Anosov representations-4

ANOSOV REPRESENTATIONS 53 Part 4. Convex projective manifolds From the geometry of his heart he mapped it out He saw the King rise, fitted with armor Set upon a white horse An immaculate cross in his right hand. He advanced toward the enemy And the symmetry, the perfection of his mathematics Caused the scattering of the enemy Agitated, broken, they fled ———–Patti Smith [193] Our first examples of Anosov representations into higher rank Lie group will be the Benoist representations. We say that a discrete, faithful representation ρ : Γ → PGL(n + 1, R) is a Benoist representation if there exists a strictly convex domain in RPn so that ρ(Γ) acts cocompactly on Ω. If Γ is torsion-free, then the quotient M = Ω/ρ(Γ) is a strictly convex real projective manifold. We will first see that in this case Γ is a hyperbolic group and Ω admits a natural Γ-invariant metric, called the Hilbert metric, which has unique geodesics and is Gromov hyperbolic. Moreover, ∂Ω is C 1 and there exists a ρ-equivariant homeomorphim ξρ : ∂Γ → ∂Ω. The most basic examples are provided by projective bending of cocompact representations into PO(n, 1) ⊂ PGL(n + 1, R). Benoist proved that the set Ben(Γ, PGL(n + 1, R)) of Benoist representations of Γ is always a collection of components of Hom(Γ, PGL(n + 1, R)). So, this situation is a natural generalization of the classical Teichmüller theory. You may find it easier when first encountering this material to always assume that we are working in PSL(n + 1, R) and that all our groups are torsion-free. Our main resources for the material in this chapter are the article “Convex divisibles I” by Yves Benoist [20] and the survey article “Around groups in Hilbert geometry” by Ludovic Marquis [158]. 14. Basic definitions Despite all the computations You could just dance to that rock’n’roll station —————–Lou Reed [210] Recall that RPn = Rn+1 /(R − {0}) and that the action of PGL(n + 1, R) on RPn is its group of projective automorphisms. If P is a 2-plane through the origin, L = P(P ) is a projective line. More generally, if Qd+1 is a (d + 1)-plane which passes through the-origin, then P(Q) is a projective d-plane. Projective automorphism of RPn naturally take projective d-planes to projective d-planes and restrict to projective automorphisms. Any hyperplane A in Rn+1 which does not pass through the origin gives rise to an affine chart for RPn such that RPn − A is the projective hyperplane of lines through the origin parallel to A. We say that E ⊂ RPn is a (projective) ellipsoid if it is an ellipsoid in some affine chart. Notice that if E is an ellipsoid in some affine chart, then it is an ellipsoid in every chart containing it. All ellipsoids are projectively equivalent to a round disk. 54 We say that a domain Ω ⊂ RPn is properly convex, if it is a bounded convex subset of some affine chart. Notice that this is equivalent to saying that the closure of Ω is disjoint from some projective hyperplane and every two points in Ω are joined by a projective line segment contained in Ω. We say that a domain Ω ⊂ RPn is strictly convex if it is properly convex and there are no projective line segments contained in ∂Ω. If Ω is properly convex, then we let Aut(Ω) denote the set of projective automorphisms which preserve Ω. A (properly) convex projective manifold M = Ω/Γ is the quotient of a properly convex domain Ω by a subgroup Γ of Aut(Ω) acting freely and properly discontinuously on Ω. If Ω is strictly convex, we say that M = Ω/Γ is a strictly convex projective manifold. The first examples of strictly convex projective manifolds are provided by hyperbolic manifolds. Let H n ⊂ Rn+1 be the two-sheeted hyperboloid whose upper sheet gives the hyperboloid model for Hn , i.e. H n = {~x | x21 + · · · + x2n − x2n+1 = −1}. Then P(H n ) is the unit disc in the affine chart given by A = {~x | xn+1 = 1} and Aut(P(H n )) = PO(n, 1). (Notice that P(H n ) is an ellipsoid in any affine chart containing it.) Then if Γ is any discrete torsion-free subgroup of PO(n, 1), M = P(H n )/Γ is a strictly convex projective n-manifold. ˆ is the positive octant in R3 , i.e. ∆ ˆ = {~x | x1 > 0, x2 > 0, x3 > 0}, then ∆ = P (∆) ˆ If ∆ is a simplex in the affine chart A = {~x | x1 + x2 + x3 = 1}. So ∆ is properly convex, but not strictly convex. Let Γ ⊂ SL(3, R) = PSL(3, R) be generated by the diagonal matrices  4 0 A = 0 21 0 0  0 0   2 0 0 B = 0 14 0 0 0 2 and 1 2 Notice that A and B commute, Γ ⊂ Aut(∆) and if m, n ∈ Z, then An B m  n m 4 2  = 0 0 0 1 2n 4m 0  0 0  2m−n so Γ ∼ = Z ⊕ Z, Γ is discrete and Γ acts freely on ∆. (We will see later that this implies that Γ acts properly discontinuously on ∆). Therefore, M = ∆/Γ is a convex projective manifold and π1 (M ) ∼ =Γ∼ = Z ⊕ Z, so M is homeomorphic to a torus. We say that a group Γ ⊂ PGL(n + 1, R) divides a properly convex domain Ω if Γ ⊂ Aut(Ω) and Γ acts properly discontinuously and cocompactly on Ω. In an abuse of notation, we say that a discrete faithful representation ρ : Γ → PGL(n + 1, R) divides a properly convex domain Ω if ρ(Γ) divides Ω. We will call a representation which divides a strictly convex domain a Benoist representation. All Benoist representations into PSL(3, R) are deformations of Fuchsian representations into PO(2, 1) ⊂ PGL(3, R). The only known Benoist representations which are not deformations of cocompact representations into PO(n, 1) ⊂ PGL(n + 1, R) are certain representations of Coxeter groups in PSL(5, R) by Benoist [24] and representations of fundamental groups of Gromov-Thurston manifolds of dimension n ≥ 4 by Kapovich [129]. We now introduce the Hilbert metric on a properly convex domain. It is a projectively invariant Finsler metric, so descends to a Finsler metric on any quotient convex projective manifold. We first need to introduce the cross-ratio. ANOSOV REPRESENTATIONS 55 Given 4 distinct points x, y, z and w in RPn which lie on a projective line L, we can choose an affine chart A containing the points and define their cross-ratio to be given by [w, x, y, z] = |w − y| · |x − z| . |w − x| · |y − z| Notice that [w, x, y, z] > 1 if the points w, x, y, and z appear in that order on the line L (either from left to right or right to left, so this statement is independent of the ordering of the line). Lemma 14.1. The cross-ratio of four collinear points in RPn is well-defined (i.e. independent of the choice of affine chart A) and invariant under projective automorphisms, i.e. if T ∈ PGL(n + 1, R) and x, y, z and w lie on a projective line then [T (w), T (x), T (y), T (z)] = [x, y, z, w]. Proof. Given two affine charts A and B containing x, y, z and w, let LA = L∩A and LB = L∩B. Then the identity map on L is a projective automorphism from LA ∪ {∞} to LB ∪ {∞}. Now notice that every projective automorphism may be written as a product of some combination of the inversion x → x1 , dilations x → λx and translations x → x + a. It is clear that the cross-ratio is invariant under dilations and translations and it is an easy calculation to check that it is invariant under inversion since 1 w − 1 w − 1 y · 1 x · 1 z − 1 z − 1 x 1 y = |w − y| · |x − z| . |w − x| · |y − z| Now suppose that T ∈ PGL(n + 1, R) and A is an affine chart containing w, x, y and z. Then T (A) is an affine chart containing T (w), T (x), T (y) and T (z). Moreover, T restricts to a projective automorphism from L to T (L). The result then follows as above.  Suppose that Ω is a properly convex domain and that A is an affine chart so that Ω is a bounded convex subset of A. If x, y ∈ Ω, let Lx,y be the line in A containing x and y and let w and z be the endpoints of the line segment Lx,y ∩ Ω̄, so that w, x, y and z occur in that order along the line Lx,y . We then define the Hilbert distance between x and y in Ω to be 1 log ([w, x, y, z])) . 2 It follows from Lemma 14.1 that dH is well-defined and invariant under Aut(Ω). We further claim that it is a metric, which we call the Hilbert metric and that projective lines are geodesics for this metric. We first notice the following obvious monotonicity property of the Hilbert metric. dH Ω (x, y) = Lemma 14.2. If Ω and Ω0 are properly convex domains, Ω ⊂ Ω0 , and x, y ∈ Ω, then H dH Ω (x, y) ≥ dΩ0 (x, y). Moreover, if Lx,y ∩ Ω 6= Lx,y ∩ Ω0 , then H dH Ω (x, y) > dΩ0 (x, y). Proof. Notice that if a ≥ b > 0 and c ≥ 0, then ab ≥ a+c b+c . Moreover, if a > b > 0 and c > 0, then a a+c 0 0 0 0 0 b > b+c . Let w and z be the endpoints of Lx,y ∩ Ω , labelled so that w , x, y and z occur in picture would help 56 that order on the line Lx,y . Then notice that |w − y| ≥ |w − x| and |w0 − x| = |w − x| + |w − w0 | |w−y| |w0 −y| |x−z| |x−z 0 | and |w0 − y| = |w − y| + |w − w0 |, so |w−x| ≥ |w 0 −x| . Similarly, |y−z| ≥ |x−z| , so
