants decom-position is symplectically dual to the twist coordinate associated to thiscurve, and the symplectic form can be expressed by Wolpert’s formula asω =

∑3g−3i=1 dli ∧ dτi, where li is the length coordinate and τi is the twist

coordinate [116]. The twist flows associated to a simple closed curve c on Sis the flow given by cutting S along c and continuously twisting around thiscurve before gluing the surface back together. It is the Hamiltonian flow as-sociated to the length coordinate defined by c. The twist flows associated tothe 3g−3 simple closed curves in a pants decomposition on S commute. Thisgives Teichmüller space the structure of a complete integrable system. Formore general reductive groups, the Hamilitonian flows associated to lengthfunctions on Hom(π1(S), G)/G have been studied by Goldman [52].

In [108] provide a new approach to compute the Goldman symplectic formon TH(S,PSL(n,R)). This, in conjunction with a companion article by Sun,Zhang and Wienhard [107], gives rise to several nice statements. Given max-imal lamination with finitely many leaves (and some additional topologicaldata) we construct in [107] new families of flows on TH(S,PSL(n,R)). Theseflows give a trivialization of the Hitchin component, which is shown to besymplectic in [108]. Consequently the flows are all Hamiltonian flows andprovide TH(S,PSL(n,R) the structure of a completely integrable system.


Task 21. The mapping class group Mod(S) acts naturally on the space ofmaximal laminations and the additional topological data, so that the symplec-tic trivializations of TH(S,PSL(n,R)) induce representations πn : Mod(S) →

Sp(R(n2−1)(2g−2)). Analyze these representations.

A special situation arises when the maximal lamination is an ideal trian-gulation subordinate to a pants decomposition. In this situation we slightlymodify the Bonahon-Dreyer coordinates, to get global Darboux coordinateson TH(S,PSL(n,R)) which consist of (3g − 3)(n − 1) length coordinates,(3g−3)(n−1) twist coordinates associated to the simple closed curves in the

pants decomposition, and 2× (n−1)(n−2)2 coordinates for each pair of pants.

The twist flows are the Hamiltonian flows associated to the length functions,

and for each pair of pants we introduce (n−1)(n−2)2 new flows, which we call

eruption flows. Their Hamiltonian functions are rather complicated. Never-theless, the twist flows and the eruption flows pairwise commute, providinga half dimensional subspace of commuting flows. In the case of PSL(3,R)the eruption flow has been defined in [115], where it admits a very geometricdescription.

Classical Teichmüller space does not only admit twist flows, but carriesseveral natural flows, for example earthquake flows, which extend twist flows,geodesic flows with respect to the Weil-Petersson metric or the Teichmüllermetric or, and even an SL(2,R)-action. None of this has yet been exploredfor higher Teichmüller spaces. A new approach for lifting Teichmüller dy-namics to representation varieties for general Lie group G has recently beendescribed by Forni and Goldman [43].

10. Geodesic flows and entropy

Representations in higher Teichmüller space, and more generally Anosovrepresentations, are strongly linked to dynamics on the surface S. Any suchrepresentation gives rise to a Hölder reparametrization of the geodesic flowon S and the representation can essentially be reconstructed from the periodsof this reparametrized geodesic flow. This dynamical point of view has beenfirst observed by Labourie [75] and applied by Sambarino [97] and has beenkey in several interesting developments.

Using the thermodynamical formalism Bridgeman, Canary, Labourie, andSambarino [23] define the pressure metric on the Hitchin component TH(S,PSL(n,R))and more generally spaces of Anosov representations. The pressure metricrestricts to a multiple of the Weil-Petersson metric on the subset of principalFuchsian representations. In the case of PSL(3,R) metrics on the space ofconvex real projective structures have also been constructed in [36, 87].

Other important quantities that have been investigated using this dynami-cal viewpoint are the critical exponent and the topological entropy of Hitchinrepresentations, which are related to counting orbit points on the symmetricspace [99, 98, 94]. Here completely new features arise that are not presentin classical Teichmüller space. On Teichmüller space both quantities areconstant, but on Hitchin components these functions vary and provide infor-mation about the geometry of the representations. There are sequences ofrepresentations along which the entropy goes to zero [117, 118]. The entropy


is in fact bounded above. Tholozan for n = 3 [109], and Potrie and Sam-barino in general, establish an entropy rigidity theorem: A representationsaturates the upper bound for the entropy if and only if it is a principalFuchsian representation [100, 96]. This has consequences for the volume ofthe minimal surface in the symmetric space associated to the representation.A key aspect in the work of Potrie and Sambarino has been the regularityof the map ξ : S1 → G/B of a Hitchin representation.

