Anosov representations-5

ANOSOV REPRESENTATIONS 83 Part 5. Anosov representations: Basic properties You can have your buildings and your heavy ‘rithmetic I don’t need no crowded streets or city slicker tricks I just need to be someplace where I can move around Look down at my toes and I can still see the ground ————Rick Hirsch [214] In this chapter we develop the basic theory of Anosov representations into SL(d, R), which we will view as a common generalization of convex cocompact representations into rank one Lie groups and Benoist representations. Anosov representations are discrete, faithful representations and their orbit maps into the associated symmetric spaces Xd = SL(d, R)/SO(d) are quasi-isometric embeddings. Moreover, they are stable in the sense that any small deformation of an Anosov representation remains an Anosov representation. They arose in Labourie’s study [143] of Hitchin representations, but are now an organizing principle for the field of Higher Teichmüller theory. Anosov representations have many flavors. Let 1 ≤ k ≤ d2 be an integer. One simple definition, which follows work of Kapovich-Leeb-Porti [133] and Bochi-Potrie-Sambarino [32] is that ρ : Γ → SL(d, R) is Pk -Anosov if there exists K and C so that if γ ∈ Γ, then   1 σk (ρ(γ)) ≥ d(1, γ) − C. Kd(1, γ) ≥ log σk+1 (ρ(γ)) K This definition immediately implies that ρ is discrete and faithful, and its associated orbit map is a quasi-isometric embedding. Moreover, one can show that it implies that Γ is a hyperbolic group. However, in our discussion we will begin with Labourie’s dynamical definition which allows us more quickly to establish stability. Notice that ρ is Pk -Anosov if and only if the k-fold exterior product Λk ρ : Γ → SL(Λk Rd ) is P1 -Anosov, so the P1 -Anosov representations are the most general class of Anosov representations. More generally, associated to any parabolic subgroup P of a semi-simple Lie group G, there is a notion of a P -Anosov representation ρ : Γ → G and there is an irreducible representation τ : G → SL(d, R) (for some d) so that ρ : Γ → G is P -Anosov if and only if τ ◦ ρ is P1 -Anosov. One might naturally ask why one doesn’t simply generalize one of the definitions of a convex cococompact representation in rank one Lie groups into the higher rank setting. Notice that quasi-isometric embedding into non-positively curved spaces are not stable, e.g. a line in the plane is a limit of circles of larger and larger radius. More convincingly, Guichard [107, Appendix A] exhibits a representation of a free group F2 of rank two into SL(4, R) whose orbit map is a quasi-isometric embedding, but is approximated by representations which aren’t even discrete. (We will discuss a variation of Guichard’s example at the end of Section 27.) So, the class of representations of a hyperbolic group into SL(d, R) whose orbit maps are quasi-isometric embeddings is not stable. One might instead consider faithful representations whose images act properly discontinuously and cocompactly on convex subsets of Xd . However, Kleiner-Leeb [140] and Quint [181] show that any Zariski dense subgroup of SL(d, R) which acts properly discontinuously and cocompactly on a convex subset of Xd is a lattice, so there will not be many examples. For 84 example, no Benoist representation, which does not have image conjugate into PO(d − 1, 1), acts properly discontinuously and cocompactly on a convex subset of Xd . Moreover, any Benoist representation with image in PO(d − 1, 1) does act properly discontinuously and cocompactly on a convex subset of Xd (which is the totally geodesic copy of Hd−1 in Xd given by SO(d − 1, 1)/(SO(d) ∩ SO(d − 1, 1))), so the phenomenon of projective bending demonstrates that this notion of convex cocompactness would also fail to be stable. In Chapter 7, we will discuss a notion of convex cocompactness for actions on projective spaces and its relationship with Anosov representations. 24. Geodesic flows and flat bundles We have already studied the geodesic flow on T 1 Hn , since we may regard Hn as a divisible strictly convex domain in RPn with its associated Hilbert metric. In particular, the geodesic flow on T 1 Hn is Anosov and the (quotient) geodesic flow on the unit tangent bundle of any closed hyperbolic manifold is Anosov and topologically transitive. We may identify T 1 Hn with (∂Hn × ∂Hn − ∆) × R where ∆ = {(z, z) | z ∈ ∂Hn } is the diagonal in Hn × Hn . First notice that a point (x, ~v ) ∈ T 1 Hn determines a point on an oriented geodesic, by considering the oriented geodesic through x in the direction ~v . The space of oriented geodesics is identified with ∂Hn × ∂Hn − ∆ by identifying a geodesic with its forward and backward endpoints. We then identify each oriented geodesic with R by an orientationpreserving isometry which takes 0 to the point on the geodesic closest to ~0 (in the ball model). The resulting parametrization is known as the Hopf parametrization. Notice that in these coordinates the geodesic flow has the simple form φt (x, y, s) = (x, y, s + t) for all t. We will choose the convention that x is the forward endpoint and that y is the backward endpoint. Notice that if M = Hn /Γ is a closed hyperbolic n-manifold, Γ acts on T 1 Hn and T 1 M = T 1 Hn /Γ. f, then M f is Gromov If M is a closed negatively curved manifold with universal cover M 1 f f f hyperbolic and one obtains a Hopf parameterization T M as (∂ M × ∂ M − ∆) × R. Anosov [8] showed that the geodesic flow on T 1 M is topologically transitive and Anosov. If Γ is a torsion-free convex cocompact subgroup of PO(n, 1) then we can consider the flowinvariant subset U (Γ) = {(z, w, s) ∈ T 1 Hn | z 6= w ∈ Λ(Γ) s ∈ R} ∼ = (∂Γ × ∂Γ − ∆) × R of T 1 Hn . Then the geodesic flow {φ̂t } on T 1 Hn /Γ descends to a flow on the compact space b (Γ) = U (Γ)/Γ U b (Γ) as the geodesic flow of the group Γ. and we may regard U More generally, Gromov [106] showed that every hyperbolic group Γ has an associated geb (Γ). We first consider the space G(Γ) of isometric embeddings c : R → CΓ of odesic flow U R into the Cayley graph CΓ . (Notice that the space of isometric embeddings of R into Hn is simply T 1 Hn , so this is a natural generalization to consider). If there is a unique geodesic in CΓ joining any two points in ∂Γ, for example if Γ is a free group, then G(Γ) is identified b (Γ) = G(Γ)/Γ. However, in general this will with (∂Γ × ∂Γ − ∆) × R) and we can define U ANOSOV REPRESENTATIONS 85 not be the case, so one must “collapse” G(Γ) by identifying geodesics with the same endpoints to (somehow) obtain a flow space U (Γ) which admits an action by Γ, which commutes with b (Γ) = U (Γ)/Γ. The details of this construction are worked out by the flow, and then define U Champetier [64] and Mineyev [166]. Theorem 24.1. (Gromov, Champetier, Mineyev) If Γ is a hyperbolic group, then there exists a complete metric on U (Γ) = (∂Γ × ∂Γ − ∆) × R, a flow {φt : U (Γ) → U (Γ)}t∈R , and a properly discontinuous cocompact action of Γ on U (Γ) by isometries such that (1) There exists K and C so that, for all z 6= w ∈ ∂Γ, the map t → (z, w, t) is