1 1 log ([w, x, y, z]) ≥ log [w0 , x, y, z 0 ] = dH Ω0 (x, y). 2 2 Notice that this inequality is strict if w 6= w0 or z 6= z 0 and that this occurs exactly when Lx,y ∩ Ω 6= Lx,y ∩ Ω0 .  dH Ω (x, y) = We use this monotonicity property in the proof that the Hilbert metric is indeed a metric. Lemma 14.3. If Ω is a properly convex domain in RPn , then dH Ω is a complete metric, Aut(Ω) ⊂ Isom(Ω, dH ) and intersections of projective lines with Ω are geodesics (Ω, dH Ω Ω ). Moreover, if p, x, y ∈ Ω and H H dH Ω (x, y) = dΩ (x, p) + dΩ (p, y), then there exists line segments (possibly degenerate) [a, c] and [b, d] in ∂Ω so that the projective lines Lp,x (through p and x) and Lp,y (through p and y) each have one endpoints in [a, c] and the other in [b, d]. In particular, if Ω is strictly convex, then projective line segments are the only geodesics in (Ω, dH Ω ). Proof. Choose an affine chart A so that Ω is a bounded convex subset of A. Notice that Ω dΩ H (x, y) ≥ 0 (since, by our choice of ordering, [w, x, y, z] ≥ 1) and that dH (x, y) = 0 if and H only if x = y (since [w, x, y, z] = 1 if and only if x = y) and dH Ω (x, y) = dΩ (y, x) (since [w, x, y, z] = [z, y, x, w]). The main difficulty is to check the triangle inequality. We first check the case that p lies on Lx,y between x and y. It is calculation to verify that [w, x, p, z] · [w, p, y, z] = |w − y| · |x − z| |w − p| · |x − z| |w − y| · |z − p| · = = [w, x, y, z] |w − x| · |z − p| |w − p| · |z − y| |w − x| · |y − z| so H H dH Ω (x, p) + dΩ (p, y) = dΩ (x, y) picture very much needed which verifies the triangle inequality for collinear points. Moreover, once we check that dH Ω is a metric, it implies that projective line segments are geodesics in (Ω, dH ). Ω Now suppose that p does not lie on Lx,y . Let a and b be the endpoints of Lx,p ∩ Ω̄ and let c and d be the endpoints of Lp,y ∩ Ω̄ (with consistent orderings a, x, p, b and c, p, y, d). Let Q be the quadrilateral spanned by {a, b, c, d} and let e = La,c ∩ Lx,y and f = Lx,y ∩ Lb,d . Notice that e lies between w and x on Lx,y and that f lies between y and z. Let r = La,c ∩ Lb,d and let q = Lp,r ∩ Lx,y . Notice that q lies between x and y on Lx,y . Since a and e lie on a line through r, q and p lie on a line through r, and b and f lie on a line through r, the projective invariance of the cross-ratio implies that [e, x, q, f ] = [a, x, p, b]. Therefore, H H H dH Q (x, q) = dQ (x, p). Similarly, dQ (q, y) = dQ (p, z), so H H dH Q (x, y) = dQ (x, p) + dQ (p, y). H H H However, by construction, dH Q (x, p) = dΩ (x, p) and dQ (p, y) = dΩ (p, y), and Lemma 14.2 implies that dH Ω (x, y) ≥ dQ (x, y). Therefore, H H dH Ω (x, y) ≥ dΩ (x, p) + dΩ (p, y) ANOSOV REPRESENTATIONS 57 which completes the verification of the triangle inequality and hence the proof that dH Ω is a metric. Lemma 14.2 also implies that the inequality above is strict if w 6= e or z 6= f . However, if w = e and z = f , then e and f lie in ∂Ω, so the line segments [a, c] and [b, d] must lie in ∂Ω. However, if Ω is strictly convex, then ∂Ω contains no line segments, so H H dH Ω (x, y) > dΩ (x, p) + dΩ (p, y) unless p lies in the line segment between x and y. It follows that if Ω is strictly convex, then projective line segments are the only geodesics. Finally, notice that if x ∈ Ω, {yn } ⊂ Ω, d(yn , ∂Ω) → 0, and wn and zn are the endpoints of Lx,yn ∩ Ω̄, with the ordering wn , x, yn , zn , then, up to subsequence {yn } converges to a point y ∈ ∂Ω, {wn } converges to a point w ∈ ∂Ω and {zn } converges to a point z ∈ ∂Ω. Notice that y = z since otherwise ∂Ω contains the line segment yz, which would imply, by convexity, that Ly,z is disjoint from Ω which would contradict the fact that x ∈ Ly,z ∩ Ω. Moreover, w 6= y since x ∈ wz. It follows that |yn − zn | → 0 and |wn − x|, |wn − zn | and |zn − x| are all bounded H away from zero, so [wn , x, yn , zn ] → ∞ which implies that dH Ω (x, yn ) → ∞. Therefore, dΩ is complete.  Remarks: (1) Notice that the proof also demonstrates that if Q is a quadrilateral, then geodesics are not unique, since one can easily check that both xy and xp ∪ py are geodesics joining x to y in the quadrilateral Q that we construct in the proof. (2) Pierre de la Harpe [111] showed that if Ω is properly convex and has the unique geodesic H property, then Aut(Ω) = Isom(Ω, dH Ω ). In particular, if Ω is strictly convex, then Aut(Ω) = Isom(Ω, dΩ ). He further shows that Ω has the unique geodesic property if and only if Ω has at most one support plane which intersects ∂Ω in more than one point. We record here the elementary observation that discrete subgroups of Aut(Ω) acts properly discontinuously on Ω. We will use this observation, usually without comment, for the remainder of the notes. Lemma 14.4. If Ω is properly convex and Γ is a discrete subgroup of Aut(Ω), then Γ acts properly discontinuously on Ω. If Γ is also torsion-free, it acts freely, so Ω/Γ is a convex projective manifold. Proof. If Γ does not act properly discontinuously on Ω, there exists a sequence {γn } in Γ and a compact set K so that γn (K)∩K is non-empty for all n. Since Γ acts as a group of isometries of Ω in the Hilbert metric, there exists a subsequence of {γn } converging to an isometry of Ω (by the −1 ) Arzela-Ascoli Theorem). Therefore there exists a sequence {βi } in Γ (each of the form γn γm which converges to an isometry which is the identity on Ω. Thus, as a sequence of elements of PGL(d, R), {βi } converges to the identity in PGL(d, R), which contradicts discreteness. If Γ is torsion-free, then every non-trivial element Γ of Γ has infinite order, so cannot fix any point in Ω (since every power of γ would fix the point, violating proper discontinuity). Therefore, if Γ is discrete and torsion-free, Ω/Γ is a manifold.  We now observe that the Hilbert metric is Finsler. We will only use the Finsler property in our discussion of the Hilbert geodesic flow in Section 22. Recall that a Finsler metric on a manifold M is a continuous family of norms on the tangent bundle of M . From a Finsler metric 58 one obtains a distance function (an actual metric) just as in the Riemannian case by using the norm to define the length of a smooth curve in M and taking the distance between two points to be the infimum of the length of smooth curves joining the two points. If x ∈ Ω and ~v ∈ Tx Ω, let p+ and p− be the endpoints of the line segment {x + t~v | t ∈ R} ∩ Ω̄. We define the Finsler norm associated to the Hilbert metric by letting FΩH (x, ~v ) = = = d dH (x, x + t~v ) dt t=0 Ω  1d (|p− − x| + t|~v |)(|x − p+ |) log 2 dt t=0 |p− x|(|p+ − x| − t|~v |)   1 |~v | 1 + 2 |x − p− | |x − p+ | We observe that if Ω = P(H n ) then PO(n, 1) = Aut(Ω) ⊂ Isom(Ω, dH Ω ), so the isometries act transitively on the space of orthonormal frames in T Ω. It follows that the Hilbert metric must be a metric of constant negative sectional curvature. (Notice that the Finsler metric is Riemannian at the origin and hence everywhere, by transitivity.) It is then a calculation to show that we have chosen the correct normalization to get the actual hyperbolic metric of constant sectional curvature −1. The round disk P(H n ) with the Hilbert metric is often called the Beltrami-Klein model (or Klein or Cayley-Klein model) for hyperbolic geometry. Remark: Kay [137] proved that a Hilbert metric is Riemannian if and only if it is an ellipsoid (i.e. projectively equivalent to P(H n )). For example, you can see that in the case of the simplex, metric -balls are hexagons, so are not asymptotically round. Therefore, the Finsler metric cannot be Riemannian (since metric -balls are always asymptotically elliptical as  → 0 in a Riemannian manifold). 