Geometric structures and representations of discrete groups
We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics “at infinity” for representations of discrete groups into Lie groups.
The goal of this survey is to report on recent results relating geometric structures on manifolds to dynamical aspects of representations of discrete groups into Lie groups, thus linking geometric topology to group theory and dynamics.
1.1. Geometric structures
The first topic of this survey is geometric structures on manifolds. Here is a concrete example as illustration (see Figure 1).
Consider a two-dimensional torus .
(1) We can view as the quotient of the Euclidean plane by , which is a discrete subgroup of the isometry group of (acting by linear isometries and translations). Viewing this way provides it with a Riemannian metric and a notion of parallel lines, length, angles, etc. We say is endowed with a Euclidean (or flat) structure, or a -structure with .
(2) Here is a slightly more involved way to view : we can see it as the quotient of the affine plane by the group generated by the translation of vector and the affine transformation with linear part and translational part . This group is now a discrete subgroup of the affine group . Viewing this way still provides it with a notion of parallel lines and even of geodesic, but no longer with a notion of length or angle or speed of geodesic. We say is endowed with an affine structure, or a -structure with .
(3) There are many ways to endow with an affine structure. Here is a different one: we can view as the quotient of the open subset of by the discrete subgroup of generated by the homothety . This still makes “locally look like” , but now the image in of an affine geodesic of pointing towards the origin is incomplete (it circles around in with shorter and shorter period and disappears in a finite amount of time).
As in Example 1.1, a key idea underlying a large part of modern geometry is the existence of model geometries which various manifolds may locally carry. By definition, a model geometry is a pair where is a manifold (model space) and a Lie group acting transitively on (group of symmetries). In Example 1.1 we encountered and , corresponding respectively to Euclidean geometry and affine geometry. Another important example is (the -dimensional real hyperbolic space) and (its group of isometries), corresponding to hyperbolic geometry. (For we can see as the upper half-plane and , up to index two, as acting by homographies.) We refer to Table 1 for more examples.
The idea that a manifold locally carries the geometry is formalized by the notion of a -structure on : by definition, this is a maximal atlas of coordinate charts on with values in such that the transition maps are given by elements of (see Figure 2). Note that this is quite similar to a manifold structure on , but we now require the charts to take values in rather than , and the transition maps to be given by elements of rather than diffeomorphisms of .
Although general -structures may display pathological behavior (see ), in this survey we will restrict to the two “simple” types of -structures appearing in Example 1.1, to which we shall give names to facilitate the discussion:
Type C (“complete”): -structures that identify with a quotient of by a discrete subgroup of acting properly discontinuously;
Type U (“incomplete but still uniformizable”): -structures that identify with a quotient of some proper open subset of by a discrete subgroup of acting properly discontinuously.
Setting or as appropriate, we then have coverings (where denotes universal covers). The charts on are obtained by taking preimages in of open subsets of . Moreover, the basic theory of covering groups gives a natural group homomorphism with image and kernel , called the holonomy.
In this survey, we use the phrase geometric structures for -structures. We shall not detail the rich historical aspects of geometric structures here; instead, we refer to the excellent surveys [57, 58, 59]. We just mention that the notion of model geometry has its origins in ideas of Lie and Klein, formulated in Klein’s 1872 Erlangen program. Influenced by these ideas and those of Poincaré, Cartan and others, Ehresmann  initiated a general study of geometric structures in 1935. Later, geometric structures were greatly promoted by Thurston’s revolutionary work .
1.2. Classifying geometric structures
The fundamental problem in the theory of geometric structures is their classification, namely:
Given a manifold ,
Describe which model geometries the manifold may locally carry;
For a fixed model , describe all possible -structures on .
Problem A.(1) asks how the global topology of determines the geometries that it may locally carry. This has been the object of deep results, among which:
the classical uniformization theorem: a closed Riemann surface may carry a Euclidean, a spherical, or a hyperbolic structure, depending on its genus;
Thurston’s hyperbolization theorem: a large class of -dimensional manifolds, defined in purely topological terms, may carry a hyperbolic structure;
more generally, Thurston’s geometrization program (now Perelman’s theorem): any closed orientable -dimensional manifold may be decomposed into pieces, each admitting one of eight model geometries (see ).