a (K, C)-quasi-isometric embedding. (2) The flow φt (z, w, s) = (z, w, s + t) for all z 6= w ∈ ∂Γ and s, t ∈ R. (3) Each φt is biLipschitz. (4) The action of Γ commutes with φt for all t and {φt } descends to a topologically transitive b (Γ) = U (Γ)/Γ. flow {φ̂t } on U (5) For all γ ∈ Γ, there exists a function cγ : ∂Γ × ∂Γ − ∆ → R so that γ(x, y, t) = (γ(x), γ(y), cγ (x, y) + t). For our purposes, it is easiest to assume that Γ is just the fundamental group of a closed b (Γ) = T 1 M . This will not impact most of our proofs. The negatively curved manifold M and U one crucial difference is that it is not known whether geodesic flows of general hyperbolic groups are metric Anosov. However, it is known that geodesic flows of groups which admit Anosov representations are metric Anosov (see [45, Section 5] and [69]). We assume throughout the next several sections that Γ is a hyperbolic group. I encourage you to also assume that Γ is torsion-free. b (Γ) which are naturally associated with linear representaThere are various bundles over U bρ over U b (Γ) with fiber Rd determined by a tions. The most classical is the flat vector bundle E representation ρ : Γ → SL(d, R). We first consider the flat vector bundle E(Γ) over U (Γ) given by E(Γ) = U (Γ) × Rd . Notice that the flow {φt } lifts to an “obvious” flow on E(Γ), given by ψt ((z, w, s), v) = ((z, w, s + t), v) or ψt (Z, v) = (φt (Z), v) for all z 6= w ∈ ∂Γ and s, t ∈ R or Z ∈ U (Γ). This flow is (confusingly) called the flow parallel to the flat connection. Suppose that ρ : Γ → SL(d, R) is a representation. Let Γ act on E(Γ) as the group of covering transformations of U (Γ) in the first factor and as ρ(Γ) in the second factor, i.e. if γ ∈ Γ, then γ(Z, v) = (γ(Z), ρ(γ)(v)) for all (Z, v) ∈ E(Γ). Then bρ = E(Γ)/Γ E is the flat vector bundle associated to ρ. Notice that the action of Γ on E(Γ) commutes with bρ . the flow {ψt }, so {ψt } descends to a flow {ψ̂t } on E d Suppose that 1 ≤ k ≤ 2 is a positive integer and ρ : Γ → SL(d, R) is a representation. We say that a pair ξρ : ∂Γ → Grk (Rd ) and θρ : ∂Γ → Grd−k (Rd ) 86 of continuous, ρ-equivariant maps are Pk -transverse limit maps for ρ if ξρ (x) ⊕ θρ (y) = Rd whenever x 6= y ∈ ∂Γ. (Notice that the limit maps of a Benoist representation are a pair of P1 -transverse limit maps.) A pair of transverse Pk -limit maps gives rise to an Γ-equivariant, flow-invariant splitting E(Γ) = Ξ ⊕ Θ where Ξ|(x,y,s) = ξρ (x) and Θρ |(x,y,s) = θρ (y). Hence, the splitting descends to a flow-invariant splitting bρ = Ξ b ⊕ Θ. b E If 1 ≤ k ≤ d2 is a positive integer, we may perform this entire construction replacing Rd with Grk (Rd ) × Grd−k (Rd ). Let E k (Γ) = U (Γ) × Grk (Rd ) × Grd−k (Rd ) and notice that the geodesic flow again lifts to an obvious flow ψtk on E k (Γ). If ρ : Γ → SL(d, R) is a representation, we let Γ act on the first factor by the group of covering transformations b (Γ) and let it act on the second and third factors by the action of ρ(Γ) on the of U (Γ) over U Grassmanian. We then form the k-Grassmannian bundle associated to ρ by letting b k = E k (Γ)/Γ. E ρ Notice that the action of Γ commutes with the flow {ψtk }, so {ψtk } descends to a flow {ψ̂tk } on bρk . E We can then define the vector bundle V k (Γ) over E k (Γ) so that V k (Γ)|(Z,P,Q) = TP Grk (Rd ) and the vector bundle V d−k (Γ) over E k (Γ) with fiber V d−k (Γ)|(Z,P,Q) = TQ Grd−k (Rd ). The flow ψtk lifts to flows ηtk and ηtd−k on V k (Γ) and V d−k (Γ). The action of Γ extends to actions on V k (Γ) and V d−k (Γ) which commute with the flows, so we get flows on the quotient vector bundles Vbρk = V k (Γ)/Γ and Vbρd−k = V d−k (Γ)/Γ bρk . over E A pair of (ξρ , θρ ) of transverse Pk -limit maps for a representation ρ : Γ → SL(d, R) gives rise to a Γ-equivariant flow-equivariant section σ : U (Γ) → E k (Γ) given by σ(x, y, s) = ((x, y, s), ξρ (x), θρ (y)) b (Γ) → E bρk . We can then consider the natural which descends to a flow-equivariant section σ̂ : U b (Γ), each of which admits a flow which pull-back vector bundles σ̂ ∗ (Vbρk ) and σ̂ ∗ (Vbρd−k ) over U b (Γ). “lifts” the flow on U Notice that if P ∈ Grk (Rd ) and Q ∈ Grd−k (Rd ) are transverse, i.e. P ⊕ Q = Rd , then we may identify TP Grk (Rd ) with Hom(P, Q). Basically, φ ∈ Hom(P, Q) is identified with the tangent vector c0 (0) to the path c(t) = graph(tφ) for all t ∈ R, where graph(tφ) is the k-plane {v + tφ(v) | v ∈ P }. Similarly, TQ Grd−k (Rd ) may be identified with Hom(Q, P ). Therefore, if (ξρ , θρ ) is a pair of transverse k-limit maps for a representation ρ : Γ → SL(d, R), then we ANOSOV REPRESENTATIONS 87 b Θ) b and the bundle σ ∗ (Vρd−k ) with the bundle may identify σ̂ ∗ (Vρk ) with the bundle Hom(Ξ, b Ξ). b Hom(Θ, Remark: If we let Pk+ be the stabilizer of the standard k-plane < e1 , . . . , ek > and let Pk− be the stabilizer of the standard complementary (d − k)-plane < ek+1 , . . . , ed >, then Pk+ and Pk− are opposite parabolic subgroups of SL(d, R). Notice that Grk (Rd ) = SL(d, R)/Pk+ and Grd−k (Rd ) = SL(d, R)/Pk− .
T If we let Grk (Rd ) × Grd−k (Rd ) denote the collection of transverse pairs of k-planes and
T d (d − k)-planes, then Grk (R ) × Grd−k (Rd ) may be identified with an open subset of SL(d, R)/Pk+ ∩ Pk− .
T So, if we let E k (Γ)T be the bundle over U (Γ) with fiber Grk (Rd ) × Grd−k (Rd ) , then E k (Γ)T is contained in the bundle over U (Γ) with fiber SL(d, R)/Pk+ ∩ Pk− . We can then pass to a bρk )T which is contained in a bundle over U b (Γ) with fibre SL(d, R)/P + ∩ P − . The quotient (E k k theory of Anosov representations is sometimes formalized with this convention in place. 25. Definitions and first principles We say that a representation ρ : Γ → SL(d, R) is Pk -Anosov if there exists a pair ξρ : ∂Γ → Grk (Rd ) and θρ : ∂Γ → Grd−k (Rd ) b (Γ) → Eρk so that the flow on of transverse Pk -limit maps which determine a section σ̂ : U σ̂ ∗ (Vbρk ) is expanding and the flow on σ̂ ∗ (Vbρd−k ) is contracting, i.e. given continuously varying family of norms on σ̂ ∗ (Vbρk ) and σ̂ ∗ (Vbρd−k ) (both denoted by || · ||), there exists C > 0 and a > 0 so that if v̂ ∈ σ̂ ∗ (Vbρk ) and ŵ ∈ σ̂ ∗ (Vbρd−k ), then k ||η̂−t (v̂)|| ≤ Ce−at ||v̂|| and ||η̂td−k (ŵ)|| ≤ Ce−at ||ŵ|| for all t > 0. We will sometimes refer to a P1 -Anosov representation as projective Anosov (just to make the whole thing seem less like jargon). b Θ) b =Θ b ⊗Ξ b ∗ (where Ξ b ∗ = Hom(Ξ, b R)) Equivalently, we may require that the flow on Hom(Ξ, ∗ b b b b∗ is expanding and that the flow on Hom(Θ, Ξ) = Ξ ⊗ Θ is contracting. Notice that since Ξ ⊗ Θ ∗ ∗ ∗ b ⊗Ξ b , then the flow on Ξ b⊗Θ b is contracting if and only if the flow on Θ b ⊗Ξ b is is dual to Θ b b expanding, so it suffices to assume that the flow on Hom(Θ, Ξ) is contracting. b⊗Θ b ∗ is contracting means “roughly” that all vectors in Ξ b are Notice that the fact that Ξ b “contracted uniformly more” than vectors in Θ. Suppose that γ ∈ Γ has infinite order. Since ξρ (γ + ) and θρ (γ − ) are both preserved by ρ(γ), this suggests that each is a product of eigenspaces and that the moduli of the eigenvalues of the generalized eigenspaces making up ξρ (γ + ) should be strictly greater than the moduli of the eigenvalues of the generalized eigenspaces making up θρ (γ − ). Moreover, the logarithm of the ratio of the eigenvalues should be roughly comparable b (Γ) associated to γ, which is itself roughly comparable to the length of the periodic orbit of U to ||γ|| (by the Milnor-Svarc Lemma). The following lemma begins to make this intuition more precise. 