Anosov representations-1

ANOSOV REPRESENTATIONS: INFORMAL LECTURE NOTES RICHARD D. CANARY Abstract. “He tried to do his best, but he could not”—Neil Young [224] Contents Preface 3 Early Reviews 5 Part 1. 6 Part 1. 2. 3. 4. 5. 6. 7. 8. Motivation 2. Hyperbolic groups Quasi-isometries and the Milnor-Svarc Lemma Gromov hyperbolic spaces The Gromov boundary Hyperbolic groups and their subgroups Dynamics on the Gromov boundary Representations of hyperbolic groups The Tits Alternative Further topics 11 11 13 17 22 25 30 31 33 Part 3. Convex cocompact representations in rank one Lie groups 9. Teichmüller space: a refresher 10. Hyperbolic geometry in dimension n > 2 11. Convex cocompact representations into Isom(Hn ) 12. Fricke’s Theorem 13. Further topics: Hyperbolic 3-manifolds 37 37 41 44 49 52 Part 4. Convex projective manifolds 14. Basic definitions 15. Geometry of properly convex domains 16. Benoist’s characterizations of strictly convex divisible domains 17. Linear algebra in GL(d, R) 18. Limit maps 19. Translation length, eigenvalues and singular values 53 53 58 62 63 65 67 2021. 1 2 20. 21. 22. 23. Benoist components Projective bending The Hilbert geodesic flow Further topics: Convex projective manifolds 70 72 75 78 Part 5. Anosov representations: Basic properties 24. Geodesic flows and flat bundles 25. Definitions and first principles 26. The symmetric space of SL(d, R) 27. Singular values and Anosov representations 28. Stability 83 84 87 93 98 101 Part 6. Anosov representations: Characterizations and Examples 29. More linear algebra in SL(d, R) 30. The Cartan property 31. Characterization of Pk -Anosov representations 32. Examples 33. Irreducible representations and Benoist representations 34. The relationship between Pk -Anosov and P1 -Anosov representations 35. A characterization in terms of singular values 36. A characterization in terms of eigenvalues 105 105 108 111 114 119 122 124 127 Part 7. Convex cocompactness revisited 37. Definitions and goals 38. First principles 39. Convex cocompact groups which are Anosov 40. Anosov groups which are convex cocompact 41. Zimmer’s criterion 42. Reducible convex cocompact representations 43. Strongly convex cocompact actions 44. Convex cocompactness: Further topics 129 129 130 133 135 137 139 143 147 Part 8. Anosov representations: Extra for Experts 45. Sambarino’s geodesic flow 46. Thermodynamic Formalism and the entropy of the geodesic flow 47. Marked length rigidity 48. Topological restrictions 49. Other Lie groups 50. Open problems References 150 150 153 157 161 164 167 172 ANOSOV REPRESENTATIONS 3 Look what we found in the park in the dark. We will take him home, we will call him Clark. He will live at our house, He will grow and grow. Will our mother like this? We don’t know. ————Dr. Suess [195] Preface The intention of the course, and these notes, was to give an introduction to the theory of Anosov representations, which would be approachable for someone with some background in the study of hyperbolic manifolds, geometric group theory, and/or Teichmüller theory. Basically, I am attempting to write the notes that I would have wanted to be available when I entered the field in 2012. I will not be assuming that you know any Lie theory (since I don’t either), although I may sometimes use aspects of this theory when convenient. Our semisimple Lie groups will almost always be SL(n, R), PSL(n, R) or well-known subgroups like SO(n, 1). In this setting, the Lie theory we use will largely be familiar facts from linear algebra rephrased in (intentionally?) confusing ways so that they hold more generally. Chapter 1 is a (hopefully) motivational introduction. Chapter 2 covers much, but not all, of the basic background theory on hyperbolic groups that we will need later in the notes. If you have some familiarity with hyperbolic groups and/or a willlingness to believe their properties, you can skip this chapter. It exists largely because not all the students in my class had seen the theory of hyperbolic groups before and requested a brief introduction to the theory. Chapter 3 gives a quick review of the theory of convex cocompact hyperbolic manifolds from a viewpoint which should be suggestive of later definitions in the theory of Anosov representations. It can be skipped by readers familiar with the basic theory. In Chapter 4, we give our first higher rank examples, Benoist representations, which