15. Geometry of properly convex domains Since our first explicit example of a Hilbert metric is the hyperbolic metric on the round disk it is natural to ask when the Hilbert metric is Gromov hyperbolic. Notice that the simplex ∆ is quasi-isometric to the Euclidean plane, since it has a quotient which is a torus, so the simplex is not Gromov hyperbolic. We first observe that if the Hilbert metric is Gromov hyperbolic, then the domain must be strictly convex. Proposition 15.1. If Ω is properly convex and its Hilbert metric is Gromov hyperbolic, then Ω is strictly convex. picture needed Proof. Suppose that Ω is Gromov hyperbolic, but not strictly convex. Let [x, y] be a maximal line segment in ∂Ω. Choose z ∈ Ω and sequences {xn } ⊂ xz and {yn } ⊂ yz so that xn → x, yn → y and xn yn is parallel to xy for all n. Let un be the midpoint of xn yn , for all n, so {un } converges to the midpoint u of xy. If Ω is δ-hyperbolic, then, for all n, there exists zn ∈ xn z ∪yn z so that d(un , zn ) ≤ δ. Without loss of generality zn ∈ xz for all n. Since d(un , z) → ∞, we must have d(zn , z) → ∞, so zn → x. Suppose that pn qn = Lun ,zn ∩ Ω̄. Since un zn → ux, we see that pn qn converges to a line segment in ∂Ω containing ux. Since xy is a maximal line segment in ∂Ω, we see that pn → x and {qn } converges up to subsequence to a point in uy. ANOSOV REPRESENTATIONS 59 But then |pn − zn | → 0, but |pn − un | and |qn − zn | are bounded away from zero, so [pn , zn , un , qn ] → ∞ which implies that d(zn , un ) → ∞ which is a contradiction. So, Ω must not be Gromov hyperbolic.  We next observe that when Ω is Gromov hyperbolic, then its Gromov boundary agrees with its topological boundary. Proposition 15.2. If Ω is a strictly convex domain and (Ω, dH Ω ) is hyperbolic, then the Gromov boundary ∂∞ Ω of Ω may be identified with the topological boundary ∂Ω of Ω. More precisely, there exists a homeomorphism from Ω̄ to Ω ∪ ∂∞ Ω which is the identity on Ω. Proposition 15.2 will be a consequence of the following more general fact which we will use on several other occasions. Lemma 15.3. Suppose that Ω is a properly convex domain, x0 ∈ Ω, w 6= z ∈ ∂Ω and Lw = [x0 , w) and Lz = [x0 , z) are projective line segments in Ω joining x0 to w and z. Then Lw and Lz are a finite Hausdorff distance apart (in the Hilbert metric on Ω) if and only z and w lie in the interior of a projective line segment in ∂Ω. Proof. First suppose that w and z lie in the interior of a (maximal) projective line segment [x, y] in ∂Ω and assume that they occur in the order x, w, z, y. If Lw does not lie in a bounded metric neighborhood of Lz , then there exists a sequence wn ∈ Lw so that d(wn , Lz ) → ∞. Choose a point zn on Lz so that [wn , zn ] is parallel to [w, z]. If xn and yn are the points of intersection of the projective line ← wn−→ zn with ∂Ω (in the order xn , wn , zn , yn ), then [xn , yn ] converges to [x, y], so [xn , wn , zn , yn ] converges to [x, y, z, w] which is finite. Therefore, d(wn , zn ) is bounded and we have achieved a contradiction. Thus, Lw lies in a bounded metric neighborhood of Lz . Symmetrically, Lz lies in a bounded metric neighborhood of Lw , so Lw and Lz are a finite Hausdorff distance apart. On the other hand, suppose that Lw and Lz are a Hausdorff distance at most K apart. Choose sequences wn in Lw and zn ∈ Lz so that wn → w and d(wn , zn ) ≤ K. Let xn and yn be the points of intersection of the projective line ← wn−→ zn with ∂Ω (in the order xn , wn , zn , yn ), and pass to a subsequence so that xn → x ∈ ∂Ω and yn → y ∈ ∂Ω. Notice that if either x = w or y = z, then [xn , wn , zn , yn ] → ∞, so d(wn , zn ) → ∞, which is a contradiction. Therefore, z and  w lie in the interior of the line segment xy contained in ∂Ω. Proof of Proposition 15.2. Fix p ∈ Ω. Every geodesic ray emanating from p is a projective line segment which intersects ∂Ω in a unique point. Moreover, every point in ∂Ω determines a geodesic ray emanating from p. Since Ω is strictly convex, Lemma 15.3 implies that two geodesic rays emanating from p lie a bounded Hausdorff distance apart if and only they agree. So, we may identify ∂∞ Ω with ∂Ω. We see that this identification is a homeomorphism by noting that if {zn } ⊂ Ω̄, then {pzn } converges to pz if and only if {zn } converge to z. 
∗ n+1 ∗ ∗ Every properly convex domain Ω admits a dual domain Ω ⊂ P (R ) where Ω denote n+1 the set of (projective classes of) linear functionals φ : R → R such
that φ(~v ) 6= 0 if ~v 6= ~0 and [~v ] ∈ Ω̄. Alternatively, we may equivariantly identify P (Rn+1 )∗ with the Grassmanian Grn (Rn+1 ) of hyperplanes in Rn+1 , by identifying a projective class of linear functional with its kernel. Then, Ω∗ may be identified with the set of projective hyperplanes disjoint from Ω̄ and ∂Ω∗ is the set of support planes to Ω. 60 Similarly if ρ : Γ →
GL(n+1, R) is a representation, then we may define dual representation ρ∗ : Γ → GL (Rn+1 )∗ . If φ ∈ (Rn+1 )∗ and γ ∈ Γ, then ρ∗ (γ)(φ) = φ ◦ ρ(γ)−1 . Notice that
ker(ρ∗ (γ)(φ))) = ρ(γ)(ker(φ)). If one identifies GL (Rn+1 )∗ with GL(n + 1, R) by choosing
⊥ n+1 )∗ , then A acts by (AT )−1 on P (Rn+1 )∗ . So the standard basis {e⊥ 1 , . . . , en+1 } for (R with this identification ρ∗ (γ) = (ρ(γ)−1 )T . Notice that this whole construction commutes with projection, so if ρ : Γ →
PGL(n + 1, R) is a representation, then we obtain a dual representation ρ∗ : Γ → PGL (Rn+1 )∗ . We record a few elementary observations in a lemma for future use. The proof uses the standard fact that if Ω is properly convex, then ∂Ω is C 1 at a point x if and only if there is a unique support plane to Ω through the point x. (In fact, this is the key property of C 1 points that we will use, so one may simply take this as the definition for our purposes.) Lemma 15.4. Suppose that Ω ⊂ RPn is a properly convex domain. Then (1) The dual domain Ω∗ is properly convex. (2) If ρ : Γ → PGL(n + 1, R) is a representation and ρ(Γ) ⊂ Aut(Ω), then ρ∗ (Γ) ⊂ Aut(Ω∗ ). (3) The boundary ∂Ω of Ω is C 1 if and only if Ω∗ is strictly convex. More precisely, if x ∈ ∂Ω is not a C 1 point, then there exists a line segment in ∂Ω∗ consisting of support planes to Ω at x. Proof. Suppose that Ω is properly convex. If x ∈ RPn , let Px∗ ⊂ ((Rn+1 )∗ ) be the projective hyperplane   Px∗ = P φ ∈ (Rn+1 )∗ ) φ(~u) = 0 if [~u] = x consisting of (projective classes of) linear functional with the line x in their kernel. Notice that if x ∈ Ω, then, by definition, Ω∗ is disjoint from Px∗ , so Ω∗ is a bounded subset of an affine
chart for P (Rn+1 )∗ . If [φ], [ψ] ∈ Ω∗ , then we may choose a component Ω0 of the pre-image of Ω in S n and linear functionals φ0 ∈ [φ] and ψ0 ∈ [ψ] so that if v ∈ Ω0 , then φ0 (v) > 0 and ψ0 (v) > 0. Then (tφ0 + (1 − t)ψ0 )(v) > 0 for all t ∈ [0, 1] and v ∈ Ω0 . It follows that {[tφ0 + (1 − t)ψ0 ] | t ∈ [0, 1]} is a projective line segment in Ω∗ joining [ψ] to [φ]. Therefore, Ω∗ is properly convex, so (1) holds. Consider a representation ρ : Γ → PGL(n + 1, R) so that ρ(Γ) preserves Ω. If γ ∈ Γ and H is a hyperplane disjoint from Ω̄, then ρ(γ)(H) is also disjoint from Ω̄. Therefore, ρ∗ (γ) preserves Ω∗ , so (2) holds. The set of support planes to x is a non-trivial convex subset Cx of the projective hyperplane ∗ Px and Cx is contained in ∂Ω∗ . Notice that ∂Ω is C 1 at x if and only if Cx is a single point, Therefore, if ∂Ω is not C 1 , then Ω∗ is not strictly convex. So, if Ω∗ is strictly convex, then Ω is C 1 . Conversely, if Ω∗ is not strictly convex, then there exists a non-trivial line segment [a, b] in ∂Ω∗ . Each point y ∈ [a, b] is associated to a support plane Hy to Ω. Suppose that Ha is a support plane at c ∈ ∂Ω and Hb is a support plane at d ∈ ∂Ω. If c 6= d, then [c, d] ∈ Ω̄ and if y ∈ (a, b), then Hy intersects [c, d] transversely. Hence, Hy must intersect Ω which contradicts the fact that it is a support plane to Ω. Therefore, c = d, so c is not a C 1 point of ∂Ω (since it admits two distinct support planes). Thus, if Ω is not strictly convex, then ∂Ω is not C 1 . So, if ∂Ω is C 1 , then Ω∗ is strictly convex, which completes the proof of (3).  