Problem A.(2) asks to describe the deformation space of -structures on . In the simple setting of Example 1.1, this space is already quite rich (see ). For hyperbolic structures on a closed Riemann surface of genus (Example 2.1), Problem A.(2) gives rise to the fundamental and wide-ranging Teichmüller theory.
1.3. Representations of discrete groups
The second topic of this survey is representations (i.e. group homomorphisms) of discrete groups (i.e. countable groups) to Lie groups , and their dynamics “at infinity”. We again start with an example.
Let where is a closed oriented Riemann surface of genus . By the uniformization theorem, carries a complete (“type C”) hyperbolic structure, which yields a holonomy representation as in Section 1.1. Embedding into , we obtain a representation , called Fuchsian, and an associated action of on the hyperbolic space and on its boundary at infinity (the Riemann sphere). The limit set of in is the set of accumulation points of -orbits of ; it is a circle in the sphere . Deforming slightly yields a new representation , called quasi-Fuchsian, which is still faithful, with discrete image, and whose limit set in is still a topological circle (now “wiggly”, see Figure 3). The action of is chaotic on the limit set (e.g. all orbits are dense) and properly discontinuous on its complement.
Example 1.2 plays a central role in the theory of Kleinian groups and in Thurston’s geometrization program; it was extensively studied by Ahlfors, Beardon, Bers, Marden, Maskit, Minsky, Sullivan, Thurston, and many others.
In this survey we report on various generalizations of Example 1.2, for representations of discrete groups into semisimple Lie groups which are faithful (or with finite kernel) and whose images are discrete subgroups of . While in Example 1.2 the group has real rank one (meaning that its Riemannian symmetric space has no flat region beyond geodesics), we also wish to consider the case that has higher real rank, e.g. with . In general, semisimple groups tend to have very different behavior depending on whether their real rank is one or higher; for instance, the lattices of (i.e. the discrete subgroups of finite covolume for the Haar measure) may display some forms of flexibility in real rank one, but exhibit strong rigidity phenomena in higher real rank. Beyond lattices, the landscape of discrete subgroups of is somewhat understood in real rank one (at least several important classes of discrete subgroups have been identified for their good geometric, topological, and dynamical properties, see Section 3.1), but it remains very mysterious in higher real rank. We shall explain some recent attempts at understanding it better.
One interesting aspect is that, even when has higher real rank, discrete subgroups of of infinite covolume may be nonrigid and in fact admit large deformation spaces. In particular, as part of higher Teichmüller theory, there has recently been an active and successful effort to find large deformation spaces of faithful and discrete representations of surface groups into higher-rank semisimple which share some of the rich features of the Teichmüller space of (see Sections 4.3 and 5, and [27, 110]). Such features also include dynamics “at infinity” as in Example 1.2, which are encompassed by a notion of Anosov representation  (see Section 4).
1.4. Flag varieties and boundary maps
Let us be a bit more precise. Given a representation , by dynamics ‘at infinity” we mean the dynamics of the action of via on some flag varieties (where is a parabolic subgroup), seen as “boundaries” of or of its Riemannian symmetric space . In Example 1.2 we considered a rank-one situation where and . A typical higher-rank situation that we have in mind is with and (the Grassmannian of -planes in ) for some .
In the work of Mostow, Margulis, Furstenberg, and others, rigidity results have often relied on the construction of -equivariant measurable maps from or to . More recently, in the context of higher Teichmüller theory [26, 52, 85], it has proved important to study continuous equivariant boundary maps which embed the boundary of a Gromov hyperbolic group (i.e. the visual boundary of the Cayley graph of ) into . Such boundary maps define a closed invariant subset of , the limit set, on which the dynamics of the action by accurately reflects the intrinsic chaotic dynamics of on . These boundary maps may be used to transfer the Anosov property of the intrinsic geodesic flow of into some uniform contraction/expansion properties for a flow on a natural flat bundle associated to and (see Section 4). They may also define some open subsets of on which the action of is properly discontinuous, by removing an “extended limit set” (see Sections 3, 5, 6); this generalizes the domains of discontinuity in the Riemann sphere of Example 1.2.
For finitely generated groups that are not Gromov hyperbolic, one can still define a boundary in several natural settings, e.g. as the visual boundary of some geodesic metric space on which acts geometrically, and the approach considered in this survey can then be summarized by the following general problem.
Given a discrete group with a boundary , and a Lie group with a boundary , identify large (e.g. open in ) classes of faithful and discrete representations for which there exist continuous -equivariant boundary maps . Describe the dynamics of on via .