88 Lemma 25.1. ([45, Proposition 2.3]) If ρ : Γ → SL(d, R) is a representation with transverse b Ξ) b is contracting if and only if given any continuous Pk -limit maps ξρ and θρ , then Hom(Θ, bρ , there exists a, C > 0 such that if Ẑ ∈ U b (Γ), v̂ ∈ Ξ| b , ŵ ∈ Θ| b , and t > 0 then norm on E Ẑ Ẑ ||ψ̂t (v̂)|| ||ψ̂t (ŵ)|| ≤ Ce−at ||v̂|| . ||ŵ|| bρ associated to ρ. Recall that {ψ̂t } is the flow on the flat bundle E bρ , which induces a continuous family of Proof. Let || · || be a continuous family of norms on E b Ξ). b Notice that, by construction, the flow {η̂ d−k } on Hom(Θ, b Ξ) b is consistent norms on Hom(Θ, t bρ . with the flow {ψ̂t } on E b b is contracting, so there exists a, C > 0 so that if  ∈ Hom(Θ, b Ξ) b Suppose that Hom(Θ, Ξ) and t > 0, then ||ηtd−k (Â)|| ≤ Ce−at ||Â||. b (Γ), v̂ ∈ Ξ b Z , and ŵ ∈ Θ b Z , choose  ∈ Hom(Θ, b Ξ) b such that Given Ẑ ∈ U Ẑ Â(ŵ) = v̂ and ||Â|| = ||v̂|| . ||ŵ|| (One may do so, by composing orthogonal projection of Θ|Ẑ onto < w >, with respect to || · ||Ẑ , with a linear map taking w to v.) We may lift this whole picture, including the norms ||·||, up to the bundle Hom(Θ, Ξ) over U (Γ), to obtain Z ∈ U (Γ), v ∈ Ξ|Z , w ∈ Θ|Z and A ∈ Hom(Θ, Ξ)|Z . Then, since ψt (v) = ψt (A(w)) = ηtd−k (A)(ψt (w)), we see that ( ) ||ψt (v)|| ||ηtd−k (A)(u)|| ||ψ̂t (v̂)|| ||ηtd−k (A)(ψt (w)))|| = = ≤ max = ||ηtd−k (A)|| = ||η̂td−k (Â)|| ||ψ (w)|| ||ψ (w)|| ||u|| u∈Θ| t t ||ψ̂t (ŵ)|| φt (Z) so ||ψ̂t (v̂)|| ≤ ||η̂t (Â)|| ≤ Ce−at ||Â|| = Ce−at ||v|| . ||w|| ||ψ̂t (ŵ)|| This establishes the forward direction of our claim. b (Γ), v̂ ∈ Ξ| b , ŵ ∈ Θ| b , On the other hand, suppose that there exists C, a > 0 so that if Ẑ ∈ U Ẑ Ẑ and t > 0 then ||ψ̂t (v̂)|| ||v̂|| ≤ Ce−at . || ŵ|| ||ψ̂t (ŵ)|| b Ξ) b and suppose that t > 0, then there exists ŵ ∈ Θ| b (Z) so that Let  ∈ Hom(Θ, t ||η̂td−k (Â)|| = ||η̂td−k (Â)(ŵ)|| ||ŵ|| . Since η̂td−k (Â)(ŵ) = ψt (Â(ψ̂−t (ŵ)), we see that ||η̂td−k (Â)(ŵ)|| Â(ψ̂−t (ŵ))|| ≤ Ceat ≤ Ce−at ||Â||. ||ŵ|| ||ψ̂−t (w)|| which establishes the reverse direction.  ANOSOV REPRESENTATIONS 89 Remark: In Labourie’s definition [143], see also Guichard-Wienhard [109], a representation ρ : Γ → SL(d, R) is Pk -Anosov, if there exists a continuous ρ-equivariant map  T α : ∂Γ × ∂Γ − ∆ → Grk (Rd ) × Grd−k (Rd ) b (Γ) → E b k so that so that the flow on δ̂ ∗ (Vb k ) which gives rise to a flow-invariant section δ̂ : U ρ ρ is expanding and the flow on δ̂ ∗ (Vbρd−k ) is contracting. He then observes that if this is the case, then α must have the form α = ξρ × θρ where ξρ and θρ are a pair of transverse Pk -limit maps for ρ. So, his definition is equivalent to the one we gave above. We can generalize the discussion in Section 17 to the setting of Pk -proximal matrices. We say that A ∈ GL(d, R) is Pk -proximal if |λk (A)| > |λk+1 (A)|. In this case, there is well-defined attracting k-plane and repelling (d − k)-plane, such that v does not lie in the repelling (d − k)-plane, then An (v) converges to the attracting k-plane (i.e. all accumulation points of {An (v)}n∈N lie in the attracting k-plane). Moreover, the attracting k-plane is an attracting fixed point for the action of A on Grk (Rd ) and any k-plane disjoint from the repelling (d − k)-plane will be attracted to the attracting k-plane. Similarly, the repelling (d − k)-plane is an attracting fixed point for the action of A−1 on Grd−k (Rd ). We say that A is Pk -biproximal if both A and A−1 are Pk -proximal. In this language, proximal elements are exactly the P1 -proximal elements. Suppose that Γ is a hyperbolic group, ρ : Γ → SL(d, R) is a representation, 1 ≤ k ≤ d2 , and ξρ : ∂Γ → Grk (Rd ) and θρ : ∂Γ → Grd−k (Rd ) are continuous ρ-equivariant maps. We say (1) ξρ and θρ are Pk -compatible if ξρ (x) ⊂ θρ (x) for all x ∈ ∂Γ, and (2) ξρ and θρ are Pk -dynamics preserving if whenever γ ∈ Γ has infinite order, then a) ρ(γ) is Pk -biproximal, b) ξρ (γ + ) is the attracting k-plane of ρ(γ), and c) θρ (γ − ) is the repelling (d − k)-plane of ρ(γ). Although such an approach involves more familiar methods, the author brutally chose to develop extra structure. ————William Thurston [203] We can now use Lemma 25.1 to show that the limit maps of an Anosov representation are well-behaved. We have chosen a brutally concrete argument, which I hope will allow you to get a hands on sense of what the Anosov condition actually means. Slicker, but less direct, proofs are available. Proposition 25.2. If ρ : Γ → SL(d, R) is a Pk -Anosov representation with transverse Pk -limit maps ξρ : ∂Γ → Grk (Rd ) and θρ : ∂Γ → Grd−k (Rd ), then ξρ and θρ are Pk -compatible and Pk -dynamics preserving. Moreover, there exists J, B > 0, so that   1 |λk (ρ(γ))| ≥ ||γ|| − B J||γ|| ≥ log |λk+1 (ρ(γ))| J 90 for any element γ ∈ Γ. In particular, ρ has finite kernel. Since fixed points of infinite order elements are dense in ∂Γ, by Proposition 5.6, Proposition 25.2 implies that limit maps for Pk -Anosov representations are unique. Proof. Suppose that γ ∈ Γ has infinite order. The flow line (γ − , γ + ) × R of U (Γ) projects to a b (Γ). Let tγ be the length of Qγ . Consider a continuous family of norms periodic orbit Qγ of U bρ which induces a continuous family of norms on Hom(Θ, b Ξ). b These norms lift to || · || on E d equivariant norms on E(Γ) and Hom(Θ, Ξ). Therefore, if ~v ∈ R and Z0 = (γ − , γ + , 0) ∈ U (Γ), then
||ψntγ (Z0 , ~v )|| = || Z0 , ρ(γ)−n (~v ) || for all n ∈ Z. In particular, if ~v is an eigenvector of ρ(γ) with eigenvalue µ, then ||ψntγ (Z0 , ~v )|| = |µ|−n ||(Z0 , ~v )||. We begin with the simplest case when k = 1. In this case, ξρ (γ + ) is an eigenline, so there exists µ so that if ~v ∈ ξρ (γ + ), then A(~v ) = µ~v , so ||ψntγ (Z0 , ~v )|| = |µ|−n ||(Z0 , ~v )||. We claim that |µ| = |λ1 (ρ(γ))| > |λ2 (ρ(γ))|. First notice that the Jordan block J of ρ(γ) whose generalized eigenspace E contains ~v must be one-dimensional. Notice that J has an eigenline, so it must be upper triangular and if [~v ] ∈ P(E), then lim ρ(γ)n ([~v ]) = ξρ (γ + ). However, if E is not one-dimensional then E ∩ θρ (γ − ) is nonempty, which would contradict our assumptions that θρ (γ − ) is ρ(γ)-invariant and transverse to ξρ (γ + ). Now suppose that the generalized eigenspace E2 of another Jordan block has eigenvalue µ2 and |µ2 | ≥ |µ|. If E2 contains an eigenline < w ~ >, then U =< ~v , w ~ > must intersect θρ (γ − ) in a non-trivial vector ~u = a~v + bw ~ (with b 6= 0). Then,   a b ||ψntγ (Z0 , ~u)|| = || Z0 , n ~v + n w ~ ||. µ µ2 So, if |µ| < |µ2 |, then ||ψntγ (Z0 , ~u)|| ∼ 1 u)||. So, in either case |µ|n ||(Z0 , ~ lim inf b ~ |µ2 |n ||(Z0 , w)||, while if |µ| = |µ2 |, then ||ψntγ (Z0 , ~u)|| = ||ψntγ (Z0 , ~v )|| ||(Z0 , ~v )|| ≥ ||ψntγ (Z0 , ~u)|| ||(Z0 , ~u)|| which contradicts Lemma 25.1, since (Z0 , ~v ) ∈ Ξ and (Z0 , ~u) ∈ Θ. If E2 does not contain an eigenline, then either there exists s > 0 so that E2 contains an eigenline for ρ(γ)s (in which case, the argument above, shows that |µ|s > |µ2 |s and hence that |µ| > |µ2 |) or E2 contains a plane W so that ρ(γ)(W ) = W and if w ∈ W − {~0}, then {ρ(γ)n (< w >)} is dense in P(W ). Notice that W ∩ θρ (γ + ) must be non-empty, so, since θρ (γ + ) is invariant under ρ(γ), W ⊂ θρ (γ + ). We can then choose a sequence nr → ∞ so that ρ(γ)nr (< w ~ >) converges to < w ~ > in P(V ). Therefore, lim ||(Z0 , ρ(γ)nr (~u))|| =1 |µ2 |nr ||(Z0 , ~u)|| which again contradicts Lemma 25.1 if |µ2 | ≥ |µ|. ANOSOV REPRESENTATIONS 91 We have established that |µ| = |λ1 (ρ(γ))| > |λ2 (ρ(γ))|, so ρ(γ) is P1 -proximal and ξρ (γ)+ is the attracting eigenline. Notice that θρ (γ − ) is a (d − 1)-plane invariant under ρ(γ) and transverse to the attracting eigenline of ρ(γ), so it must be the repelling hyperplane of ρ(γ). Notice that ξρ (γ + ) ⊂ θρ (γ + ), since ξρ (γ + ) is the repelling eigenline of ρ(γ −1 and θρ (γ + ) is the repelling hyperplane of ρ(γ 1 ). Since ξρ and θρ are continuous and attracting fixed points of infinite order elements are dense in ∂Γ, it follows that ξρ (x) ⊂ θρ (x) for all x ∈ ∂Γ. Therefore, ξρ and θρ are P1 -dynamics preserving and P1 -compatible. Let C and a be the constants provided by Lemma 25.1. If θρ (γ − ) contains an eigenline < w ~> with eigenvalue λ2 (ρ(γ)), then, ||ψtγ (Z0 , ~v )|| |λ2 (ρ(γ))| ||(Z0 , ~v )|| ||(Z0 , ~v )|| = ≤ Ce−atγ ~ ||ψtγ (Z0 , w)|| |λ1 (ρ(γ))| ||(Z0 , w)|| ~ ||(Z0 , w)|| ~ so  log |λ1 (ρ(γ))| |λ2 (ρ(γ))|  ≥ atγ − log(C). If not, there exists ~u ∈ θρ (γ − ) − {0} and nr → ∞ so that   ||(Z0 , ρ(γ)nr (~u))|| =1 lim |λ2 (ρ(γ)|)nr ||(Z0 , ~u)|| which allows us to obtain the same estimate. Notice that this lower bound implies that every element in the kernel of ρ has finite order (since if γ has infinite order, tγ n = ntγ → ∞), so ρ has finite kernel, since every subgroup of a hyperbolic group consisting of finite order elements is finite, see [49, Prop. 2.22]. We now observe that tγ is comparable to ||γ||. The Milnor-Svarc Lemma shows that the orbit map τρ : CΓ → U (Γ) is a (K, C 0 )-quasi-isometry, so, just as in the proof of Corollary 19.2, we see that, if Z ∈ U (Γ), then d(Z, γ(Z)) ≥ 1 ||γ|| − 3C 0 K so tγ ≥ d(Z0 , γ(Z0 )) ≥ 1 ||γ|| − 3C 0 K and  log |λ1 (ρ(γ))| |λ2 (ρ(γ))|  ≥ a||γ|| − 3C 0 K which gives the lower bound we want. (It suffices to establish the lower bound for infinite order elements, since every hyperbolic group contains only finitely many conjugacy classes of finite order elements, see Bridson-Haefliger [49, Thm. 3.Γ.3.2], so we may assume that B is greater than a||α|| K for any finite order element α ∈ Γ.) To get the upper bound, let M = sup{log σ1 (ρ(s)) | s ∈ S} and notice that there exists γ̂ which is conjugate to γ so that d(1, γ̂) = ||γ||. Then,         σ1 (ρ(γ̂)) |λ1 (ρ(γ̂))| |λ1 (ρ(γ))| |λ1 (ρ(γ))| 2M ||γ|| = 2M d(1, γ̂) ≥ 2 log ≥ 2 log = 2 log ≥ 2 log . σd (ρ(γ̂)) |λd (ρ(γ̂))| |λd (ρ(γ))| |λ2 (ρ(γ))| 92 (Notice that the exact same argument shows that   |λk (ρ(γ))| 2M ||γ|| ≥ 2 log |λk+1 (ρ(γ))| for all k.) This upper bound completes the proof when k = 1. We now discuss the alterations needed for the case of general k. (It would probably be fine to skip this portion of the argument when reading through the notes for the first time.) We could also handle the general case by first proving that ρ is Pk -Anosov if and only if Λk ρ is P1 -Anosov and applying the above argument to Λk ρ and translating the results back to the setting of ρ. Let A be the subspace of Rd , spanned by all the generalized eigenspaces with eigenvalue of modulus at least |λk (ρ(γ))| and let B be the subspace of Rd , spanned by all the generalized eigenspaces with eigenvalue of modulus at most |λk+1 (ρ(γ))|. If ρ(γ) is not Pk -proximal, then |λk (ρ(γ))| = |λk+1 (ρ(γ))|, so A has dimension at least k + 1 and B has dimension at least d − k + 1. Therefore, there exist non-trivial vectors in ~a ∈ A ∩ θρ (γ − ) and ~b ∈ B ∩ ξρ (γ + ). Let < ~u > be an accumulation point of {< ρ(γ)n (~a) >}n∈N in P(A ∩ θρ (γ − )), then, by considering the Jordan normal form, one can see that there exists j ≥ k + 1 and nk → ∞ so that   ||(Z0 , ρ(γ)nr (~u))|| = 1. lim |λj (ρ(γ))|nr ||(Z0 , ~u)|| Similarly, if < ~v > be an accumulation point of {< ρ(γ)nr (~b) >} in P(B ∩ ξρ (γ + )) then there exists m ≤ k and a subsequence {n` } of {nk } such that   ||(Z0 , ρ(γ)n` (~u))|| lim = 1. |λm (ρ(γ))|n` ||(Z0 , ~u)|| Then, Lemma 25.1, implies that ||φn` tγ (Z0 , ~u)|| lim = ||φn` tγ (Z0 , ~v )||  λm (ρ(γ))n` λj (ρ(γ))n`  ||(Z0 , ~u)|| =0 ||(Z0 , ~v )|| which contradicts the face that m > k. Therefore, ρ(γ) is Pk -proximal, A is the attracting k-plane and B is the repelling (d − k)-plane. The same argument gives, more generally, that at least one of A ∩ θρ (γ − ) or B ∩ ξρ (γ + ) is trivial. First suppose that A ∩ θρ (γ − ) is trivial, then we see that if ~v ∈ ξρ (γ + ) − A, then {ρ(γ)−n (~v )}n∈N accumulates at θρ (γ − ). Since A and θρ (γ − ) are ρ(γ)-invariant and transverse, this is impossible. Thus, ξρ (γ + ) = A is the attracting k-plane for ρ(γ). If w ~ ∈ θρ (γ − ) − B, then {ρn (w)} accumulates on A = ξρ (γ + ) which is again impossible, since ξρ (γ + ) and θρ (γ − ) are transverse, so θρ (γ − ) = B is the repelling (d − k)-plane. We argue similarly in the case that B∩ξρ (γ + ) is trivial. If ~v ∈ θρ (γ − )−B, then {ρ(γ)n (~v )} accumulates at ξρ (γ + ), which is impossible, so θρ (γ − ) = B. On the other hand, if w ~ ∈ ξρ (γ + ) − A, then {ρ(γ)−n (w)} ~ accumulates at B = θρ (γ − ) which is impossible, so ξρ (γ + ) = A as required. We complete the remainder of the argument, just as in the case where k = 1.  ANOSOV REPRESENTATIONS 93 26. The symmetric space of SL(d, R) Ink mathematics, grey mass ecstatics Noggin elastics, cerebral tactics Cranium classics, brainium domics Denizen omics, grey massmatistics Quantum puree, it’s plain to feel, hard to see Fission antics, abombastics Death antiques, wrong deductions Poor instructions, mass destructions —————–Don Van Vliet [63] In analogy with the action of PO(n, 1) on Hn = PO(n, 1)/O(n), the Lie group SL(d, R) naturally acts on its associated symmetric space Xd = SL(d, R)/SO(d). We will recall the basic theory of this symmetric space and observe that the orbit map of a Pk -Anosov representation is a quasi-isometric embedding. (We will base our discussion on lecture notes by Rich Schwartz [187]. One can also look at the treatment of Bridson and Haefliger [49, Section II.10] which begins with GL(d, R)/O(d) which turns out to be a metric product Xd × R.) It is convenient to regard Xd as the space of symmetric, positive definite, matrices in SL(d, R). (Recall that A ∈ SL(d, R) is positive definite if A(~v ) · ~v > 0 for every non-trivial vector in Rd .) 