For maximal representations which are not in the Hitchin component muchless is known. One obstacle is the missing regularity of the boundary mapξ : S1 → G/S, which has rectifiable image, but is in general not smooth.In [50] Glorieux and Monclair study the entropy of Anti-de-Sitter Quasi-Fuchsian representations π1(S) → SO(2, n), some of their methods methodsmight also be useful to investigate maximal representation π1(S) → SO(2, n).

Task 22. Investigate the topological entropy of maximal representations.Find and characterize sequences along which the entropy goes to zero. Findbounds for the topological entropy and geometrically characterize the repre-sentations that saturate these bounds.

A lot of the geometry of Teichmüller space can be recovered from geodesiccurrents associated to the representations and from their intersection [15].Recently Martone and Zhang [91] have associated geodesic currents to pos-itively ratioed representations, a class that includes Hitchin representationsand maximal representations but should include also Θ-positive represen-tations. In [24] Bridgeman, Canary, Labourie, and Sambarino define theLiouville current for a Hitchin representation which they use to constructthe Liouville pressure metric. From the intersection of the geodesic cur-rents one can recover the periods of the reparametrization of the geodesicflow and the periods of crossratios associated to the representation [78, 66].These geodesic currents and the corresponding crossratio functions also playan important role for the next topic we discuss.

11. Compactifications

Classical Teichmüller space admits various non-homeomorphic compact-ifications. One such compactification is the marked length spectrum com-pactification. The marked length spectrum of a representation ρ : π1(S) →PSL(2,R) associates to any conjugacy class in [γ] in π1(S) the translationlength of the element ρ(γ) in H

2. It is a basic result that for Teichmüllerspace the map Φ : T (S) → P(RC) provides an embedding. The closure ofΦ(T (S)) is the marked length spectrum compactification. It is homeomor-phic to the Thurston compactification of T (S) by the space of projectivizedmeasured laminations. This compactification has been reconstructed usinggeodesic currents by Bonahon [15].

The marked length spectrum compactification has been generalized tocompactifications of spaces of representations of finitely generated groupsinto reductive Lie groups by Parreau [93], where the translation length in H

2

is replaced by the vector valued translation length in the symmetric spaceassociated to G. This can be applied to Hitchin components and spacesof maximal representations to provide marked length spectrum compactifi-cations. The investigation of the fine structure of these compactifications


has just begun. A key ingredient are generalizations of the Collar Lemmafrom hyperbolic geometry to Hitchin representations [85] and to maximalrepresentations [30]. See also [81, 113, 41] for generalizations of cross-ratioidentities in the context of higher Teichmüller spaces. In [26] Burger, Iozzi,Parreau and Pozzetti establish a new decomposition theorem for geodesiccurrents, which will play a crucial role in their program to understand themarked length spectrum compactification of maximal representations.

Compactifications of the space of positive representations (when S haspunctures) have been constructed using explicit parametrizations and thetheory of tropicalizations [42, 1, 84].

In all these constructions it is a challenge to give a geometric interpreta-tion of points in the boundary of the compactification. The most naturalis in terms of actions on R-buildings [93, 84, 30]. This naturally general-izes the description of boundary points of Teichmüller space by actions onR-trees. However, Thurston’s compactification gives an interpretation ofboundary points by measured laminations on S. It would be interestingto get a description of boundary points of Hitchin components or spaces ofmaximal representations in terms of geometric objects on S that generalizemeasured laminations, and to relate the compactifications to degenerationsof the geometric structures associated to Hitchin components and maximalrepresentations in low dimensions (see Section 5). For convex real projectivestructures such an interpretation of boundary points in terms of a mixedstructure consisting of a measured lamination and a HEX-metric has beenannounced by Cooper, Delp, Long, and Thistlethwaite [34].

12. Arithmetics

. Discrete Zariski dense subgroups of semisimple Lie groups have beenstudied for a long time. In recent years there has been a revived interest fromnumber theory in discrete Zariski dense subgroups, which are contained inarithmetic lattices without being lattices themselves. Such groups have beencoined thin groups [101]. Hitchin representations and maximal representa-tions, and more generally Anosov representations, provide many examples ofdiscrete Zariski dense subgroups of higher rank Lie groups which are not lat-tices. They are not necessarily contained in (arithmetic) lattices. However,there are many Hitchin representations and maximal representations thatare even integral, i.e. up to conjugation contained in the integral points ofthe group G. This is a new feature that arises for higher Teichmüller spacesand is not present in classical Teichmüller space. The group PSL(2,Z) isa non-uniform lattice in PSL(2,R), consequently there is no discrete em-bedding of the fundamental group π1(S) of a closed oriented surface intoPSL(2,R) which takes values in the integer points PSL(2,Z).

For n = 3 the first examples of integral Hitchin representations were givenby Kac and Vinberg [112], infinite families were constructed by Long, Reid,and Thistlethwaite with an explicit description using triangle groups.