are discrete, faithful representations of a hyperbolic group into PSL(d, R) whose images preserve and act cocompactly on strictly convex subset of RPd−1. We will see many key phenomena which occur for Anosov representations first in this more concrete setting. In Chapter 5, we define and develop the basic properties of Anosov representations of hyperbolic groups into SL(d, R). Readers with some background in hyperbolic groups, can begin here if they want to get to the general theory of Anosov representations immediately. In particular, the material in Chapters 5 and 6 does not rely on Chapter 4 (except perhaps for the discussion of 4 basic linear algebra in SL(d, R) which occurs there). In Chapter 6, we derive several characterizations of Anosov representations. In particular, we see that convex cocompact representations into rank one Lie groups and Benoist representations are Anosov. We also introduce Hitchin representations. In chapter 7, we prove some of the foundational results in the emerging study of convex cocompact actions on projective spaces. In chapter 8, we give brief introductions, with occasional proof, to some more advanced topics in the theory. Our approach does have its limitations and here is a (very) partial list of important topics we will not cover, which might naturally be part of an introduction to the field. • Maximal representations of surface groups into Lie groups of Hermitian type as developed by Burger, Iozzi and Wienhard [52]. • The Higgs bundle approach pioneered by Hitchin [114]. • Labourie’s proof [143] that Hitchin representations are Borel Anosov. • The characterizations of Anosov representations in terms of their actions on the associated symmetric spaces and their boundaries by Kapovich, Leeb and Porti [132, 133] • The theory of positive representations developed by Fock and Goncharov [97]. • The symplectic structure on character varieties of surface groups introduced by Goldman [103] and the recent work of Sun, Wienhard, and Zhang ([198, 199, 215]) on the symplectic structure of the Hitchin component. • The compactification of character varieties due to Parreau [174] and more recent work on compactifications of Hitchin components and components of maximal representations by Burger-Iozzi-Parreau-Pozzetti [50]. • Compactifications of the locally symmetric spaces associated to Anosov representations, see Guéritaud-Guichard-Kassel-Wienhard [108] and Kapovich-Leeb [130]. • The collar lemmas for Hitchin representations, discovered by Lee and Zhang [147], and maximal representations, by Burger and Pozzetti [54], see also Martone-Zhang [160]. ANOSOV REPRESENTATIONS Early Reviews You got a way of saying something everyone wants to hear In a way that no one wants to hear it ———–Becky Warren [213] I never knew a man could tell so many lies He had a different story for every set of eyes How can he remember who he’s talking to? ’Cause I know it ain’t me, and I hope it isn’t you —————–Neil Young [221] Them that don’t know him won’t like him and them that do, Sometimes won’t know how to take him. He ain’t wrong, he’s just different but his pride won’t let him, Do things to make you think he’s right. ————Ed and Patsy Bruce [123] 5 6 Part 1. Motivation I’m climbing this ladder My head in the clouds I hope that it matters I’m having my doubts ———–Neil Young [223] Anosov representations were introduced by François Labourie [143] in his study of Hitchin representations. They give a flexible generalization of the theory of convex cocompact representations of hyperbolic groups into rank one Lie groups into the setting of representations of hyperbolic groups into semi-simple Lie groups. Their theory was further developed by GuichardWienhard [109], Kapovich-Leeb-Porti [132, 133], Guéritaud-Guichard-Kassel-Wienhard [107], Bochi-Potrie-Sambarino [32], and others. They now serve as an organizing principle for the geometric and dynamical approach to the so-called Higher Teichmüller theory. The most basic example of a convex cocompact representation into a rank one Lie group is a discrete faithful representation of a surface group into PSL(2, R). Here we view PSL(2, R) as the group Isom+ (H2 ) of orientation-preserving isometries of the hyperbolic plane H2 . So, a discrete faithful representation of π1 (S) into PSL(2, R) gives rise to a hyperbolic structure on the closed surface S. Recall that the upper half-plane model