2 It is then obvious that Xd is a submanifold of Rd . The Lie group SL± (d, R) (of linear transformations of Rd with determinant ±1) acts on Xd by letting A(M ) = AM AT for all A ∈ SL± (d, R) and M ∈ Xd . One may check that AM AT is positive definite, by noticing that if ~v is non-trivial AM AT (~v ) · ~v = M AT ~v · AT ~v = M (AT ~v ) · (AT ~v ) > 0 where the last inequality follows from the fact M is positive definite. Since it is obvious that AM AT is symmetric and has determinant 1, AM AT ∈ Xd . One may also check that this a group action by noticing that if A, B ∈ SL± (d, R) and M ∈ Xd , then AB(M ) = (AB)M (AB)T = ABM B T AT = A(BM B T ) = A(B(M )). We notice that A ∈ SL± (d, R) fixes I ∈ Xd if and only if AAT = I which occurs if and only if A ∈ O(d). Every positive-definite symmetric matrix M can be written as M = gDg −1√where g ∈ SO(d) and D is a diagonal matrix with positive entries and determinant 1. If we let D be the diagonal √ −1matrix whose entries are theTsquare roots of the corresponding entries of D, then A = g Dg ∈ SL(d, R) and A(I) = AA = M . Therefore, SL(d, R) acts transitively on Xd . So, we see that, Xd = SL(d, R)/SO(d) = SL± (d, R)/O(d). The tangent space TI (Xd ) is naturally identified with the space xd of symmetric matrices with trace zero. One can see this concretely by noting that if c : (−, ) → Xd is a smooth path 94 and c(0) = I, then we may write c(s) = gs Ds gs−1 where gs ∈ SO(d) and the diagonal matrices Ds both vary smoothly, then       d d d T T 0 T c (o) = g0 Ds g0 + gs g0 + g0 g . ds 0 ds 0 ds 0 s We then notice that the last two terms cancel since    
d d d d T T T 0= I= gs g0 + g0 g gs gs = ds 0 ds 0 ds 0 ds 0 s d and that E = ds D is a diagonal matrix with trace zero, so c0 (0) = g0 Eg0T ∈ xd . (We will use, 0 s mostly without saying so, the fact that if g ∈ SO(d), then g −1 = g T .) So we define a symmetric blinear from on xd , 1 < V, W >= trace(AB) 4 which is called the Killing form. Notice that the Killing form is positive definite, since if V = gEg −1 ∈ xd is non-zero, then 4 < V, V >= trace(E 2 ) > 0 since trace is conjugacy invariant and E is a non-zero diagonal matrix. Moreover, the Killing form is invariant under the action of O(d), since if g ∈ O(d) and V, W ∈ xd , then 4 < g(V ), g(W ) >= 4 < gV g T , gW g T >= trace(gV g T gW g T ) = trace(gV W g −1 ) = trace(V W ) = 4 < V, W > . (Notice that since g is a linear transformation dg = g.) If M ∈ Xd , then TM Xd = dM (xd ) = M xd , so it is natural to transport the Killing form on xd to TM Xd by letting 1 < V, W >M =< M −1 V, M −1 W >= trace(M −1 V M −1 W ) 2 for all V, W ∈ M xd . So we now have a Riemannian metric on Xd . Notice that SL± (d, R) acts by isometries of this metric, since if M, N ∈ Xd , A ∈ SL± (d, R) and A(M ) = N , then AM AT = N and 1 < A(V ), A(W ) >N = < AV AT , AW AT >N = trace(N −1 AV AT N −1 AW AT ) 2 1 = trace((AT )−1 M −1 A−1 AV AT (AT )−1 M −1 A−1 AW AT ) 4 1 = trace((AT )−1 M −1 V M −1 W AT ) 4 1 = trace(M −1 V M −1 W ) =< V, W >M 4 for all V, W ∈ TM Xd . The following is our key estimate for the distance function on Xd . Proposition 26.1. If M ∈ Xd , then 1p dXd (I, M ) = (log σ1 (M ))2 + · · · + (log σd (M ))2 . 2 Notice that if A ∈ SL(d, R), then A(I) = AAT and σi (AAT ) = σi (A)2 for all i. Therefore, we see that the singular values of A record the translation distance of A at the origin. ANOSOV REPRESENTATIONS 95 Corollary 26.2. If A ∈ SL(d, R), then p dXd (I, A(I)) = (log σ1 (A))2 + · · · + (log σd (A))2 . Let’s warm-up by showing that the “obvious shortest path” from I to M has the expected length. If you feel comfortable with the machinery here, you can skip this and move on to the proof. If F ∈ Xd is a diagonal matrix, let c : [0, 1] → Xd be the smooth path joining I to F given by  t log f  11 e 0 0···0 0  0 et log f22 0 · · · 0 0    c(t) =  . ..  .  . .  0 0 0 · · · 0 et log fdd for all t ∈ [0, 1]. Then  log f11 et log f11  0  c0 (t) =  ..  . 0 0 log f22 et log f22 0 0···0 0···0 0 0 .. . 0 · · · 0 log fdd et log fdd      so 4 < c0 (t), c0 (t) >c(t) = (log f11 )2 + · · · + (log fdd )2 = (log σ1 (F ))2 + · · · + (log σd (F ))2 for all t, so c([0, 1]) has length Z 1p 1p hc0 (t), c0 (t)i dt = (log σ1 (F ))2 + · · · + (log σd (F ))2 . 2 0 In general, since M is symmetric and positive definite, we can write M = KDK T where K ∈ SO(d) and F ∈ Xd is diagonal. Then, since K act on Xd as an isometry and stabilizes I, the path ĉ(t) = Kc(t)K T joins I to M and has length r 1p 1 (log σ1 (F ))2 + · · · + (log σd (F ))2 = (log σ1 (M ))2 + · · · + (log σd (M ))2 . 2 2 2 However, this is not really a proof, since we don’t yet know that these paths are lengthminimizing. Proof of Proposition 26.1: Let Fd ⊂ Xd be the set of diagonal matrices with positive diagonal entries and determinant 1. Then TI Fd = fd is the set of diagonal matrices with trace 0. The restriction of the Killing form to TI Fd is then just 4 < A, B >I = a11 b11 + · · · + add bdd . and the restriction of the Killing form to TF Fd is simply
a11 b11 add bdd 4 < A, B >F = trace F −1 AF −1 B = + ··· + . 2 2 f11 fdd (Notice that TF Fd is also identified with the space of trace-free diagonal matrices so the F −1 term just introduces a scaling factor.) So the map τ : Fd → Ed given by τ (F ) = 2(log f11 , . . . , log fdd ) 96 is an isometry from Fd , with its intrinsic metric, to the hyperplane {~x ∈ Rd | x1 + · · · + xd = 0} with its usual Euclidean metric. We claim that Fd is totally geodesic, so its intrinsic and extrinsic metrics agree. Since Fd asks transitively on itself, it suffices to check that if α : [0, 1] → Xd is a path joining I to F in Xd , then there is a path β : [0, 1] → Fd joining I to F which is no longer than α. We may assume that α(s) = g(s)D(s)g(s)T where g(s) ∈ SO(d) and D(s) ∈ Fd both vary smoothly and g(0) = g(1) = I. So, we simply take β(s) = D(s) and check that ||α0 (s)|| ≥ ||β 0 (s)|| for all s. Notice that α0 (s) = g(s)D0 (s)g(s)T + g 0 (s)D(s)g(s)T + g(s)D(s)g 0 (s)T and β 0 (s) = D0 (s). Since 4 < g(s)D0 (s)g(s)T , g(s)D0 (s)g(s)T ) >α(s) =   trace g(s)D(s)−1 g(s)T g(s)D0 (s)g(s)T g(s)D(s)−1 g(s)T g(s)D0 (s)g(s)T   trace (g(s)D(s)−1 D0 (s)D(s)−1 D0 (s)g(s)T