ANOSOV REPRESENTATIONS 61 The following result, concerning endpoints of geodesics in properly convex domains, will only be used in Section 39, so can be ignored for now if you want to move on immediately to the study of Benoist representations. Notice that this result is obvious for strictly convex domains, since geodesics in strictly convex domains are projective line segments. Proposition 15.5. (Foertsch-Karlsson [98, Theorem 3], see also [83, Lemma 2.6]) Suppose that Ω ⊂ RPn is a properly convex domain. (1) If r : [0, ∞) → Ω is a geodesic ray, then limt→∞ r(t) = r(∞) exists and r(∞) ∈ ∂Ω. (2) If s : R → Ω is a bi-infinite geodesic, then s(∞) = lim s(t) 6= lim s(t) = s(−∞). t→∞ t→−∞ Proof. Let r : [0, ∞) → Ω be a geodesic ray in a properly convex domain Ω. For all t > 0, −−−−−→ let zt = r(0)r(t) ∩ ∂Ω and let Pt be the facet of ∂Ω containing zt (i.e. the intersection of a supporting hyperplane to ∂Ω through zt with ∂Ω). Lemma 14.3 implies that if 0 < s < t, then zs ⊂ Pt . Therefore, Pt = Ps , so zt lies in a single facet P for all t > 0. −−−−−→ Let wt = r(t)r(0) ∩ ∂Ω and Qt be the facet of ∂Ω containing wt . We argue just as above to show that Qt = Qs if 0 < s < t, so wt lies in a single facet Q for all t > 0. Moreover, P and Q are disjoint. Let C be the set of accumulation points of r([0, ∞)) in ∂Ω. Since any accumulation in C is also an accumulation point of {zt }, we see that C ⊂ P . Suppose that C contains more than one point. Then, there exist sequence {sn } and {tn } so that lim r(sn ) = v 6= w = lim r(tn ). Notice that the line Ln joining r(sn ) to r(tn ) converges to the line L containing [v, w]. It follows that L intersects ∂Ω only at P . −−−−−−−→ Lemma 14.3 implies that if 0 < sn < tn , then ztn and r(sn )r(tn ) ∩ ∂Ω span a line segment in −−−−−−−→ ∂Ω and wtn and r(tn )r(sn ) ∩ ∂Ω span a line segment in ∂Ω, so Ln ∩ Ω̄ has endpoints in P and Q. If 0 < tn < sn , we argue similarly that Ln ∩ Ω̄ has endpoints in P and Q. Since P and Q are closed and disjoint this implies that L ∩ Ω̄ has endpoints in P and Q. We have achieved a contradiction, since we previously saw that L intersected ∂Ω only in P . Therefore, C is a single point {c} and lim limt→∞ r(t) = c. Now suppose that s : R → Ω is a geodesic, the previous argument implies that c = limt→∞ s(t) and d = limt→−∞ s(t) exist. Moreover, we may check, just as above that c lies in the same −−−−−−→ −−−−−−→ facet of ∂Ω as s(−1)s(1) ∩ ∂Ω and that d lies in the same facet of ∂Ω as s(1)s(−1) ∩ ∂Ω. Since ←−−−−−→ s(−1)s(1) must intersect ∂Ω in distinct facets, c and d must be distinct.  Karlsson and Noskov proved that that Gromov hyperbolic properly convex domains have C 1 boundary. We will not use this fact in these notes, but I feel like it illuminates the general picture. Feel free to skip it if you prefer. Proposition 15.6. (Karlsson-Noskov [135]) If Ω ⊂ RPn is strictly convex and Gromov hyperbolic, then ∂Ω is a C 1 -submanifold of RPn . Remarks: (1) If Ω is properly convex and has at least two line segments in its boundary, then the Hilbert metric is not even CAT(0) since geodesics are not unique. However, they still exhibit some of the geometric flavor of non-positively curved manifolds. Ludovic Marquis [158] refers to them as “damaged non-positively curved manifolds.” 62 (2) Benoist [19] showed that a properly convex domain is Gromov hyperbolic in its Hilbert metric if and only if it is quasisymmetrically convex. Colbois, Vernicos and Verovic [67] showed that Ω is Gromov hyperbolic if and only if there is an upper bound on the area of all ideal triangles. (3) Benzecri [26] proved that if Ω ⊂ RP2 is properly convex and divisible, but not strictly convex, then Ω is projectively equivalent to the simplex ∆. Benoist [23] produced and studied more interesting examples of properly convex divisible domains in RP3 which are not strictly convex (see also Ballas-Danciger-Lee [10])). 16. Benoist’s characterizations of strictly convex divisible domains Yves Benoist established the following beautiful characterization of strictly convex divisible domains. Theorem 16.1. (Benoist [20]) If a discrete group Γ ⊂ PGL(n + 1, R) acts cocompactly on a properly convex domain Ω ⊂ RPn , then the following are equivalent. (1) (2) (3) (4) (Ω, dH Ω ) is Gromov hyperbolic. Γ is Gromov hyperbolic. Ω is strictly convex. ∂Ω is a C 1 submanifold of RPn . Proof. Notice that the equivalence of (1) and (2) is an immediate consequence of the MilnorSvarc Lemma and Proposition 4.1. Proposition 15.1 asserts that if Ω is not strictly convex, then Ω is not Gromov hyperbolic, so (1) implies (3). We next show that (3) implies (2). Proposition 16.2. (Benoist [20]) If Γ ⊂ PGL(n + 1, R) divides a strictly convex domain Ω, then (Ω, dH Ω ) is Gromov hyperbolic. Proof. Recall that if Ω is strictly convex, then all geodesics are projective line segments, see Lemma 14.3. If Ω is not Gromov hyperbolic, then there exists sequences {xn }, {yn }, {zn } and {un } of points in Ω such that un ∈ xn yn for all n, and d(un , xn zn ∪ yn zn ) ≥ n for all n. (For the remainder of this chapter d will be the relevant Hilbert metric unless we say otherwise.) Since Γ acts cocompactly on Ω, there is a compact set K such that Γ(K) = Ω. So, we can always choose γn ∈ Γn so that γn (un ) ∈ K. So, after replacing xn , yn , zn , un with γ(xn ), γ(yn ), γ(zn ), γ(un ), we may assume that un ∈ K for all n. We may then pass to a subsequence so that un → u, xn → x, yn → y and zn → z. Since d(un , xn zn ∪ yn zn ) → ∞, we must have x, y, z ∈ ∂Ω. Since u ∈ xy, x 6= y. If x = z, then yn zn → yz, so d(un , yn zn ) → 0, which is a contradiction, so x 6= z. Similarly y 6= z. But then, d(u, xz) < ∞ since Ω is strictly convex. (This is the only usage of strict convexity in the proof.) But d(un , xn zn ) → d(u, xz), so we have again achieved a contradiction. Therefore, Ω is Gromov hyperbolic.  The following result shows that (3) holds if and only if (4) holds and completes the proof of Theorem 16.1.  Proposition 16.3. (Benoist [20]) If Γ ⊂ PGL(n + 1, R) divides a properly convex domain Ω ⊂ RPn , then Ω is strictly convex if and only ∂Ω is a C 1 submanifold of RPn . ANOSOV REPRESENTATIONS 63 Proof. Suppose that Ω is strictly convex. Then, by Proposition 16.2, Ω, and hence Γ, is Gromov hyperbolic. Let Γ∗ = {(γ −1 )T | γ ∈ Γ} (so Γ∗ is the image of the dual to the inclusion map of Γ into PGL(n + 1, R)). Lemma 15.4 implies that Ω∗ is properly convex and Γ∗ ⊂ Aut(Ω∗ ). Since Γ∗ is discrete, it acts properly discontinuously on Ω∗ . Since M = Ω/Γ is a closed manifold and is homotopy equivalent to the manifold M ∗ = Ω∗ /Γ∗ , M ∗ is also closed, so Γ∗ divides Ω∗ . Since Γ∗ , and hence Ω∗ is Gromov hyperbolic, Proposition 15.1 implies that Ω∗ is strictly convex. Lemma 15.4 then implies that Ω is a C 1 submanifold of RPn . (If Γ has torsion, we first pass to a finite index torsion-free subgroup of Γ and then apply the above argument.) On the other hand, if Γ divides a properly convex domain Ω so that ∂Ω is a C 1 submanifold of RPn , then we argue as above to show that Γ∗ divides the strictly convex domain Ω∗ . It follows, from Proposition 16.2, that Γ∗ , and hence Γ and Ω, is Gromov hyperbolic. Proposition 15.1 then implies that Ω is strictly convex.  Remarks: Benzecri [26] proved that if a group Γ divides a strictly convex domain and ∂Ω is a C 2 -submanifold, then Ω is an ellipsoid. Benoist [20] proved that ∂Ω is always C 1+α for some α > 0 if Ω is a strictly convex divisible domain. 17. Linear algebra in GL(d, R) The country I come from Is called the Midwest I was taught and brought up there The laws to abide ————Bob Dylan [92] We say that A ∈ GL(d, R) is proximal if it has an attracting fixed point for its action on RPn , i.e. there exists a point x ∈ RPn such that A(x) = x and an open neighborhood U of x so that if u ∈ U , then An (u) → x uniformly on compact subsets of U . We say that A is biproximal if both A and A−1 are proximal. We will see how the linear algebra reflects this property. Consider the (real) Jordan canonical form for A ∈ GL(d, R), i.e. A = BJB −1 for some B ∈ GL(n + 1, R) and J is a block diagonal matrix   J1 0 · · · 0  0 J2 · · · 0     .. .. ..  . . .  0 0 · · · Jm where each Ji is either a single real entry ji or a (real) Jordan block of the form   Ci 1 0 · · · 0  0 Ci 1 · · · 0     0 0 Ci · · · 0  Ji =    .. .. ..  . . . 