for the hyperbolic plane is given by H2 = {z ∈ C | Im(z) > 0} with line element 1 p 2 dx + dy 2 y In this metric, the geodesics are lines and semi-circles perpendicular to the real line and the group of Möbius transformations with real co-efficients acts as the group of orientationpreserving isometries of H2 , i.e. dshyp = Isom+ (H2 ) = PSL(2, R). The classical Teichmüller space T (S) is the space of all discrete, faithful representations of π1 (S) into PSL(2, R) (up to conjugacy in PGL(2, R)). We may also think of Teichmüller space as the space of all (marked) hyperbolic structures on S (up to isotopy). It is well-known that T (S) is homeomorphic to R6g−6 and is identified with a component of the space of conjugacy classes of representations of π1 (S) into PSL(2, R). Moreover, the mapping class group of (isotopy classes of) orientation-preserving homeomorphisms acts properly discontinuously on T (S). Teichmüller space is rich with structure and can be studied from a multitude of perspectives. Teichmüller space is the motivating example for Higher Teichmüller theory and Anosov representations. The first natural generalization of this theory was to the study of higher-dimensional hyperbolic manifolds. We recall that SO0 (n, 1) ⊂ SL(n + 1, R) is the group of orientation-preserving isometries of hyperbolic space Hn . Here SO(n, 1) is the group of matrices in SL(n+1, R) preserving the indefinite quadratic form with associated diagonal matrix J with entries (1, 1, . . . , 1, −1), i.e. A ∈ SO(n, 1) if and only if JAT J = A−1 , while SO0 (n, 1) is the index two subgroup of ANOSOV REPRESENTATIONS 7 SO(n, 1) preserving the upper sheet of the hyperboloid ~ ∈ Rn+1 | x2 + · · · + x2 − x2 = −1}. H n = {X 1 n n+1 We may then identify Hn with the upper sheet of H n with the metric induced by the indefinite quadratic form. (If this is not familiar to you, we will review this theory in Chapter 3). If ρ : Γ → SO0 (n, 1) is a representation of a finitely presented torsion-free group Γ, then there is an associated orbit map τρ : Γ → Hn given by γ 7→ ρ(γ)(x0 ) where x0 is some pre-chosen point in Hn . Recall that, given a fixed finite presentation of the group, Γ admits a word metric where dΓ (α, β) is the minimal word length of a representative of α−1 β. Then Γ acts by isometries on itself by left multiplication and the orbit map is equivariant in the sense that τρ (αβ) = ρ(α)(τρ (β)) for all α, β ∈ Γ. We say that ρ is convex cocompact if this orbit map is a quasi-isometric embedding, i.e. is a K-bilpschitz embedding on large enough scale. More formally, τρ is a quasi-isometric embedding if there exist K and C so that 1 dΓ (α, β) − C ≤ dHn (τρ (α), τρ (β)) ≤ KdΓ (α, β) + C K for all α, β ∈ Γ. Notice that if τρ is a quasi-isometric embedding then ρ is discrete and faithful. Moreover, it immediately follows, since Hn is Gromov hyperbolic, that Γ is a Gromov hyperbolic group. On the other hand, if ρ is discrete and faithful and ρ(Γ) acts cocompactly on Hn , then the Milnor-Svarc Lemma implies that τρ is a quasi-isometric embedding, so ρ is convex cocompact. However, Mostow proved that if n ≥ 3 and ρ(Γ) acts cocompactly, then any other discrete, faithful representation is conjugate to ρ. So hyperbolic lattices in dimensions above two will not interest us here. In general, there exist discrete, faithful representations which are not convex cocompact. For example, consider   the representation ρ of the free group F2 =< a, b > into PSL(2, R) given 1 1 by ρ(a) = while ρ(b) takes the interior of a disk of radius 18 based at 14 to the exterior 0 1 of a disk of radius 18 based at 34 . One may make an elementary ping-pong argument to see that ρ is discrete and faithful. However, an has word length n, but d(ρ(a)n (i), i) = O(2 log n). Therefore, the orbit map cannot be a quasi-isometric embedding. So, in general the space of convex cocompact representations will not constitute an entire component of the character variety. In fact, we can easily see that ρ is a limit of convex cocompact representations but also a limit of indiscrete representations (since ρ(a) can be approximated by irrational rotations). However, convex cocompact representations are stable, in the sense that if ρ0 ∈ Hom(Γ, SO0 (n, 1)) is convex cocompact, then it has an open neighborhood U so that if ρ ∈ U , then ρ is also convex cocompact. On the other hand, it is a consequence of a lemma of