trace D(s)−1 D0 (s)D(s)−1 D0 (s) = 4 < D0 (s), D0 (s) >β(s) = 4||β 0 (s)||2 = = it suffices to check that C(s) =< g(s)D0 (s)g(s)T , g 0 (s)D(s)g(s)T + g(s)D(s)g 0 (s)T >α(s) = 0. This is an unpleasant calculation which we include for completeness (but you should feel free to skip it). First notice that α(s)−1 g(s)D0 (s)g(s)T = g(s)D(s)−1 g(s)T g(s)D0 (s)g(s)T = g(s)Ω(s)g(s)T where Ω(s) = D(s)−1 D0 (s) is diagonal, and α(s)−1 (g 0 (s)D(s)g(s)T + g(s)D(s)g 0 (s)T ) = g(s)D(s)−1 g(s)T g 0 (s)D(s)g(s)T + g(s)D(s)−1 g(s)T g(s)D(s)g 0 (s)T = g(s)D(s)−1 g(s)T g 0 (s)D(s)g(s)T + g(s)g 0 (s)T So 4C(s) = = = = =   trace g(s)Ω(s)g(s)T g(s)D(s)−1 g(s)T g 0 (s)D(s)g(s)T + g(s)Ω(s)g(s)T g(s)g 0 (s)T   trace g(s)Ω(s)D(s)−1 g(s)T g 0 (s)D(s)g(s)T + g(s)Ω(s)g 0 (s)T   trace D(s)g(s)T g(s)Ω(s)D(s)−1 g(s)T g 0 (s) + Ω(s)g 0 (s)T g(s)   trace D(s)Ω(s)D(s)−1 g(s)T g 0 (s) + Ω(s)g 0 (s)T g(s)   trace Ω(s)g(s)T g 0 (s) − Ω(s)g(s)T g 0 (s) = 0 (In the transition from the second line to the third line we used the fact that trace(AB +CD) = trace(BA + DC) and in the transition from the fourth line to the fifth line we used the facts that diagonal matrices commute and that g 0 (s)T g(s) + g(s)T g 0 (s) = 0, which one obtains by ANOSOV REPRESENTATIONS 97 differentiating the equation g(s)T g(s) = I.) This completes our proof that Fd is totally geodesic, so if F ∈ Fd , then 1p d(I, F ) = (log σ1 (F ))2 + · · · + (log σd (F ))2 . 2 In general, we can write M ∈ Xd as M = KF K T where K ∈ SO(d) and F ∈ Fd . Then, since K acts on Xd as an isometry and stabilize I, p p 2d(I, A) = 2d(I, F ) = (log σ1 (F ))2 + · · · + (log σd (F ))2 = (log σ1 (A))2 + · · · + (log σd (A))2 .  We have proven that Fd is a totally geodesic (d−1)-submanifold of Xd isometric to Euclidean space. It is called a maximal flat, since it is known that every isometrically embedded, totally geodesic copy of Euclidean space has dimension at most d−1. The symmetric space Xd contains many maximal flats, since every translate of Fd by an element of SL(d, R) is a maximal flat. (In fact, all maximal flats are of this form, see [49, Proposition 10.45].) For example, any two points in Xd are contained in a single maximal flat. The quotient SO0 (d − 1, 1)/SO(d − 1) is a totally geodesic copy of Hd−1 in Xd . If we embed SL(2, R) in the upper left-hand corner, then SL(2, R)/SO(2) gives a totally geodesic copy of 21 H2 within Xd . (Here 12 Hd−1 is a simply connected Riemannian manifold with constant sectional curvature −4.) One can compute the intrinsic metric on these submanifolds directly and see that it has the claimed form. One can then use Proposition 26.1 to see that this agrees with the extrinsic metric, and hence that the submanifolds are totally geodesic (see also [49, Theorem 10.58]). One may further show that Xd has non-positive sectional curvature, see [49, Theorem 10.39]. Given an Pk -Anosov representation ρ : Γ → SL(d, R) we can define an orbit map τρ : Γ → Xd by letting τρ (γ) = ρ(γ)(I). We can reprise the proof of Corollary 19.3 to show that τρ is a quasiisometric embedding. Corollary 26.3. If ρ : Γ → SL(d, R) is a Pk -Anosov representation, then the orbit map τρ : Γ → Xd is a quasi-isometric embedding. In particular, ρ(Γ) is discrete. Proof. Let S be a finite symmetric generating set for Γ and let M = max{log σ1 (ρ(s)) | s ∈ S}. Recall that if A, B ∈ SL(d, R), then 1 σ1 (A)σd (B)σ1 (AB) ≤ σ1 (A)σ1 (B) and σd (A−1 ) = . σd (A) Therefore, if γ ∈ Γ, then σ1 (A) ≤ M d(1, γ) where d(1, γ) is the minimal word length of γ in the generating set S. Thus,   σ1 (ρ(γ)) 1 1 2M d(1, γ) ≥ log ≥ dXd (I, ρ(γ)(I)) = dXd (τρ (id), τρ (γ)). σd (ρ(γ)) d d By Proposition 8.4, there exists η ∈ {1, α, β} so that d(1, γ) ≤ 3||γη|| + K. If J and B are the constants from Proposition 25.2, then     |λ1 (ρ(γη))| |λk (ρ(γη))| 1 1 K log ≥ log ≥ ||γη|| − B ≥ d(1, γ) − − B. |λd (ρ(γη))| |λk+1 (ρ(γη))| J 3J J 98 So,  log σ1 (ρ(γ)) σd (ρ(γ))   ≥ log σ1 (ρ(γη)) σd (ρ(γη))   − 2M d(1, η) ≥ log |λ1 (ρ(γη))| |λd (ρ(γη))|  − 2M d(1, η) so, if we set G = max{d(1, α), d(1, β)}, we may combine the last two inequalities to see that     1 1 1 σ1 (ρ(γ)) K ≥ dXd (τρ (id), τρ (γ)) ≥ log d(1, γ) − − B − 2M G . 2 σd (ρ(γ)) 2 3J J Since τρ is ρ-equivariant, this suffices to show that τρ is a (K, C)-quasi-isometric embedding where K = max{3J, 2dM } and C = K J + B + 2M G. Notice that this implies that ρ is discrete, since if {γn } is a sequence of distinct elements in Γ, then d(id, γn ) → ∞, so d(I, ρ(γn )(I)) → ∞.  27. Singular values and Anosov representations In this section we show that if ρ is a Pk -Anosov representation, the ratio of the k th and (k + 1)st singular values grows uniformly exponentially in the word length. One may view this as a strengthening of the fact that the orbit map is a quasi-isometric embedding. Kapovich-LeebPorti [133] and Bochi-Potrie-Sambarino [32] showed that this property characterizes Pk -Anosov representations. Proposition 27.1. (Guichard-Wienhard [109]) If ρ : Γ → SL(d, R) is a Pk -Anosov representation, then there exists D ≥ 1 and L ≥ 0 so that   1 σk (ρ(γ)) ≥ d(1, γ) − L Dd(1, γ) ≥ log σk+1 (ρ(γ)) D for all γ ∈ Γ. Notice that one can use Proposition 27.1 to give a more direct proof of Corollary 26.3, which asserts that orbit maps of Anosov representations are quasi-isometric embeddings into Xd . bρ = Ξ b ⊕Θ b be the splitting of the flat bundle associated to ρ over U b (Γ) and let || · || Proof. Let E bρ (which we may assume all come from bilinear forms on be a continuous family of norms on E the fibers). We can then lift the norms || · || and the splitting E(Γ) = Ξ ⊕ Θ equivariantly to the cover, which is a vector bundle over U (Γ). We will assume, for simplicity, that Γ is torsion-free. By Lemma 25.1, there exist a, C > 0 so that if t > 0, Z ∈ U (Γ), (Z, ~v ) ∈ Ξ|Z and (Z, w) ~ ∈ Θ|Z , then ||(Z, ~v )|| ||ψt (Z, ~v )|| ≤ Ce−at . ||ψt (Z, w)|| ~ ||(Z, w)|| ~ Recall that ||(Z, ~v )|| is just the norm of the vector ~v in the norm on E(Γ)|Z , so we could also re-write ||(Z, ~v )|| = ||~v ||Z , in which case we would write ||ψt (Z, ~v )|| = ||~v ||φt (Z) , since ψt (Z, ~v ) = (φt (Z), ~v ). The above inequality can thus be rewritten as ||~v ||φt (Z) ||~v ||Z ≤ Ce−at ||w|| ~ φt (Z) ||w|| ~ Z which I hope will offer more intuition. We now observe that there exists a compact subset R of U (Γ) so that Γ(R) = U (Γ) and if f/Γ γ ∈ Γ, then there exists Z ∈ R and sγ > 0 so that φsγ (Z) ∈ γ(R). If Γ = π1 (M ) and M = M ANOSOV REPRESENTATIONS 99 f be a compact submanifold is a closed negatively curved manifold, then we can simply let S ⊂ M 1 f f (with boundary) so that Γ(S) = M and then let R ⊂ T M = U (Γ) be the set of all unit tangent vectors at points in S. The same argument works whenever Γ acts properly discontinuously and cocompactly on a CAT(−1)-space X, for example, if Γ is a convex cocompact subgroup of a rank one Lie group, where X is the convex core of the limit set of Γ. One may establish this fact more generally for the Gromov geodesic flow, but we will omit details here. Let γ be a non-trivial element of Γ and suppose that Z ∈ R, sγ > 0 and φsγ (Z) ∈ γ(R). Let W = γ −1 (φsγ (Z)), so, by the equivariance of the norms, ||ρ(γ)−1 (~u)||W = ||~u||φsγ (Z) for all ~u ∈ Rd . So, in this case, the inequality above becomes ||~v ||φsγ (Z) ||ρ(γ)−1 (~v )||W ||~v ||Z = ≤ Ce−asγ −1 ||ρ(γ) (w)|| ~ W ||w|| ~ φsγ (Z) ||w|| ~ Z if ~v ∈ Ξ|Z and w ~ ∈ Θ|Z . However, since R is compact, there exists B > 0 so that if || · ||0 is the standard norm on Rd , then the identity map id : (Rd , || · ||0 ) → (Rd , || · ||Z ) is B-bilipschitz for all Z ∈ R. Therefore, ||ρ(γ)−1 (~v )||0 ||~v ||0 ≤ B 4 Ce−asγ −1 ||ρ(γ) (w)|| ~ 0 ||w|| ~ 0 if ~v ∈ Ξ|Z and w ~ ∈ Θ|Z . Notice that, by equivariance, ρ(γ −1 )(ΞZ ) = ΞW and ρ(γ)−1 (Θ|Z ) = ΘW So we may rewrite the inequality above to say that if ~x ∈ ρ(γ)−1 (Ξ|Z ) = ΞW and ~y ∈ ρ(γ)−1 (Θ|Z ) = ΘW , then ||ρ(γ)(~x)||0 ||~x||0 ≤ B 4 Ce−asγ ||~y ||0 ||ρ(γ)(~y )||0 so ||~x||0 ||ρ(γ)(~x)||0 1 ≥ 4 easγ . ||ρ(γ)(~y )||0 B C ||~y ||0 Now consider the following alternative formulation of the definition of singular values     ||A(v)||0 d σk (A) = sup inf P ∈ Grk (R ) . ||v||0 v∈P −{0} It follows immediately from this definition, since Ξ|W ∈ Grk (Rd ), that σk (ρ(γ)) ≥ σk (ρ(γ)|Ξ|W ) On the other hand, if P ∈ Grk+1 (Rd ), then P ∩ Θ|W is non-trivial, so σk+1 (ρ(γ)) ≤ σ1 (ρ(γ)|Θ|W ). By combining we see that σk (ρ(γ)|Ξ|W ) σk (ρ(γ)) 1 asγ ≥ ≥ e . σk+1 (ρ(γ)) σ1 (ρ(γ)|Θ|W ) BC 4 We now see that the Milnor-Svarc lemma implies that sγ is coarsely comparable to d(id, γ) (although we will only keep track of the lower bound we need). Choose Z0 ∈ R and r > 0 so 100 that R ⊂ B(r, Z0 ). The Milnor-Svarc Lemma implies that there exists (K, C) so that the orbit map τ : Γ → U (Γ) given by τ (γ) = γ(Z0 ) is a quasi-isometric embedding, so d(Z0 , γ(Z0 )) ≥ 1 d(1, γ) − C K which implies that 1 d(1, γ) − C − 2r. K On the other hand, by Theorem 24.1, there exists (K̂, Ĉ) so that all flow-lines are (K̂, Ĉ)-quasiisometric embeddings, so   C + 2r 1 1 d(id, γ) − sγ ≥ d(Z, φsγ (Z)) − Ĉ ≥ + Ĉ K K̂ K̂ K̂ which implies our desired lower bound 