1.5. Goal of the paper
We survey recent results on -structures (Problem A) and on representations of discrete groups (Problem B), making links between the two topics. In one direction, we observe that various types of -structures have holonomy representations that are interesting for Problem B. In the other direction, starting with representations that are interesting for Problem B (Anosov representations), we survey recent constructions of associated -structures. These results tend to indicate some deep interactions between the geometry of -manifolds and the dynamics of their holonomy representations, which largely remain to be explored. We hope that they will continue to stimulate the development of rich theories in the future.
Organization of the paper
In Section 2 we briefly review the notion of a holonomy representation. In Section 3 we describe three important families of -structures for which boundary maps into flag varieties naturally appear. In Section 4 we define Anosov representations and give examples and characterizations. In Section 5 we summarize recent constructions of geometric structures associated to Anosov representations. In Section 6 we discuss a situation in which the links between geometric structures and Anosov representations are particularly tight, in the context of convex projective geometry. In Section 7 we examine an instance of -structures for a nonreductive Lie group , corresponding to affine manifolds and giving rise to affine Anosov representations. We conclude with a few remarks.
I would like to heartily thank all the mathematicians who helped, encouraged, and inspired me in the past ten years; the list is too long to include here. I am very grateful to all my coauthors, in particular those involved in the work discussed below: Jeffrey Danciger (§5, 6, 7), François Guéritaud (§4, 5, 6, 7), Olivier Guichard (§4, 5), Rafael Potrie (§4), and Anna Wienhard (§4, 5). I warmly thank J.-P. Burelle, J. Danciger, O. Guichard, and S. Maloni for reading earlier versions of this text and making many valuable comments and suggestions, and R. Canary and W. Goldman for kindly answering my questions.
2. Holonomy representations
Let be a real Lie group acting transitively, faithfully, analytically on a manifold , as in Table 1. In Section 1.1 we defined holonomy representations for certain types of -structures. We now give a short review of the notion in general.
|Type of geometry|
|Real projective||stab. in of a line of|
|Complex projective||stab. in of a line of|
Let be a -manifold, i.e. a manifold endowed with a -structure. Fix a basepoint and a chart with . We can lift any loop on starting at to a path on starting at , using successive charts of which coincide on their intersections; the last chart in this analytic continuation process coincides, on an open set, with for some unique ; we set where is the homotopy class of the loop (see Figure 4). This defines a representation called the holonomy (see [57, 59] for details); it is unique modulo conjugation by . This coincides with the notion from Section 1.1; in particular, if with open in and discrete in , and if is simply connected, then is just the natural identification of with .
We shall define the deformation space to be the quotient of the set of marked -structures on (i.e. pairs where is a -manifold and a diffeomorphism) by the group of diffeomorphisms of isotopic to the identity (acting by precomposition). The holonomy defines a map from to the space of representations of to modulo conjugation by . This map may be bijective in some cases, as in Example 2.1 below, but in general it is not. However, when is closed, the so-called Ehresmann–Thurston principle  states that the map is continuous (for the natural topologies on both sides), open, with discrete fibers; in particular, the set of holonomy representations of -structures on is then stable under small deformations.
Let where is the isometry group of the real hyperbolic plane . Let be a closed oriented connected surface of genus . All -structures on are complete. Their holonomy representations are the Fuchsian (i.e. faithful and discrete) representations from to . The deformation space is the Teichmüller space . The holonomy defines a homeomorphism between and the space of Fuchsian representations from to modulo conjugation by .
3. Examples of -structures and their holonomy representations
In this section we introduce three important families of -structures, which have been much studied in the past few decades. We observe some structural stability for their holonomy representations, and the existence of continuous equivariant boundary maps together with expansion/contraction properties “at infinity”. These phenomena will be captured by the notion of an Anosov representation in Section 4.
3.1. Convex cocompact locally symmetric structures in rank one
Let be a real semisimple Lie group of real rank one with Riemannian symmetric space (i.e. is a maximal compact subgroup of ). E.g. for . Convex cocompact groups are an important class of discrete subgroups of which generalize the uniform lattices. They are special cases of geometrically finite groups, for which no cusps appear; see Bowditch [20, 21] for a general theory.