Margulis, that a limit of discrete, faithful representations is itself discrete and faithful, so being discrete and faithful is a closed condition. Teichmüller space has the special property of being both open and closed, since discrete, faithful representation of closed surface groups into PSL(2, R) always have images which act cocompactly on H2 . The key properties we will look for in a generalization of the theory of convex compact representations into the setting of higher rank Lie groups are: 8 (1) Our representations should induce quasi-isometric orbit maps into the quotient symmetric spaces. (2) Our representations should form an open subset of the representation variety. In order to give a quick definition of Anosov representations, it will be useful to recall the singular value decomposition. If A ∈ SL(n, R), then we may write A = LDK where L, K ∈ SO(n) and D is a diagonal matrices with positive entries in descending order along the diagonal, i.e. d11 ≥ d22 ≥> · · · ≥ dnn > 0 and d11 d22 · · · dnn = 1. The matrix D depends only on A, but L and K need not be unique when some of the diagonal entries agree. We let σi (A) = dii and call it the ith singular value of A. More geometically, σi (A) is (half) the length of the ith axis of the ellipsoid A(S n−1 ). In the hyperboloid model for Hn we get a nice simple formula for translation distance in terms of singular values: dHn (en+1 , A(en+1 )) = log σ1 (A) = log σ1 (A) σ2 (A) if A ∈ SO0 (n, 1). (The second equality holds, since σ2 (A) = 1 if A ∈ O(n, 1).) Therefore, a representation ρ : Γ → PSO(n, 1) of finitely generated group is convex cocompact if and only if there exists K, C > 0 so that   σ1 (ρ(g)) 1 d(id, g) − C ≤ log ≤ K d(id, g) + C K σ2 (ρ(g)) for all g ∈ Γ. This observation hopefully motivates a particularly simple definition of an Anosov representation into SL(d, R), due to Kapovich-Leeb-Porti [132] and Bochi-Potrie-Sambarino [32]. If 1 ≤ k ≤ d2 and k ∈ Z, we say that a representation ρ : Γ → SL(d, R) of a finitely generated group into SL(n, R) is Pk -Anosov if there exists K and C so that   1 σk (ρ(g)) d(id, g) − C ≤ log ≤ K d(id, g) + C K σk+1 (ρ(g)) for all g ∈ Γ. A representation is called Anosov if it is Pk -Anosov for some k and is called Borel Anosov if it is Pk -Anosov for all 1 ≤ k ≤ d2 . Notice that these same definitions apply if the image Lie group is PSL(d, R), PGL(d, R), SL± (d, R) or even GL(d, R). It is immediate from this definition that the associated orbit map into the symmetric space Xd = SL(d, R)/SO(d) is a quasi-isometric embedding. (We will review the structure of Xd in an elementary manner later.) It also turns out that this definition again implies that Γ is a Gromov hyperbolic group. We note that this definition is simple to state but when we get down to business we will work more directly with Labourie’s original more dynamical definition. Labourie’s definition more directly indicates that Anosov representations are stable as a simple consequence of the usual stability results for hyperbolic dynamical systems. One key feature of Anosov representations, which we will establish later, is that they admit limit maps into Grassmanians. In particular, if ρ : Γ → SL(d, R) is Pk -Anosov then there exist continuous, ρ-equivariant maps ξρk : ∂Γ → Grk (Rd ) and ξρd−k : ∂Γ → Grd−k (Rd ) ANOSOV REPRESENTATIONS 9 where Gri (Rd ) is the Grassmanian of i-planes in Rd and ∂Γ is the Gromov boundary of the group Γ. Moreover, these limit maps are transverse, i.e. if x 6= y ∈ ∂Γ, then ξρk (x) ⊕ ξρd−k (y) = Rd , compatible, i.e. if x ∈ ∂Γ, then ξρk (x) ⊂ ξρd−k (x), and dynamics-preserving, i.e. if γ + ∈ ∂Γ is the attracting fixed point of an infinite order element γ ∈ Γ, then ξρi (γ + ) is the attracting i-plane of ρ(γ). (We will discuss the Gromov boundary of a hyperbolic group in the next chapter, but in the case of the surface group the boundary is just the circle which one can visualize as the boundary of H2 by embedding π1 (S) into H2 via the orbit map of a Fuchsian representation.) If Γ is finitely presented and ρ : Γ → SO0 (n, 1) is a representation, then ρ is convex cocompact if and only if ρ is P1 -Anosov. Notice that no representation into SO0 (n, 1) can be Pk -Anosov for any 2 ≤ k ≤ n+1 2 , since σ2 (A) = · · · = σn (A) = 1 if A ∈ SO0 (n, 1). So, the theory of Anosov representations is indeed a generalization of the theory of convex cocompact representations. Anosov