0 0 0 · · · Ci   p a −bi and Ci is either a single (real) entry ji , or Ci = i , in which case ji = ± |det(Ci )|ei arccos(ai ) . bi ai 64 We will call ji a generalized eigenvalue, and we call the subspace preserved by Ji its generalized eigenspace Ei . We may assume that |j1 | ≥ |j2 | ≥ · · · ≥ |jm |. If Ji is a Jordan block, then its generalized eigenspace contains a vector ~vi so that kJ n (~vi )k = |ji |n k~vi k| for all n. If Ji is upper-triangular, but not a single entry, then Ei contains an eigenvector w ~ i so that if ~u ∈ Ei , then J n ([~u]) → [wi ] in RPn . However, the convergence will not be n uniform. For example, if J = Ji = [ 10 11 ], then < e1 > is the only eigenline, but if vn = [ −1 ], n 0 then {< vn >} converges to < e1 > and J (vn ) = −1 for all n. If Ji is Jordan block which is not upper triangular, then its generalized eigenspace contains no eigenlines. Notice that J is proximal if and only if |j1 | > |j2 | and J1 is a single entry j1 . In this case, the eigenspace E1 is called the attracting eigenline, and is an attracting fixed point for the action of J on RPd−1 . Moreover, J n (v) → [e1 ] if and only if v does not lie in < e2 , . . . , en , en+1 >, so we call < e2 , . . . , en , en+1 > the repelling hyperplane of J. If A = BJB −1 , then A is proximal if and only if J is proximal. Then, B(< e1 >) is the attracting eigenline of A and B(< e2 , . . . , en+1 >) is the repelling hyperplane of A. If A is biproximal, the repelling hyperplane of A−1 is sometimes called the attracting hyperplane of A. In this case, in our notation, Jm is a single real number and the attracting hyperplane has the form B(< e1 , . . . , en >). We may also rephrase this in terms of the eigenvalues of A. If we let {λi (A)}n+1 i=1 denote the complex eigenvalues of A, then we may order them so that |λ1 (A)| ≥ |λ2 (A)| ≥ · · · ≥ |λd (A)|. Notice that each block Ji gives rise to dim(Ji ) eigenvalues of modulus |ji |. Therefore, A is proximal if and only if |λ1 (A)| > |λ2 (A)|. Notice that the notions of proximality, biproximality, attracting eigenline, repelling hyperplane all make sense in PGL(n + 1, R) since they are unaffected by the choice of element in the projective class. Moreover, the tuple (|λ1 (A)|, . . . , |λd (A)|) is well-defined. We also recall the singular value decomposition, which we discussed briefly in the introduction. If B ∈ GL(d, R), then we may write B = LAK where L, K ∈ O(d) and A is a diagonal matrices with positive entries in descending order along the diagonal, i.e. a11 ≥ a22 ≥ · · · ≥ ann > 0. The matrix A depends only on B, but L and K need not be unique when some of the diagonal entries agree. We let σi (B) = aii and call it the ith singular value of B. It is (half) the length of the ith minor of the ellipse B(S n−1 ). In the literature, you will often see references to the Cartan projection which is given by µ : SL± (d, R) → a+ where a+ = {~x ∈ Rd | x1 ≥ x2 ≥ · · · ≥ xd and x1 + · · · + xn = 0} so that µ(B) = (log σ1 (B), . . . , log σd (B)). We will probably not use this language ourselves. ANOSOV REPRESENTATIONS 65 It is important to remember that singular values and eigenvalues are rather different animals (as I have learned to my dismay in the past). For example,   1 n ∼n σ1 0 1 However, one does have the relationship log σi (B n ) . n Notice that since A and −A have the same singular values it makes sense to talk about σi (X) when X ∈ PSL± (n, R), which may be identified with PGL(d, R). Moreover, if i 6= j and |λi (A)| A ∈ PGL(d, R), then both σσji (A) (A) and |λj (A)| are well-defined. log |λi (B)| = lim 18. Limit maps Told my little Pollyanna There’s a place for you and me We’ll go down to Transverse City Life is cheap, and death is free ————Warren Zevon [229]
In this section, we produce ρ-equivariant limit maps ξρ : ∂Γ → ∂Ω and θρ : ∂Γ → P (Rn+1 )∗ associated to a Benoist representation ρ : Γ → PGL(n + 1, R). These limit maps generalize the limit maps we saw in the study of convex cocompact representations into O0 (n, 1) and foreshadow the limit maps which will play a crucial role in the theory of Anosov representations. Suppose that Γ is a hyperbolic group and ρ : Γ → PGL(n + 1, R) is a representation and that
Aρ : ∂Γ → RPn and Bρ : ∂Γ → Grn (Rn+1 ) ∼ = P (Rn+1 ))∗ are continuous ρ-equivariant maps. We say that Aρ and Bρ are compatible if Aρ (z) ⊂ Bρ (z) for all z ∈ ∂Γ. They are said to be transverse if Aρ (w) ⊕ Bρ (z) = Rn+1 for all z 6= w ∈ ∂Γ. Proposition 18.1. If a Benoist representation ρ : Γ → PGL(n + 1, R) divides a strictly convex domain Ω, then it admits ρ-equivariant maps ξρ : ∂Γ → RPm and θρ : ∂Γ → Grn (Rn+1 ) which are compatible and transverse. Proof. Theorem 16.1 implies that Γ and Ω are Gromov hyperbolic and that Ω is strictly convex and has C 1 boundary. The Milnor-Svarc Lemma implies that the orbit map τρ : Γ → Ω is a quasi-isometry. Proposition 15.2 allows us to identify the Gromov boundary ∂∞ Ω with the topological boundary ∂Ω. Proposition 3.5 and Corollary 3.6 then show that τρ extends to a homeomorphism ξρ : ∂Γ → ∂Ω so that if {γn } ⊂ Γ, lim γn = z ∈ ∂Γ and x ∈ Ω, then lim ρ(γn )(x) = ξρ (z).
We define θρ : ∂Γ → P (Rn+1 )∗ by letting θρ (z) be the tangent plane (unique support plane) to ∂Ω at ξρ (z). By construction, θρ is a ρ-equivariant embedding. Notice that ξρ and θρ are compatible by definition. Since θρ (γ + ) is invariant under ρ(γ) and does not contain the repelling eigenline of ρ(γ) it must be the attracting n-plane of ρ(γ). Since Ω is strictly convex, if z ∈ ∂Γ, then θρ (z) is the tangent plane to ∂Ω at ξρ (z), so it cannot intersect ∂Ω at any other point. In particular, if w 6= z, then the line ξρ (w) ∈ ∂Ω − {ξρ (z)} is 66 not contained in the hyperplane θρ (z). Therefore, ξρ (w) ⊕ θρ (z) = Rn+1 if z 6= w ∈ ∂Γ, so ξρ and θρ are transverse.  We next show that if ρ is a Benoist representation, then all images of infinite order elements are biproximal and the limit maps are dynamics-preserving. Suppose that Γ is a hyperbolic group, ρ : Γ → PGL(n + 1, R) is a representation and
Aρ : ∂Γ → RPn and Bρ : ∂Γ → Grn (Rn+1 ) ∼ = P (Rn+1 ))∗ are continuous ρ-equivariant maps. The map Aρ is said to be dynamics preserving if whenever γ is an infinite order element of Γ, then ρ(γ) is biproximal and Aρ (γ + ) is the attracting eigenline of ρ(γ). Similarly, Bρ (γ + ) is said to be dynamics preserving if whenever γ is an infinite order element of Γ, then ρ(γ) is biproximal and Bρ (γ + ) is the attracting hyperplane of ρ(γ). Proposition 18.2. If ρ : Γ → PGL(n + 1, R) is a Benoist representation, then ξρ and θρ are dynamics preserving. In particular, if γ ∈ Γ has infinite order, then ρ(γ) is biproximal. We derive Proposition 18.2 from a more general statement which will be useful in Section 42. Lemma 18.3. Suppose that ρ : Γ → PGL(n + 1, R) is a discrete, almost faithful representation, ρ(Γ) preserves a properly convex domain Ω and there exist continuous, transverse ρ-equivariant maps ξ : ∂Γ → RPn and θ : ∂Γ → Grn (Rn+ ). If γ ∈ Γ has infinite order and {ρ(γ)n (x)}n∈N converges to ξ(γ + ) if x ∈ Ω, then ρ(γ) is proximal, ξ(γ + ) = ρ(γ)+ and θρ (γ − ) is the repelling hyperplane of ρ(γ). Proof. We may assume without loss of generality that J = ρ(γ) is in Jordan canonical form. Notice that ξ(γ + ) and θ(γ − ) are transverse and invariant under ρ(γ), so every generalized eigenspace of a Jordan block of J is contained in either ξ(γ + ) or θ(γ − ). Therefore, the Jordan block J+ whose generalized eigenspace contains the line ξ(γ + ) must be one-dimensional. Let j+ be the eigenvalue of J+ and let j− be the largest generalized eigenvalue of a Jordan block contained in θ(γ − ). Let ~v1 be a vector in J+ and let ~v2 be a vector in θ(γ − ) so that ||J n (~v2 )|| = (j− )n ||~v2 ||. Let p+ : Rn+1 → J+ , p− : Rn+1 and p2 : Rn+1 →< ~v2 > be projection maps. Since o n ||p− (J n (~v ))|| Ω is open, there exists x = [~v ] ∈ Ω, so that p2 (~v ) 6= 0. If |j− | ≥ |j+ |, then ||p+ (J n (~v))|| n∈N does not converge to 0, so ρ(γ)n (x) does not converge to ξ(γ + ). This violates our assumption, so |j+ | > |j− | which implies that ρ(γ) is proximal and ξ(γ + ) = ρ(γ)+ .  Proof of Proposition 18.2. Proposition 18.1 implies that ξρ and θρ are continuous, transverse ρ-equivariant maps, such that if γ has infinite order then {ρ(γ)n (x)}n∈N converges to ξ(γ + ). Lemma 18.3 then implies that if γ has infinite order, then ρ(γ) is proximal and ξ(γ + ) = ρ(γ + ). Similarly, ρ(γ −1 ) is proximal, so ρ(γ) is biproximal and θρ ((γ −1 )− ) is the repelling hyperplane of ρ(γ −1 ). Since (γ −1 )− = γ + , this implies that θρ (γ + ) is the attracting hyperplane of ρ(γ). This completes the proof that ξρ and θρ are dynamics-preserving.  