By definition, a discrete subgroup of is convex cocompact if it preserves and acts with compact quotient on some nonempty convex subset of ; equivalently, the complete -manifold (or orbifold) has a compact convex subset (namely ) containing all the topology. Such a group is always finitely generated. A representation is called convex cocompact if its kernel is finite and its image is a convex cocompact subgroup of .
For instance, in Example 1.2 the quasi-Fuchsian representations are exactly the convex cocompact representations from to ; modulo conjugation, they are parametrized by . Another classical example of convex cocompact groups in rank-one is Schottky groups, namely free groups defined by the so-called ping pong dynamics of their generators in .
Here denotes the visual boundary of , yielding the standard compactification of ; for we can see in projective space as in Example 3.2.(1) below. The -action on extends continuously to , and identifies with where is a minimal parabolic subgroup of .
For a convex cocompact representation , the existence of a cocompact invariant convex set implies (by the Švarc–Milnor lemma or “fundamental observation of geometric group theory”) that is a quasi-isometric embedding. This means that the points of any -orbit in go to infinity at linear speed for the word length function : for any there exist such that for all . (This property does not depend on the choice of finite generating subset of defining .) A consequence “at infinity” is that any -orbital map extends to a -equivariant embedding , where is the boundary of the Gromov hyperbolic group . The image of is the limit set of in . The dynamics on is decomposed as in Example 1.2: the action of is “chaotic” on (e.g. all orbits are dense if is nonelementary), and properly discontinuous, with compact quotient, on the complement .
Further dynamical properties were studied by Sullivan: by , the action of on is expanding at each point , i.e. there exist and such that multiplies all distances by on a neighborhood of in (for some fixed auxiliary metric on ). This implies that the group is structurally stable, i.e. there is a neighborhood of the natural inclusion in consisting entirely of faithful representations. In fact, admits a neighborhood consisting entirely of convex cocompact representations, by a variant of the Ehresmann–Thurston principle. For , a structurally stable subgroup of is either locally rigid or convex cocompact, by .
3.2. Convex projective structures: divisible convex sets
Let be the projective linear group and the projective space , for . Recall that a subset of is said to be convex if it is contained and convex in some affine chart, properly convex if its closure is convex, and strictly convex if it is properly convex and its boundary in does not contain any nontrivial segment.
Any properly convex open subset of admits a well-behaved (complete, proper, Finsler) metric , the Hilbert metric, which is invariant under the subgroup of preserving (see e.g. ). In particular, any discrete subgroup of preserving acts properly discontinuously on .
By definition, a convex projective structure on a manifold is a -structure obtained by identifying with for some properly convex open subset of and some discrete subgroup of . When is closed, i.e. when acts with compact quotient, we say that divides . Such divisible convex sets are the objects of a rich theory, see . The following classical examples are called symmetric.
(1) For , let be a symmetric bilinear form of signature on , and be the projective model of the real hyperbolic space . It is a strictly convex open subset of (an ellipsoid), and any uniform lattice of divides .
(2) For , let us see as the space of symmetric real matrices, and let be the image of the set of positive definite ones. The set is a properly convex open subset of ; it is strictly convex if and only if . The group acts on by , which induces an action of on . This action is transitive and the stabilizer of a point is , hence identifies with the Riemannian symmetric space . In particular, any uniform lattice of divides . (A similar construction works over the complex numbers, the quaternions, or the octonions: see .)
Many nonsymmetric strictly examples were also constructed since the 1960s by various techniques; see [13, 35] for references. Remarkably, there exist irreducible divisible convex sets which are not symmetric and not strictly convex: the first examples were built by Benoist  for . Ballas–Danciger–Lee  generalized Benoist’s construction for to show that large families of nonhyperbolic closed -manifolds admit convex projective structures. Choi–Lee–Marquis  recently built nonstrictly convex examples of a different flavor for .
For strictly convex , dynamics “at infinity” are relatively well understood: if divides , then is Gromov hyperbolic  and, by cocompactness, any orbital map extends continuously to an equivariant homeomorphism from the boundary of to the boundary of in . This is similar to Section 3.1, except that now itself is a flag variety (see Table 1). The image of the boundary map is again a limit set on which the action of is “chaotic”, but is now part of a larger “extended limit set” , namely the union of all projective hyperplanes tangent to at points of . The space is the disjoint union of and . The dynamics of on are further understood by considering the geodesic flow on , defined using the Hilbert metric of Remark 3.1; for as in Example 3.2.(1), this is the usual geodesic flow. Benoist  proved that the induced flow on is Anosov and topologically mixing; see  for further properties.