Theorem 23. [89] Let Γ = 〈a, b | a3 = b3 = (ab)4 = 1〉 be the (3,3,4) trianglegroup. There is an explicit polynomial map φ : R → Hom(Γ,PSL(3,R))whose image lies in the Hitchin component. For all t ∈ Z, the image of φ(t)


give a Zariski-dense subgroup of PSL(3,Z). The representations φ(t), t ∈ Zare pairwise not conjugate in PGL(3,R).

Since Γ contains subgroups of finite index which are isomorphic to the fun-damental group π1(S) of a closed oriented surface S, Theorem 23 gives rise toinfinitely many, non conjugate integral representations in TH(π1(S),PSL(n,R)).The representations lie on different mapping class group orbits.

In unpublished work with Burger and Labourie we use bending to showthe following

Theorem 24. For n ≥ 5 and odd there are infinitely many pairwise non con-jugate integral representations in the Hitchin component TH(π1(S),PSL(n,R)).These representations lie on different mapping class group orbits.

Task 25. Develop tools to count integral representations in TH(π1(S),PSL(n,R))modulo the action of the mapping class group. Investigate the counting func-tions and their asymptotics.

A first step to start counting is to find appropriate height functions onTH(π1(S),PSL(n,R)) such that there are only finitely many integral rep-resentations of finite height. A height function, inspired by Thurston’sasymmetric metric on Teichmüller space has been proposed by Burger andLabourie.

For more general constructions of surface subgroups in lattices of Liegroups, following the construction of Kahn and Markovic surface subgroupsin three-manifold groups [71] we refer to work of Hamenstädt and Kahn[65], and forthcoming work of Labourie, Kahn, and Mozes [80]. Examples ofintegral maximal representations are constructed in [110].

13. Complex Analytic Theory

In classical Teichmüller theory complex analytic methods and the theoryof quasi-conformal mappings play a crucial role. These aspects are so farlargely absent from higher Teichmüller theory. Dumas and Sanders startedexploring the complex analytic aspects of discrete subgroups of complex Liegroups of higher rank in [40]. They investigate in particular deformations ofHitchin representations and maximal representations in the complexificationsof G, and establish important properties of the complex compact manifoldsM that arise as quotients of domains of discontinuity of these representations(see Section 5). In a forthcoming paper [39] the complex deformation theoryof these representations will be investigated further.

14. Higher dimensional higher Teichmüller spaces

Fundamental groups of surfaces are not the only finitely generated groupsfor which there are special connected components in the representation va-riety Hom(π1(S), G)/G, which consist entirely of discrete and faithful rep-resentations. This phenomenon also arises for fundamental groups of higherdimensional manifolds, and even for more general finitely generated hyper-bolic groups. The main examples are convex divisible representations, whichhave been introduced and studied by Benoist in a series of papers, starting


with [12]. They are generalizations of convex real projective structures onsurfaces, and exist in any dimension.

Let N be a compact hyperbolic manifold of dimension n and π1(N) itsfundamental group. A representation ρ : π1(N) → PGL(n+ 1,R) is convexdivisible if there exists a strictly ρ(π1(N))- invariant convex domain in RP

n,on which ρ(π1(N)) acts cocompactly.

Theorem 26. [13] The set of convex divisible representations is a union ofconnected components of Hom(π1(N),PGL(n + 1,R))/PGL(n + 1,R) con-sisting entirely of discrete and faithful representations.

Barbot [10] showed that a similar phenomenon arise for representationsρ : π1(N) → SO(2, n) that are Anti-de-Sitter Quasi-Fuchsian representation.In fact these are precisely Anosov representations of π1(N) into SO(2, n)with respect to the parabolic subgroup that stabilizes an isotropic line [11].

Theorem 27. [10] The set of Quasi Fuchsian AdS representations in Hom(π1(N),SO(2, n))/SO(2, n)is a connected component consisting entirely of discrete and faithful repre-sentations.

In view of Definition 2 we might call these connected components of therepresentation variety Hom(π1(N), G)/G containing only discrete and faith-ful representations higher dimensional higher Teichmüller spaces.

When N is of dimension two, the notion of Θ-positivity gives us a conjec-tural criterion why and when higher Teichmüller spaces exist. It would beinteresting to find a unifying principle behind the existence of such specialconnected components in Hom(π1(N), G)/G.

Task 28. Find the underlying principle for the existence of connected compo-nents of the representation variety Hom(π1(N), G)/G which consist entirelyof discrete and faithful representations.

A first test case could be to analyze deformations of the representationρ0 : π1(N) → SO(1, n) → SO(k, n), or more generally deformations of repre-sentations ρ0 : π1(N) → SO(1, n) → G, where the centralizer of SO(1, n) inG is compact and contained in the maximal compact subgroup of the Levigroup of a parabolic subgroup containing the parabolic subgroup defined bySO(1, n). We expect all such deformations to be discrete and faithful.

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