We may use the analysis above to check that Benoist representations are strongly irreducible, which is a special case of a result of Vey [211]. Recall that a representation ρ : Γ → PGL(d, R) is said to be irreducible if every subspace of Rd invariant under every element of ρ(Γ) is either trivial or all of Rd . It is called strongly irreducible if the restriction of ρ to any finite index subgroup is still irreducible. ANOSOV REPRESENTATIONS 67 Proposition 18.4. (Vey [211]) If ρ : Γ → PGL(n + 1, R) is a Benoist representation, then ρ is strongly irreducible. Proof. Suppose that a proper linear subspace V is invariant under ρ(Γ). If γ is any infinite order element of Γ, then the translates Γ(γ + ) of γ + are dense in ∂Γ, by Proposition 5.6. Therefore, the attracting eigenlines of conjugates of ρ(γ) are dense in ∂Ω and hence span Rn+1 . It follows that there exists an infinite order element γ ∈ Γ so that V does not contain the attracting eigenline ξρ (γ + ) of ρ(γ). Thus, V must be contained in the repelling hyperplane of ρ(γ), since otherwise there exists v ∈ V , so that ρ(γ)n (v) → ξρ (γ + ). So V is contained in the hyperplane P tangent to ∂Ω at ξρ (γ − ). If α ∈ Γ, then α(P ) is the tangent plane to ∂Ω. If α ∈ Γ, then α(P ) is the tangent plane to ∂Ω at α(ξρ (γ − )), and since V is Γ-invariant, we see that V ⊂ α(P ). Since translates of γ − are dense in ∂Γ, again by Proposition 5.6, we see that V is contained in the tangent plane to every point of ∂Ω. Now notice that every hyperplane in the affine chart is parallel to exactly two distinct tangent planes to Ω, since Ω is strictly convex and ∂Ω is C 1 . Therefore, the intersection of all the projective hyperplanes tangent to ∂Ω is trivial, so V must be trivial, which is again a contradiction. Therefore, ρ must be irreducible. Since the restriction of every Benoist representation to a finite index subgroup is a Benoist representation, we see immediately that Benoist representations are strongly irreducible.  Remarks: 1) Vey [211] further proved that if a discrete subgroup Γ of PGL(n + 1, R) divides a properly convex, irreducible domain in RPn , then Γ is strongly irreducible. 2) Benoist [18] proved that the Zariski closure of the image of a Benoist representation into PSL(n + 1, R) is either PSO(n, 1) or PSL(n + 1, R). Benoist [21] proved, more generally, that if a discrete group Γ ⊂ PGL(n + 1, R) divides a properly convex domain Ω, then Γ is Zariski dense unless Ω is a product or symmetric cone. (The symmetric cones have been completely classified and are all associated to semi-simple Lie groups, and PO(n, 1) is the only rank one Lie group which arises. We will later see the symmetric domain associated to PO(n, n) in Section 41.) 3) If ρ : Γ → PGL(n + 1, R) is a Benoist representation and ρ(Γ) preserves Ω, then, by Lemma 15.4, ρ∗ (Γ) preserves Ω∗ . We saw, in the proof of Proposition 16.3, that ρ∗ (Γ) acts cocompactly on Ω∗ and that Ω∗ is strictly convex, so ρ∗ is also a Benoist representation. If z ∈ ∂Γ, then θρ (z) is the unique support plane to Ω at ξρ (z), so θρ (z) lies in ∂Ω∗ . So, θρ is a ρ∗ -equivariant homeomorphism from ∂Γ to ∂Ω∗ which suggest that θρ = ξρ∗ . If γ ∈ Γ has infinite order, then θρ (γ + ) is the attracting n-plane of ρ(γ), so it is the attracting eigenline of ρ∗ (γ). Therefore, ξρ∗ (γ + ) = θρ (γ + ) and since both θρ and ξρ∗ are continuous, we see that θρ = ξρ∗ . We will later establish a generalization of this observation in the context of Anosov representations, see Corollary 32.4. 19. Translation length, eigenvalues and singular values If ρ : Γ → PGL(n + 1, R) is a Benoist representation dividing a domain Ω, we define the translation length of an element ρ(γ) by  `ρ (γ) = inf dH Ω (x, ρ(γ)(x)) | x ∈ Ω . Since Ω/ρ(Γ) is compact, the translation length is always achieved. ANOSOV REPRESENTATIONS 65 It is important to remember that singular values and eigenvalues are rather different animals (as I have learned to my dismay in the past). For example,   1 n ∼n σ1 0 1 However, one does have the relationship log σi (B n ) . n Notice that since A and −A have the same singular values it makes sense to talk about σi (X) when X ∈ PSL± (n, R), which may be identified with PGL(d, R). Moreover, if i 6= j and |λi (A)| A ∈ PGL(d, R), then both σσji (A) (A) and |λj (A)| are well-defined. log |λi (B)| = lim 18. Limit maps Told my little Pollyanna There’s a place for you and me We’ll go down to Transverse City Life is cheap, and death is free ————Warren Zevon [229]
In this section, we produce ρ-equivariant limit maps ξρ : ∂Γ → ∂Ω and θρ : ∂Γ → P (Rn+1 )∗ associated to a Benoist representation ρ : Γ → PGL(n + 1, R). These limit maps generalize the limit maps we saw in the study of convex cocompact representations into O0 (n, 1) and foreshadow the limit maps which will play a crucial role in the theory of Anosov representations. Suppose that Γ is a hyperbolic group and ρ : Γ → PGL(n + 1, R) is a representation and that
Aρ : ∂Γ → RPn and Bρ : ∂Γ → Grn (Rn+1 ) ∼ = P (Rn+1 ))∗ are continuous ρ-equivariant maps. We say that Aρ and Bρ are compatible if Aρ (z) ⊂ Bρ (z) for all z ∈ ∂Γ. They are said to be transverse if Aρ (w) ⊕ Bρ (z) = Rn+1 for all z 6= w ∈ ∂Γ. Proposition 18.1. If a Benoist representation ρ : Γ → PGL(n + 1, R) divides a strictly convex domain Ω, then it admits ρ-equivariant maps ξρ : ∂Γ → RPm and θρ : ∂Γ → Grn (Rn+1 ) which are compatible and transverse. Proof. Theorem 16.1 implies that Γ and Ω are Gromov hyperbolic and that Ω is strictly convex and has C 1 boundary. The Milnor-Svarc Lemma implies that the orbit map τρ : Γ → Ω is a quasi-isometry. Proposition 15.2 allows us to identify the Gromov boundary ∂∞ Ω with the topological boundary ∂Ω. Proposition 3.5 and Corollary 3.6 then show that τρ extends to a homeomorphism ξρ : ∂Γ → ∂Ω so that if {γn } ⊂ Γ, lim γn = z ∈ ∂Γ and x ∈ Ω, then lim ρ(γn )(x) = ξρ (z).
We define θρ : ∂Γ → P (Rn+1 )∗ by letting θρ (z) be the tangent plane (unique support plane) to ∂Ω at ξρ (z). By construction, θρ is a ρ-equivariant embedding. Notice that ξρ and θρ are compatible by definition. Since θρ (γ + ) is invariant under ρ(γ) and does not contain the repelling eigenline of ρ(γ) it must be the attracting n-plane of ρ(γ). Since Ω is strictly convex, if z ∈ ∂Γ, then θρ (z) is the tangent plane to ∂Ω at ξρ (z), so it cannot intersect ∂Ω at any other point. In particular, if w 6= z, then the line ξρ (w) ∈ ∂Ω − {ξρ (z)} is 66 not contained in the hyperplane θρ (z). Therefore, ξρ (w) ⊕ θρ (z) = Rn+1 if z 6= w ∈ ∂Γ, so ξρ and θρ are transverse.  We next show that if ρ is a Benoist representation, then all images of infinite order elements are biproximal and the limit maps are dynamics-preserving. Suppose that Γ is a hyperbolic group, ρ : Γ → PGL(n + 1, R) is a representation and
Aρ : ∂Γ → RPn and Bρ : ∂Γ → Grn (Rn+1 ) ∼ = P (Rn+1 ))∗ are continuous ρ-equivariant maps. The map Aρ is said to be dynamics preserving if whenever γ is an infinite order element of Γ, then ρ(γ) is biproximal and Aρ (γ + ) is the attracting eigenline of ρ(γ). Similarly, Bρ (γ + ) is said to be dynamics preserving if whenever γ is an infinite order element of Γ, then ρ(γ) is biproximal and Bρ (γ + ) is the attracting hyperplane of ρ(γ). Proposition 18.2. If ρ : Γ → PGL(n + 1, R) is a Benoist representation, then ξρ and θρ are dynamics preserving. In particular, if γ ∈ Γ has infinite order, then ρ(γ) is biproximal. We derive Proposition 18.2 from a more general statement which will be useful in Section 42. Lemma 18.3. Suppose that ρ : Γ → PGL(n + 1, R) is a discrete, almost faithful representation, ρ(Γ) preserves a properly convex domain Ω and there exist continuous, transverse ρ-equivariant maps ξ : ∂Γ → RPn and θ : ∂Γ → Grn (Rn+ ). If γ ∈ Γ has infinite order and {ρ(γ)n (x)}n∈N converges to ξ(γ + ) if x ∈ Ω, then ρ(γ) is proximal, ξ(γ + ) = ρ(γ)+ and θρ (γ − ) is the repelling hyperplane of ρ(γ). Proof. We may assume without loss of generality that J = ρ(γ) is in Jordan canonical form. Notice that ξ(γ + ) and θ(γ − ) are transverse and invariant under ρ(γ), so every generalized eigenspace of a Jordan block of J is contained in either ξ(γ + ) or θ(γ − ). Therefore, the Jordan block J+ whose generalized eigenspace contains the line ξ(γ + ) must be one-dimensional. Let j+ be the eigenvalue of J+ and let j− be the largest generalized eigenvalue of a Jordan block contained in θ(γ − ). Let ~v1 be a vector in J+ and let ~v2 be a vector in θ(γ − ) so that ||J n (~v2 )|| = (j− )n ||~v2 ||. Let p+ : Rn+1 → J+ , p− : Rn+1 and p2 : Rn+1 →< ~v2 > be projection maps. Since o n ||p− (J n (~v ))|| Ω is open, there exists x = [~v ] ∈ Ω, so that p2 (~v ) 6= 0. If |j− | ≥ |j+ |, then ||p+ (J n (~v))|| n∈N does not converge to 0, so ρ(γ)n (x) does not converge to ξ(γ + ). This violates our assumption, so |j+ | > |j− | which implies that ρ(γ) is proximal and ξ(γ + ) = ρ(γ)+ .  