For nonstrictly convex , the situation is less understood. Groups dividing are never Gromov hyperbolic ; for they are relatively hyperbolic , but in general they might not be (e.g. if is symmetric), and it is not obvious what type of boundary (defined independently of ) might be most useful in the context of Problem B. The geodesic flow on is not Anosov, but Bray  proved it is still topologically mixing for . Much of the dynamics remains to be explored.
By Koszul , discrete subgroups of dividing are structurally stable; moreover, for a closed manifold with fundamental group , the set of holonomy representations of convex -structures on is open in . It is also closed in as soon as does not contain an infinite normal abelian subgroup, by Choi–Goldman  (for ) and Benoist  (in general). For , when is a closed surface of genus , Goldman showed that is homeomorphic to , via an explicit parametrization generalizing classical (Fenchel–Nielsen) coordinates on Teichmüller space.
3.3. AdS quasi-Fuchsian representations
We now discuss the Lorentzian counterparts of Example 1.2, which have been studied by Witten  and others as simple models for -dimensional gravity. Let be as in Example 1.2. Instead of taking , we now take and
In other words, we change the signature of the quadratic form defining from (as in Example 3.2.(1)) to . This changes the natural -invariant metric from Riemannian to Lorentzian, and the topology of from a ball to a solid torus. The space is called the anti-de Sitter -space.
The manifold does not admit -structures of type C (see Section 1.1), but it admits some of type U, called globally hyperbolic maximal Cauchy-compact (GHMC). In general, a Lorentzian manifold is called globally hyperbolic if it satisfies the intuitive property that “when moving towards the future one does not come back to the past”; more precisely, there is a spacelike hypersurface (Cauchy hypersurface) meeting each inextendible causal curve exactly once. Here we also require that the Cauchy surface be compact and that be maximal (i.e. not isometrically embeddable into a larger globally hyperbolic Lorentzian -manifold).
To describe the GHMC -structures on , it is convenient to consider a different model for , which leads to beautiful links with -dimensional hyperbolic geometry. Namely, we view as the space of real matrices, and the quadratic form as minus the determinant. This induces an identification of with sending to , and a corresponding group isomorphism from the identity component of acting on , to acting on by right and left multiplication: . It also induces an identification of the boundary with the projectivization of the set of rank-one matrices, hence with (by taking the kernel and the image); the action of on corresponds to the natural action of on .
With these identifications, Mess  proved that all GHMC -structures on are obtained as follows. Let be a pair of Fuchsian representations from to . The group preserves a topological circle in , namely the graph of the homeomorphism of conjugating the action of to that of . For any , the orthogonal of for is a projective hyperplane tangent to at . The complement in of the union of all for is a convex open subset of contained in (see Figure 5) which admits a -invariant Cauchy surface. The action of on via is properly discontinuous and the convex hull of in (called the convex core) has compact quotient by . The quotient is diffeomorphic to , and this yields a GHMC -structure on .
Such -structures, or their holonomy representations , are often called AdS quasi-Fuchsian, by analogy with Example 1.2. Their deformation space is parametrized by , via . Their geometry, especially the geometry of the convex core and the way it determines , is the object of active current research (see [7, 19]). Generalizations have recently been worked out in several directions (see [6, 8, 18] and Section 6.2).
As in Section 3.1, the compactness of the convex core of an AdS quasi-Fuchsian manifold implies that any orbital map extends “at infinity” to an equivariant embedding with image . Here is still a flag variety , where is the stabilizer in of an isotropic line of for . Although has higher rank, the rank-one dynamics of Section 3.1 still appear through the product structure of acting on .
4. Anosov representations
In this section we define and discuss Anosov representations. These are representations of Gromov hyperbolic groups into Lie groups with strong dynamical properties, defined using continuous equivariant boundary maps. They were introduced by Labourie  and further investigated by Guichard–Wienhard . They play an important role in higher Teichmüller theory and in the study of Problem B. As we shall see in Section 4.5, most representations that appeared in Section 3 were in fact Anosov representations.
4.1. The definition
Let be a Gromov hyperbolic group with boundary (e.g. a surface group and a circle, or a nonabelian free group and a Cantor set). The notion of an Anosov representation of to a reductive Lie group depends on the choice of a parabolic subgroup of up to conjugacy, i.e. on the choice of a flag variety (see Section 1.4). Here, for simplicity, we restrict to . We choose an integer and denote by the stabilizer in of an -plane of , so that identifies with the Grassmannian .