 a 1 − C+2r +Ĉ σk (ρ(γ)) K̂ ≥ e e K K̂ d(id,γ) . 4 σk+1 (ρ(γ)) BC d(Z, φsγ (Z)) ≥ The proof of the upper bound is standard at this point. Let M = sup{σ1 (ρ(s)) | s ∈ S}. Then σ1 (ρ(γ)) σk (ρ(γ)) M 2d(1,γ) ≥ ≥ . σd (ρ(γ)) σk+1 (ρ(γ))  Kapovich, Leeb and Porti [133, Theorem 1.5] proved that uniform growth of the k th singular value gap implies that a representation is Pk -Anosov. Moreover, they show that one doesn’t need to assume that the domain group is Gromov hyperbolic. Bochi, Potrie and Sambarino [32] reproved this result using the theory of dominated splittings. Guéritaud, Guichard, Kassel and Wienhard [107] obtained closely related results. Theorem 27.2. If Γ is a finitely generated group, ρ : Γ → SL(d, R) is a representation, and there exists L > 1 and D ≥ 0 so that   σk (ρ(γ)) 1 Dd(1, γ) ≥ log ≥ d(1, γ) − L σk+1 (ρ(γ)) D for all γ ∈ Γ, then Γ is Gromov hyperbolic and ρ is a Pk -Anosov representation. The key point of the approach of Bochi, Potrie and Sambarino [32] is that if ρ : Γ → SL(d, R) bρ = Ξ b ⊕Θ b is a dominated splitting of the flat bundle is Pk -Anosov, then the resulting splitting E (which can be viewed as a strong form of the contraction property in Lemma 25.1). They use the singular value gap property to show that Γ is hyperbolic and to produce limit maps which give rise to a splitting. They then use work of Bochi and Gourmelon [31] to show that the resulting splitting is dominated and to recover that ρ is Pk -Anosov. Kapovich, Leeb and Porti [133] work more directly with the action of the group on the symmetric space and its boundary. Remarks: One can derive the estimate on the eigenvalue gap in Theorem 25.2 from Proposition n) 27.1, by noticing that if A ∈ SL(d, R), then log λi (A) = lim log σi (A and that ||γ|| is uniformly n d(1,γ n ) comparable to lim n (see [72, Proposition 10.6.4]). ANOSOV REPRESENTATIONS 101 On the other hand if ρ is irreducible (or, more generally, a direct sum of irreducible representations), then one may use a result of Benoist [16], which is stated (and reproven) in the form needed here in [107, Theorem 4.12], to derive Proposition 27.1 from the estimate in Theorem 25.2. Examples: Proposition 27.1 allows us to give a simple example of a discrete faithful representataion ρ : F2 → SL(4, R) whose orbit map is a quasi-isometric embedding, but which is not P1 -Anosov or P2 -Anosov. Since these are the two flavors of Anosov available for SL(4, R), we say that ρ is not Anosov. Our example is based on an example of Guichard [107, Appendix A]. Let F2 =< a, b > be the free group on two generators. Let ρ1 : F2 → SL(2, R) be a representation which projects to a convex cocompact representation of F2 into PSL(2, R). Therefore, log σ1 (µ(ρ1 (γ)) grows linearly in the word length of γ, i.e. there exist J > 0 and B so that 1 Jd(1, γ) ≥ log σ1 (µ(ρ1 (γ)) ≥ d(1, γ) − B J for all γ ∈ Γ. Then let ρ2 : F2 → SL(2, R) be defined so that ρ2 (a) = ρ1 (a) and ρ2 (b) = I. Let ρ : F2 → SL(4, R) be defined by ρ = ρ1 ⊕ ρ2 (where we consider the natural embedding of SL(2, R) ⊕ SL(2, R) into SL(4, R) obtained by putting the first factor in the upper left hand corner and the second factor in the lower right hand corner.) Then, since √ 2 log σ1 (ρ1 (γ)) ≥ σ1 (ρ(γ) ≥ σ1 (ρ1 (γ)) for all γ ∈ F2 . Proposition 26.1 then implies that the orbit map τρ is a quasi-isometric embedding. On the other hand, σ1 (ρ(an )) = σ1 (ρ1 (an )) = σ2 (ρ(an )) for all n, so ρ is not P1 -Anosov. Moreover, σ2 (ρ(bn )) = 1 = σ3 (ρ(bn )) for all n, so ρ is not P2 -Anosov. Therefore, ρ is not an Anosov representation into SL(4, R). However, if we view ρ as a representation into SL(2, R) × SL(2, R) then it is Anosov with respect to the parabolic subgroup {I} × SL(2, R). (See Section 49 for a discussion of what it means to be an Anosov representation into a semi-simple Lie group other than SL(d, R).) In Guichard’s example [107, Appendix A], he chooses ρ1 to be a geometrically finite representation where a is taken to a parabolic (and every parabolic in ρ1 (Γ) is conjugate, in ρ1 (Γ), to a power of ρ(a)). One can then take ρ2 = ρ1 ◦ ι where ι : F2 → F2 is the involution taking a to b, and let ρ = ρ1 ⊕ ρ2 . The orbit map of ρ is a quasi-isometric embedding, yet fails to be Anosov. Moreover, ρ is a limit of P2 -Anosov representations, but also a limit of indiscrete representations. Guichard’s example also fails to be an Anosov representation into SL(2, R) × SL(2, R). Quite recently, Tsouvalas [207] has produced representations into SL(d, R), for d ≥ 4, whose orbit maps are quasi-isometric embeddings into SL(d, R), but which are not Pk -Anosov for any k and are not limits of Pk -Anosov representations for any k. 28. Stability We are now ready to establish the stability of Pk -Anosov representations. Labourie [143, b (Γ)) of Proposition 2.1] first established stability by showing that the image of the section σ̂(U 102 bρk is an isolated hyperbolic set and applying standard results from hyperbolic dynamics. We E will give a more self-contained proof which follows the treatment of Guichard and Wienhard [109, Theorem 5.13]. This type of argument is fairly standard in hyperbolic dynamics, going back at least to the 1960’s. However, it may be best to skip this argument if it feels alien to you, since we will not be using similar arguments elsewhere. Theorem 28.1. If ρ0 : Γ → SL(d, R) is a Pk -Anosov representation, then there exists a neighborhood U of ρ0 in Hom(Γ, SL(d, R)) so that if ρ ∈ U , then ρ is Pk -Anosov. Moreover, the maps Xk : U → C 0 (∂Γ, Grk (Rd )) and Xd−k : U → C 0 (∂Γ, Grd−k (Rd )) given by Xk (ρ) = ξρ and Xd−k (ρ) = θρ are continuous. Proof. Suppose that W is a neighborhood of ρ0 in Hom(Γ, SL(d, R)). We can form the associated k and V b d−k over W × U b (Γ) by first considering flat Grassmanian bundles VbW W j VW = W × U (Γ) × Grj (Rd ) j → W × U (Γ), which is a fibre bundle for all j = 1, . . . , d − 1, and the projection map πj : VW j W W map. Let the flow {ψt }t∈R on VW be given by ψt (ρ, Z, P ) = (ρ, φt (Z), P ) for all t. We let j by the group Γ act on VW γ(ρ, Z, P ) = (ρ, γ(Z), ρ(γ)(P )) and set j j VbW = VW /Γ. k → W ×U b (Γ) with fiber Grj (Rd ). Since all the Then πj descends to give a fiber bundle π̂j : VbW W commute with the action of Γ, they descend to a flow {ψ̂ W } bj t∈R on VW . (Notice that we t t are really just collecting all the Vρj for all ρ ∈ W .) Let ξ0 : ∂Γ → Grk (Rd ) and θ0 : ∂Γ → Grd−k (Rd ) be the limit maps for ρ0 . Let b (Γ) → V d−k | σ̂0 : {ρ0 } × U b (Γ) W {ρ0 }×U be the associated section. Let || · || be a continuous family of norms on the tangent spaces to d−k . Notice that each ψ̂tW is differentiable on every tangent space to every fiber the fibers of VW since it restricts to a map on Grassmanians induced by a linear map. Recall that, by definition, b (Γ)) in the sense that there exists t0 > 0 so that if t ≥ t0 the flow ψ̂tW is contracting on σ̂0 (U d−k W v )|| ≤ 1 ||~ and ~v ∈ Tσ̂0 (Z) b VW |π̂ −1 (ρ0 ,Z) b , then ||D ψ̂t (~ 4 v ||. k We can choose a closed neighborhood W1 of σ0 , contained in W , a section b (Γ) → Vb d−k σ̂ : W1 × U W b (Γ) in V d−k so that and a closed neighborhood B of σ̂0 ({ρ0 } × U W1 (1) σ̂|{ρ0 }×Ub (Γ)) = σ0 , b (Γ)) ⊂ B, (2) σ̂(W1 × U d−k (3) ||Dψ̂tW (~v )|| ≤ 21 ||~v || for all ~v ∈ Tb VW |π̂−1 (π̂j (b)) where b ∈ B and t ≥ t0 , and j (4) ψ̂tW (B) ⊂ B for all t ≥ t0 . ANOSOV REPRESENTATIONS 103 b (Γ) → Vb d−k so that s(W1 × U b (Γ)) ⊂ B. Let S(W1 , B) be the set of continuous sections s : W1 ×U W r For any r ≥ t0 , we can define a uniform contraction F on S(W1 , B), given by saying that F r (s) is the section so that b = ψ̂rW (s(ρ, φ−r (Z))). b F r (s)(ρ, Z) The contraction mapping theorem, then assures that F r has a unique fixed point ν̂ r = lim(F r )n (σ̂). (Here, the norm on the tangent spaces to the fibers gives rise to a continuous family of metrics on the fibers. We then get a metric on S(W1 , B) by considering b s2 (ρ, Z)) b | (ρ, Z) b ∈ W1 × U b (Γ)} d(s1 , s2 ) = max{d(s1 (ρ, Z), where s1 , s2 ∈ S(W1 , B) and the distance is measured in the fibers. Our assumptions imply that d(F r (s1 ), F r (s2 )) ≤ 21 d(s1 , s2 ) if r ≥ t0 and s1 , s2 ∈ S(W1 , B).) r , ν̂ r | r1 and F r2 commute for all Notice that since σ̂0 is fixed by ψ̂W b (Γ) = σ̂0 . Since F {ρ0 }×U r1 , r2 ≥ t0 and the fixed points are unique, ν̂ r1 = ν̂ r2 . So we can simply define ν̂ = ν̂ r for some r ≥ t0 and ν̂ is fixed by all ψ̂tW with t ≥ t0 and hence for all t ∈ R. Moreover, ψ̂tW is contracting b (Γ)). on ν̂(W1 × U d−k d−k We now lift ν̂ to a section ν : W1 × U (Γ) → VW . Since VW = W × U (Γ) × Grd−k (Rd ), we may write ν(ρ, (x, y, s)) = (ρ, (x, y, s), θ(ρ, x, y, s)). In order to fit with our definition of an Anosov representation, we need to check that θ(ρ, x, y, s) depends only on ρ and y. The map θ does not depend on s, since ν is flow-invariant. First suppose that y = γ − for some infinite order element γ ∈ Γ and tγ ≥ t0 , where tγ is the translation distance of γ on U (Γ). If z ∈ ∂Γ − {y}, there exists some tz so that (γ + , γ − , 0) and (z, γ − , tz ) are in the same leaf of the unstable foliation of U (Γ), so

d γ −n (γ + , γ − , 0), (z, γ − , tz − ntγ ) = d (γ + , γ − , −ntγ ), (z, γ − , tz − ntγ ) → 0 where d is some Γ-invariant metric on U (Γ). (We established the existence of an unstable foliation for closed hyperbolic manifolds, and more generally closed strictly convex projective manifolds in Section 22. Mineyev [166, Theorem 60(h)] establishes the existence of tz for all geodesic flows of hyperbolic groups.) So,
d (γ + , γ − , 0), γ n (z, γ − , tz − ntγ ) → 0. But then, since θ is continuous,
d θ(ρ, γ + , γ − , 0), θ(γ n (ρ, z, γ − , z − ntγ ) → 0 where d is our usual metric on Grk (Rd ). Since

1 ||Dφ−ntγ ν(ρ, (γ + , γ − , 0)) || = ||Dρ(γ −n ) ν(ρ, (γ + , γ − , 0)) || ≤ n 2 (where D(ρ(γ −1 ))(ν(ρ, Z)) is the derivative of the action of ρ(γ −1 ) on the fiber {ρ} × {Z} × Grd−k (Rd ) at the point ν(ρ, Z)), and θ is Γ-equivariant, we see that 


d γ −n θ(ρ, γ + , γ − , 0) , γ −n θ(γ n (ρ, z, γ − , z−ntγ ) = d θ(ρ, γ + , γ − , −ntγ , θ(ρ, z, γ − , z−ntγ ) → 0.
Therefore, since θ(ρ, x, y, s) does not depend on s, d θ(ρ, γ + , γ − , 0), θ(ρ, z, γ − , 0) = 0, so θ(ρ, γ + , γ − , 0) = θ(ρ, z, γ − , 0). So, if y = γ − , then θ(ρ, x, y, s) does not depend on x. Since θ is 104 continuous and repelling fixed points of conjugates of γ are dense in ∂Γ, θ(ρ, x, y, s) does not depend on x for any y. If ρ ∈ W1 , we define the limit map θρ : ∂Γ → Grd−k (Rd ) by setting θρ (y) = θ(ρ, x, y, 0) for any choice of x. It is then obvious that the map Xd−k (ρ) = θρ is continuous on W1 . W } is b (Γ) → Vbρk so that the inverse flow {ψ̂−t Now notice that ξ0 determines a section η0 : U b (Γ)). We can then run the whole argument above to obtain a neighborhood contracting on η0 (U b (Γ) → Vb k so that the inverse flow W2 of ρ0 and an inverse flow-invariant section µ̂ : W2 × U W b (Γ)). We then lift µ̂ to µ : W2 × U (Γ) → V k , write it in the form is contracting on µ̂(W2 × U W µ(ρ, (x, y, s)) = (ρ, (x, y, s), ξ(ρ, x, y, s)) and show that ξ depends only on ρ and x. So we define ξρ : ∂Γ → Grk (Rd ), by letting ξρ (x) = ξ(ρ, x, y, 0) for any y and notice that the resulting map Xk (ρ) = ξρ is continuous on W2 . Since transversality is an open condition, and ξ0 and ρ0 are transverse, we can find an open neighborhood U ⊂ W1 ∩ W2 of ρ0 , so that if ρ ∈ U , then θρ and ξρ are transverse. Moreover, the sections constructed from ξρ and θρ have the required contraction properties. Therefore, if ρ ∈ U , ρ is Pk -Anosov.  Hirsch, Pugh and Shub [115, Theorem 3.8] proved that any section obtained in this manner is actually Hölder, which implies that our limit maps are Hölder. Guichard and Wienhard [109, Theorem 5.14] note that since one can choose the contraction constants to be uniform over the neighborhood U in the proof above, one may also choose uniform constants D and L so that if ρ ∈ U and γ ∈ Γ, then   1 λk (ρ(γ)) ≥ ||γ|| − L. D||γ|| ≥ log λk+1 (ρ(γ)) D Along with Proposition 8.3 this is all one needs to generalize Fricke’s Theorem into the setting of Pk -Anosov representations. Let Anosovk (Γ, SL(d, R)) ⊂ X(Γ, SL(d, R) = Hom(Γ, SL(d, R))/SL(d, R) be the set of conjugacy classes of Pk -Anosov representations. Theorem 28.2. (Guichard-Wienhard [109, Corollary 5.4], see also [55, Theorem 6.2]) If Γ is a torsion-free hyperbolic group, d ≥ 2 and 1 ≤ k ≤ d2 , then Out(Γ) acts properly discontinuously on Anosovk (Γ, SL(d, R)). Remarks: (1) Bridgeman, Canary, Labourie and Sambarino [45, Theorem 6.1] show that if ρ0 is a smooth point of the (real algebraic) variety Hom(Γ, SL(d, R)), then one can choose U so that Xk and Xd−k are real analytic on U . The proof is really just a more technically complicated version of the argument above. (2) Kapovich, Leeb and Porti [133, Theorem 1.5] develop a notion of Pk -Morse quasigeodesic and show that a representation is Pk -Anosov if and only if the orbit map takes geodesic in CΓ to (uniform) Pk -Morse-quasigeodesics. They also establish a local-to-global principle [134, Theorem 7.18] which allows them to establish stability of Pk -Anosov representations with a proof which is reminiscent of the proof in rank one, see [134, Theorem 7.33].