Proof of Proposition 18.2. Proposition 18.1 implies that ξρ and θρ are continuous, transverse ρ-equivariant maps, such that if γ has infinite order then {ρ(γ)n (x)}n∈N converges to ξ(γ + ). Lemma 18.3 then implies that if γ has infinite order, then ρ(γ) is proximal and ξ(γ + ) = ρ(γ + ). Similarly, ρ(γ −1 ) is proximal, so ρ(γ) is biproximal and θρ ((γ −1 )− ) is the repelling hyperplane of ρ(γ −1 ). Since (γ −1 )− = γ + , this implies that θρ (γ + ) is the attracting hyperplane of ρ(γ). This completes the proof that ξρ and θρ are dynamics-preserving.  We may use the analysis above to check that Benoist representations are strongly irreducible, which is a special case of a result of Vey [211]. Recall that a representation ρ : Γ → PGL(d, R) is said to be irreducible if every subspace of Rd invariant under every element of ρ(Γ) is either trivial or all of Rd . It is called strongly irreducible if the restriction of ρ to any finite index subgroup is still irreducible. ANOSOV REPRESENTATIONS 67 Proposition 18.4. (Vey [211]) If ρ : Γ → PGL(n + 1, R) is a Benoist representation, then ρ is strongly irreducible. Proof. Suppose that a proper linear subspace V is invariant under ρ(Γ). If γ is any infinite order element of Γ, then the translates Γ(γ + ) of γ + are dense in ∂Γ, by Proposition 5.6. Therefore, the attracting eigenlines of conjugates of ρ(γ) are dense in ∂Ω and hence span Rn+1 . It follows that there exists an infinite order element γ ∈ Γ so that V does not contain the attracting eigenline ξρ (γ + ) of ρ(γ). Thus, V must be contained in the repelling hyperplane of ρ(γ), since otherwise there exists v ∈ V , so that ρ(γ)n (v) → ξρ (γ + ). So V is contained in the hyperplane P tangent to ∂Ω at ξρ (γ − ). If α ∈ Γ, then α(P ) is the tangent plane to ∂Ω. If α ∈ Γ, then α(P ) is the tangent plane to ∂Ω at α(ξρ (γ − )), and since V is Γ-invariant, we see that V ⊂ α(P ). Since translates of γ − are dense in ∂Γ, again by Proposition 5.6, we see that V is contained in the tangent plane to every point of ∂Ω. Now notice that every hyperplane in the affine chart is parallel to exactly two distinct tangent planes to Ω, since Ω is strictly convex and ∂Ω is C 1 . Therefore, the intersection of all the projective hyperplanes tangent to ∂Ω is trivial, so V must be trivial, which is again a contradiction. Therefore, ρ must be irreducible. Since the restriction of every Benoist representation to a finite index subgroup is a Benoist representation, we see immediately that Benoist representations are strongly irreducible.  Remarks: 1) Vey [211] further proved that if a discrete subgroup Γ of PGL(n + 1, R) divides a properly convex, irreducible domain in RPn , then Γ is strongly irreducible. 2) Benoist [18] proved that the Zariski closure of the image of a Benoist representation into PSL(n + 1, R) is either PSO(n, 1) or PSL(n + 1, R). Benoist [21] proved, more generally, that if a discrete group Γ ⊂ PGL(n + 1, R) divides a properly convex domain Ω, then Γ is Zariski dense unless Ω is a product or symmetric cone. (The symmetric cones have been completely classified and are all associated to semi-simple Lie groups, and PO(n, 1) is the only rank one Lie group which arises. We will later see the symmetric domain associated to PO(n, n) in Section 41.) 3) If ρ : Γ → PGL(n + 1, R) is a Benoist representation and ρ(Γ) preserves Ω, then, by Lemma 15.4, ρ∗ (Γ) preserves Ω∗ . We saw, in the proof of Proposition 16.3, that ρ∗ (Γ) acts cocompactly on Ω∗ and that Ω∗ is strictly convex, so ρ∗ is also a Benoist representation. If z ∈ ∂Γ, then θρ (z) is the unique support plane to Ω at ξρ (z), so θρ (z) lies in ∂Ω∗ . So, θρ is a ρ∗ -equivariant homeomorphism from ∂Γ to ∂Ω∗ which suggest that θρ = ξρ∗ . If γ ∈ Γ has infinite order, then θρ (γ + ) is the attracting n-plane of ρ(γ), so it is the attracting eigenline of ρ∗ (γ). Therefore, ξρ∗ (γ + ) = θρ (γ + ) and since both θρ and ξρ∗ are continuous, we see that θρ = ξρ∗ . We will later establish a generalization of this observation in the context of Anosov representations, see Corollary 32.4. 19. Translation length, eigenvalues and singular values If ρ : Γ → PGL(n + 1, R) is a Benoist representation dividing a domain Ω, we define the translation length of an element ρ(γ) by  `ρ (γ) = inf dH Ω (x, ρ(γ)(x)) | x ∈ Ω . Since Ω/ρ(Γ) is compact, the translation length is always achieved. ANOSOV REPRESENTATIONS 69 Corollary 19.2. If ρ : Γ → PGL(n + 1, R) is a Benoist representation, then there exists J and B so that   |λ1 (ρ(γ))| 1 J||γ|| + B ≥ log ≥ ||γ|| − B |λn+1 (ρ(γ))| J for all γ ∈ Γ. Proof. The Milnor-Svarc Lemma implies that the orbit map τρ : Γ → Ω (which is given by τρ (γ) = ρ(γ)(x0 ) for some fixed x0 ∈ Ω) is a (K, C)-quasi-isometry for some K and C. Suppose that γ ∈ Γ. Then there exists α ∈ Γ so that d(γα, α) = ||γ||. Therefore,   |λ1 (ρ(γ))| 1 K||γ|| + C ≥ d(τρ (γα), τρ (α)) = d(ρ(γ)(τρ (α)), τρ (α)) ≥ `ρ (γ)) = log 2 |λn+1 (ρ(γ))| On the other hand, there exists β ∈ Γ and a point x on the axis Aγ of γ so that d(τρ (β), y) ≤ C. Since τρ is a (K, C)-quasi-isometry, d(ρ(γ)(τρ (β)), τρ (β)) = d(τρ (γβ), τρ (β)) ≥ 1 ||γ|| − C. K Applying the triangle inequality, we see that   1 |λ1 (ρ(γ))| 1 log = `ρ (γ) = d(ρ(γ)(x)), x ≥ ||γ|| − 3C. 2 |λn+1 (ρ(γ))| K Therefore, the desired inequality holds with J = 2K and B = 6C.    1 (ρ(γ))| We would also like to show, in analogy with Lemma 11.2, that log |σ|σn+1 (ρ(γ))| grows linearly in the word length of γ. In order to do so, we use the U property of hyperbolic groups. Corollary 19.3. If ρ : Γ → PGL(n + 1, R) is a Benoist representation, then there exists L and D so that   1 σ1 (ρ(γ)) ≥ d(1, γ) − D Ld(1, γ) ≥ log σn+1 (ρ(γ)) L for all γ ∈ Γ. Proof. Let S be the finite generating set for Γ, which we assume is symmetric, i.e. s ∈ S if and only if s−1 ∈ S, which we have been using in the background. Let M = max{log σ1 (ρ(s)) | s ∈ S}. Since Γ has the U property, see Proposition 8.4, there exists α, β ∈ Γ and K ≥ 0, so that if γ ∈ Γ, then there exists η ∈ {1, α, β} such that d(1, γ) ≤ 3||γη|| + K. If γ ∈ Γ, then, since σ1 (AB) ≤ σ1 (A)σ1 (B) and σn+1 (A) = σ1 (A−1 )−1 , we see that   σ1 (ρ(γ)) log ≤ M 2 d(1, γ). σn+1 (ρ(γ)) Now choose η ∈ {1, α, β} so that d(1, γ) ≤ 3||γη|| + K. If J and B are the constants from Corollary 19.2,then   |λ1 (ρ(γη))| 1 1 K log ≥ ||γη|| − B ≥ d(1, γ) − − B. |λn+1 (ρ(γη))| J 3J J 70 But, since σ1 (A) ≥ |λ1 (A)|, σ1 (AB) ≤ σ1 (A)σ1 (B) and σn+1 (A) = σ1 (A−1 )−1 ,       σ1 (ρ(γ)) 1 1 σ1 (ρ(γη)) |λ1 (ρ(γη))| log ≥ ≥ log log σn+1 (ρ(γ)) max{1, 2M 2 d(1, η)} σn+1 (ρ(γη)) max{1, 2M 2 d(1, η)} |λn+1 (ρ(γη))| so, if we set G = max{d(1, α), d(1, β)}, we may combine the last two inequalities to see that   1 σ1 (ρ(γ)) K ≥ log d(1, γ) − −B σn+1 (ρ(γ)) 2M 2 GJ 2JM 2 G so our result holds with L = M 2 JG and D = K JM 2 G + B.  A more honest way to prove Corollary 19.3 would be to first relate the singular values to the translation length of the point. Once one has done this, Corollary 19.3 follows immediately from the Milnor-Svarc Lemma. The key result here is due to Danciger, Guéritaud and Kassel: Proposition 19.4. (Danciger-Guéritaud-Kassel [84, Proposition 10.1]) If Ω is a properly convex domain and x0 ∈ Ω, then there exists κ > 0 so that if A ∈ Aut(Ω), then   σ1 (A) ≥ 2d(x0 , γ(x0 )) − κ. log σn+1 (A) Remarks: (1) The converses to Corollaries 19.2 and 19.3 fail, since if ρ is a Benoist representation and H ⊂ Γ is a quasiconvex subgroup, then the conclusions are satisfied by ρ|H , but ρ|H is Benoist if and only if H has finite index in Γ. However, the converses hold if n ≥ 3 and we assume that Γ is hyperbolic and has (virtual) cohomological dimension n, see Corollary 23.3. (2) If Xn+1 = PGL(n + 1, R)/PO(n + 1) is the symmetric space associated to PGL(n + 1, R), then Corollary 19.3 implies that the orbit map τρ : Γ → Xn+1 is a quasi-isometric embedding if ρ is a Benoist representation. One need only recall that if x0 = [PO(n + 1)], then d(x0 , A(x0 )) = ||(log σ1 (A), . . . , log σn+1 (A))||. See Section 26 for a discussion of Xn+1 and a proof of the distance formula above. (3) See Cooper-Long-Tillman [70, Section 2] for a more general treatment of projective isometries of properly convex domains. 