By definition, a representation is -Anosov if there exist two continuous -equivariant maps and which are transverse (i.e. for all in ) and satisfy a uniform contraction/expansion condition analogous to that defining Anosov flows.
Let us state this condition in the original case considered by Labourie , where for some closed negatively-curved manifold . We denote by the universal cover of , by the unit tangent bundle, and by the geodesic flow on either or . Let
be the natural flat vector bundle over associated to , where acts on by . The geodesic flow on lifts to a flow on , given by . For each , the transversality of the boundary maps induces a decomposition , where are the forward and backward endpoints of the geodesic defined by , and this defines a decomposition of the vector bundle into the direct sum of two subbundles and . This decomposition is invariant under the flow . By definition, the representation is -Anosov if the following condition is satisfied.
The flow uniformly contracts with respect to , i.e. there exist such that for any , any , and any nonzero and ,
where is any fixed continuous family of norms on the fibers .
See  for an interpretation in terms of metric Anosov flows (or Smale flows).
Condition 4.1 implies in particular that the boundary maps , are dynamics-preserving, in the sense that the image of the attracting fixed point in of any infinite-order element is an attracting fixed point in or of . Thus and are unique, by density of such fixed points in .
We note that -Anosov is equivalent to -Anosov, as the integers and play a similar role in the definition (up to reversing the flow, which switches contraction and expansion). In particular, we may restrict to -Anosov for .
Guichard–Wienhard  observed that an analogue of Condition 4.1 can actually be defined for any Gromov hyperbolic group . The idea is to replace by where is the space of pairs of distinct points in the Gromov boundary of , and the flow by translation by along the factor. The work of Gromov  (see also [31, 92, 95]) yields an appropriate extension of the -action on to , which is properly discontinuous and cocompact. This leads to a notion of an Anosov representation for any Gromov hyperbolic group .
4.2. Important properties and examples
A fundamental observation motivating the study of Anosov representations is the following: if is a semisimple Lie group of real rank one, then a representation is Anosov if and only if it is convex cocompact in the sense of Section 3.1.
Moreover, many important properties of convex cocompact representations into rank-one groups generalize to Anosov representations. For instance, Anosov representations are quasi-isometric embeddings [69, 85]; in particular, they have finite kernel and discrete image. Also by [69, 85], any Anosov subgroup (i.e. the image of any Anosov representation ) is structurally stable; moreover, admits a neighborhood in consisting entirely of Anosov representations. This is due to the uniform hyperbolicity nature of the Anosov condition.
Kapovich, Leeb, and Porti, in a series of papers (see [73, 77, 67]), have developed a detailed analogy between Anosov representations into higher-rank semisimple Lie groups and convex cocompact representations into rank-one simple groups, from the point of view of dynamics (e.g. extending the expansion property at the limit set of Section 3.1 and other classical characterizations) and topology (e.g. compactifications).
Here are some classical examples of Anosov representations in higher real rank.
Let where is a closed orientable surface of genus .
(1) (Labourie ) For , let be the irreducible representation (unique up to conjugation by ). For any Fuchsian representation , the composition is -Anosov for all . Moreover, any representation in the connected component of in is still -Anosov for all . These representations were first studied by Hitchin  and are now known as Hitchin representations.
(3) (Barbot  for ) Let . Any Fuchsian representation , composed with the standard embedding (given by the direct sum of the standard action on and the trivial action on ), defines a -Anosov representation .
In (2), we say that is maximal if it maximizes a topological invariant, the Toledo number , defined for any simple of Hermitian type. If is the Riemannian symmetric space of , then the imaginary part of the -invariant Hermitian form on defines a real -form , and by definition where is any -equivariant smooth map. For , this coincides with the Euler number of . In general, takes discrete values and where is the Euler characteristic of (see ).
While (1) and (3) provide Anosov representations in two of the three connected components of for , it is currently not known whether Anosov representations appear in the third component.
4.3. Higher Teichmüller spaces of Anosov representations
Anosov representations play an important role in higher Teichmüller theory, a currently very active theory whose goal is to find deformation spaces of faithful and discrete representations of discrete groups into higher-rank semisimple Lie groups which share some of the remarkable properties of Teichmüller space. Although various groups may be considered, the foundational case is when for some closed connected surface of genus (see [27, 110]); then one can use rich features of Riemann surfaces, explicit topological considerations, and powerful techniques based on Higgs bundles as in Hitchin’s pioneering work .