representations also arise classically as the holonomy maps of strictly convex projective structures on closed n-manifolds. Concretely, a strictly convex (real) projective n-manifold M is the quotient of a strictly convex domain Ω in RPn by a discrete group Γ of projective automorphisms of RPn which preserve Ω. The holonomy map is then a representation of π1 (M ) into the group of projective automorphisms of RPn , i.e. PSL(n + 1, R). There is a natural metric, typically just Finsler, on Ω, called the Hilbert metric, so that the stabilizer of Ω in PSL(n + 1, R) acts as a group of isometries of Ω. Therefore, a convex projective manifold inherits a natural Finsler metric. This generalizes the theory of Fuchsian groups, since Hn may be embedded as a round disk ∆ in RPn , so that the Hilbert metric is the classical hyperbolic metric and the stabilizer of ∆ in PSL(n + 1, R) is PSO(n, 1) which may be identified with Isom+ (Hn ). We will see that if ρ : Γ → PSL(n + 1, R) is the holonomy map of a closed strictly convex projective manifold, then ρ is P1 -Anosov. We will review some of Benoist’s beautiful work on strictly convex projective manifolds in Chapter 4. Hitchin representations are one of the most prominent classes of Anosov representations. A representation ρ : π1 (S) → PSL(d, R) is said to be d-Fuchsian if it can be written in the form τd ◦ σ where σ : π1 (S) → PSL(2, R) is discrete and faithful and τd : PSL(2, R) → PSL(d, R) is the irreducible representation. A representation ρ : π1 (S) → PSL(d, R) is said to be Hitchin if it lies in the same component of Hom(π1 (S), PSL(d, R)) as a d-Fuchsian representation. Hitchin [114] showed that the space of (conjugacy classes of) Hitchin representations of a closed surface 2 of genus g in PSL(n, R) admits a real analytic diffeomorphism to R(n −1)(2g−2) . Labourie [143] showed that Hitchin representations are Borel Anosov. We will introduce Hitchin representations, but Labourie’s seminal result will be beyond our purview. With this definition, it is an easy exercise to check that ρ : Γ → SL(d, R) is Pk -Anosov if and only if Λk ρ : Γ → SL(Λk Rd ) is P1 -Anosov (where Λk ρ is the k th exterior power representation.) So P1 -Anosov representations are in some sense the general case. On the other hand, there are very few known Borel Anosov representations, and we now know many restrictions on such representations. 10 One might naturally ask why we don’t simply look at representations into SL(n, R) whose orbit maps are quasi-isometric embeddings. The basic answer is that quasi-isometric embeddings into non-positively curved spaces are not stable. A very basic example of this phenomenon is a sequence of circles through the origin in the plane with larger and larger radius which converge to a line through the origin. So, there is a sequence of rotations in the plane which converge to a translation. So, there are a sequence of representations of Z into the isometry group of Euclidean space whose limit is a quasi-isometric embedding while none of the representations in the sequence is a quasi-isometric embedding. We will later give examples of quasi-isometric embeddings of the free group on 2 generators into SL(d, R) which fail to be Anosov. A representation ρ : Γ → SO0 (n, 1) is known to be convex cocompact if and only if there exists a convex ρ(Γ)-invariant subset C of Hn , so that C/ρ(Γ) is compact. (This definition is responsible for the name convex cocompact.) One might attempt to generalize this definition, but Kleiner-Leeb [140] and Quint [181] showed that this does not result in a robust theory, since the only Zariski dense subgroups of SL(d, R) which are convex cocompact in this sense are uniform lattices, i.e. act cocompactly on Xd (or on SL(d, R) itself). The basic issue is that the convex hull construction is “too big” in a non-positively curved space. For√example, the convex hull of the boundary of the positive quadrant in the plane (which is itself a 2-bilipschitz embedding of a line) is the entire positive quadrant, while in the hyperbolic plane the convex hull of the image of any bilipschitz embedding of a line is always contained in a bounded metric neighborhood of the image. However, in Chapter 7, we will see that Danciger-Guéritaud-Kassel [84] and Zimmer [232] have recently pioneered a way to think of Anosov representations as convex cocompact actions on projective spaces.