Strikingly similar properties to have been found for two types of higher Teichmüller spaces: the space of Hitchin representations of into a real split simple Lie group such as , modulo conjugation by ; and the space of maximal representations of into a simple Lie group of Hermitian type such as or , modulo conjugation by . Both these spaces are unions of connected components of , consisting entirely of Anosov representations (see Examples 4.2.(1)–(2)). Similarities of these spaces to include:
the proper discontinuity of the action of the mapping class group ;
for Hitchin representations to : the topology of ;
Various characterizations of Anosov representations have been developed in the past few years, by Labourie , Guichard–Wienhard , Kapovich–Leeb–Porti [73, 74, 75], Guéritaud–Guichard–Kassel–Wienhard , and others. Here are some characterizations that do not involve any flow. They hold for any reductive Lie group , but for simplicity we state them for . For and , we denote by (resp. ) the logarithm of the -th singular value (resp. eigenvalue) of .
For a Gromov hyperbolic group , a representation , and an integer , the following are equivalent:
is -Anosov (or equivalently -Anosov, see Section 4.1);
there exist continuous, -equivariant, transverse, dynamics-preserving boundary maps and , and
there exist continuous, -equivariant, transverse, dynamics-preserving boundary maps and , and
there exist such that for all ;
there exist such that for all .
Here we denote by the word length with respect to some fixed finite generating subset of , and by the translation length in the Cayley graph of for that subset, i.e. . In a Gromov hyperbolic group the translation length is known to differ only by at most a uniform additive constant from the stable length , and so we may replace by in Conditions (3) and (5).
Condition (4) is a refinement of the condition of being a quasi-isometric embedding, which for is equivalent to the existence of such that for all . We refer to  (CLI condition) or  (Morse condition) for further refinements satisfied by Anosov representations.
By [75, 15], if is any finitely generated group, then the existence of a representation satisfying Condition (4) implies that is Gromov hyperbolic. The analogue for (5) is more subtle: e.g. the Baumslag–Solitar group , which is not Gromov hyperbolic, still admits a faithful representation into satisfying Condition (5) for the stable length , see .
Kapovich–Leeb–Porti’s original proof  of (1) (4) uses the geometry of higher-rank Riemannian symmetric spaces and asymptotic cones. Bochi–Potrie–Sambarino’s alternative proof  is based on an interpretation of (1) and (4) in terms of partially hyperbolic dynamics, and more specifically of dominated splittings for locally constant linear cocycles over certain subshifts. Pursuing this point of view further,  shows that the equivalence (4) (5) of Theorem 4.3 implies the equivalence between nonuniform hyperbolicity (i.e. all invariant measures are hyperbolic) and uniform hyperbolicity for a certain cocycle naturally associated with on the space of biinfinite geodesics of . In general in smooth dynamics, nonuniform hyperbolicity does not imply uniform hyperbolicity.
4.5. Revisiting the examples of Section 3
The boundary maps and dynamics “at infinity” that appeared in most examples of Section 3 are in fact explained by the notion of an Anosov representation:
5. Geometric structures for Anosov representations
We just saw in Section 4.5 that various -structures described in Section 3 give rise (via the holonomy) to Anosov representations; these -structures are of type C or type U (terminology of Section 1.1). In this section, we study the converse direction. Namely, given an Anosov representation , we wish to find:
homogeneous spaces on which acts properly discontinuously via ; this will yield -manifolds (or orbifolds) of type C;
proper open subsets (domains of discontinuity) of homogeneous spaces on which acts properly discontinuously via ; this will yield -manifolds (or orbifolds) of type U.
5.1. Cocompact domains of discontinuity
Domains of discontinuity with compact quotient have been constructed in several settings in the past ten years.
Guichard–Wienhard  developed a more general construction of cocompact domains of discontinuity in flag varieties for Anosov representations into semisimple Lie groups . Here is one of their main results. For , we denote by the closed subspace of the Grassmannian consisting of -planes that are totally isotropic for the standard symmetric bilinear form of signature .
Theorem 5.1 (Guichard–Wienhard ).
Let with and . For any -Anosov representation with boundary map