20. Benoist components If Γ is a hyperbolic group, we consider the set Ben(Γ, n) ⊂ Hom(Γ, PGL(n + 1, R)) of Benoist representations of Γ into PGL(n + 1, R). Koszul used the technology of (G, X)-structures to show that Ben(Γ, n) is open. We will later be able to give a simple proof that uses the stability of Anosov representations and a little convex geometry, see Theorem 33.6. Theorem 20.1. (Koszul [141]) If Γ is a hyperbolic group, then Ben(Γ, n) is an open subset of Hom(Γ, PGL(n + 1, R)). Choi and Goldman [66] showed that Ben(Γ, 2) is closed, and Benoist [22] proved Ben(Γ, n) is closed for all n. ANOSOV REPRESENTATIONS 71 Theorem 20.2. (Choi-Goldman [66], Benoist [22]) If Γ is a hyperbolic group, then Ben(Γ, n) is a closed subset of Hom(Γ, PGL(n + 1, R)). As a corollary of Theorems 20.1 and 20.2 we see that Ben(Γ, n) is a collection of components of Hom(Γ, PGL(n + 1, R)). Corollary 20.3. If Γ is a hyperbolic group, then Ben(Γ, n) is a collection of components of Hom(Γ, PGL(n + 1, R)). We will give a somewhat sketchy proof of the following weaker fact which gives some indication why Corollary 20.3 is true. This is essentially the argument given by Choi and Goldman [66], see also Marquis [158, Lemma 7.4]. Lemma 20.4. If {ρk : Γ → PGL(n + 1, R)} is a sequence in Ben(Γ, n) which converges to ρ ∈ Hom(Γ, SL(n + 1, R)) and ρ is strongly irreducible, then ρ ∈ Ben(Γ, n). Proof. Let Ωk be the strictly convex domain divided by ρk (Γ). We may pass to a subsequence so that {Ω̄k } converges to a closed subset K of RPn in the Hausdorff topology on the set of closed subsets of RPn . (We say that {Ωk } converges to K in the Hausdorff topology if the Hausdorff distance between Ωk and K converges to 0. One may easily check that the Hausdorff topology is compact.) Notice that K must be convex, since any two points in K are approximated by points in Ω̄k and hence by a projective line segment in Ω̄k . If K is properly convex and has non-empty interior Ω, then ρ(Γ) ⊂ Aut(Ω). Corollary 6.2 implies that ρ is discrete and faithful. Since ρ(Γ) is discrete and faithful, the quotient M = Ω/Γ is a manifold (or orbifold if Γ has torsion) homotopy equivalent to Mk = Ωk /ρk (Γ). Since Mn is compact, it follows that M is compact. Since Γ is hyperbolic, Ω is strictly convex. Therefore, we would see that ρ is a Benoist representation. If K has empty interior, then since it is convex, it lies in a proper projective sub-plane. Since ρ(Γ) preserves the proper subspace spanned by K, ρ is reducible, which contradicts our assumptions. Now suppose that K is not properly convex. The pre-image of each Ωk in S n is a pair of copies Ω± k of Ωk . Since each Ωk is properly convex, the closure of the two is separated by a disjoint hyperplane Pk . We may pass to a subsequence of {Pk } which converges to a hyperplane ± P separating S n into two hemispheres H + and H − . Then {Ω± k } converges to K . where each ± + + − K is a copy of K. If we label consistently throughout, K ⊂ H̄ and K ⊂ H̄ − . Notice that, by construction, K + ∩ P is the set of points which are antipodal to points in K − ∩ P . If K + and K − are disjoint, we may find v ∈ S n − (K + ∪ K − ) ∩ P and a plane P̂ through v ∪ −v which is disjoint from K + and K − , so K would be properly convex. Therefore, K + ∩ K − must be non-empty. Let pn : GL(n + 1, R) → PGL(n + 1, R) be the obvious projection map and let Θk be the intersection of all index two subgroups of p−1 n (ρk (Γ)) ∩ SL(n + 1, R). Then Θk has finite index in + − p−1 (ρ (Γ)) and preserves Ω and Ω . Therefore, the intersection Θ of all index two subgroups k n k k + and K − , and hence preserves K + ∩ K − . Therefore, Θ is of p−1 (ρ(Γ)) preserves both K n reducible, which contradicts the strong irreducibility of ρ.  Choi and Goldman complete the argument in dimension n = 2, by using special properties of low-dimensional representations to show that any limit of Benoist representation is strongly irreducible. This portion of the proof does not generalize, so we will not go into details here. 72 Remarks: Koszul [141] and Benoist [22] further showed that if Γ does not contain an infinite normal nilpotent sugbroup, then the space of representations of Γ which divide a properly convex set is a collection of components of Hom(Γ, PGL(n + 1, R)). 21. Projective bending The most general known way of constructing deformations of Benoist representations is the procedure known as projective bending. We begin with an algebraic description of projective bending, our “proof” that the projective bending of a lattice in PSO(n, 1) remains a Benoist representation will hopefully give some intuition for the geometry of this construction. Suppose that ρ0 : Γ → PSO(n, 1) is a discrete, faithful representation, so that Γ is torsion-free and M = Hn /ρ(Γ) is a closed manifold which contains a separating, connected, totally geodesic submanifold X. If we let M1 and M2 be the result of cutting the manifold M along X, then Γ = π1 (M ) = π1 (M1 ) ∗π1 (X) π1 (M2 ) = Γ1 ∗ΓX Γ2 and we may assume that ρ0 (ΓX ) ⊂ PSO(n − 1, 1) ⊂ PSO(n, 1)   1 ~0 T where PSO(n − 1, 1) is identified with a subgroup of PSO(n, 1) by identifying A with ~ 0 A (i.e. by sticking A in the lower righthand corner). Notice that if Bt ∈ PSL(n + 1, R) is the diagonal matrix with diagonal entries (ent , e−t , . . . , e−t ), then each Bt centralizes PSO(n − 1, 1) within PSO(n, 1). We then define, for all t ∈ R, the projective bending ρt : Γ → PSL(n, R) of ρ0 along X, by letting ρt (γ1 ) = ρ0 (γ1 ) for all γ1 ∈ Γ1 and letting ρt (γ2 ) = Bt ◦ ρ0 (γ2 ) ◦ B−t for all γ2 ∈ Γ2 . There is a similar construction, involving the associated HNN decomposition, when X is a totally geodesic, connected non-separating codimension one submanifold of M . Corollary 20.3 implies that the result of projective bending is always a Benoist representation, but we will later sketch a geometric proof. The motivation for projective bending was provided by the bending construction in hyperbolic geometry. We will describe the geometry of the hyperbolic construction first, since it is easier to visualize and hopefully will provide some intuition for the projective setting. We will restrict to the setting of surface groups (although the construction generalizes to higher dimensions). Let S be a closed surface and let ρ : π1 (S) → PSL(2, R) be a Fuchsian representation. Suppose that a simple closed curve C separates S into two components, S1 and S2 , and consider the associated group-theoretic decomposition Γ = π1 (S) = π1 (S1 ) ∗π1 (C) π1 (S2 ) = Γ1 ∗ΓC Γ2 . Let β be a generator of ΓC . We may assume that   λ 0 ρ(β) = 0 λ−1 (where λ > 1). Notice that ρ(β) commutes with  iθ  e 0 Bθ = ∈ PSL(2, C) 0 e−iθ and define ρθ : Γ → PSL(2, C) by letting ρθ (γ1 ) = ρ(γ1 ) for all γ1 ∈ Γ1 and letting ρθ (γ2 ) = Bθ ◦ ρ(γ2 ) ◦ B−θ for all γ2 ∈ Γ2 . Theorem 11.4 implies that ρθ is convex cocompact for small ANOSOV REPRESENTATIONS 73 values of θ, but it will not be convex cocompact for all θ. (One may see this by noticing that ρπ (π1 (S)) ⊂ PSL(2, R), but is clearly not Fuchsian.) There are pretty pictures of the result of bending a Fuchsian representation available on the web. See, for example, the bottom two pictures here: https://gauss.math.yale.edu/~yhm3/research/limset/pictures.html or here: https://www.dumas.io/poster/ We now give a geometric description of bending in the hyperbolic setting. (Here I really need a GIF of me waving my hands appropriately.) First notice that if Sρ = H2 /ρ(Γ) and β ∗ is the geodesic representative of C on Sρ , then the pre-image of β ∗ is an infinite collection of geodesics (each of which is the axis of an element of ρ(α) where α is conjugate to β) which divides H2 into infinite-sided polygons. Let C1 and C2 be components of the complement stabilized by ρ(Γ1 ) and ρ(Γ2 ). We will describe a ρθ -equivariant map hθ : H2 → H3 . Recall that H2 sits inside H3 , call this copy of H2 by the name P1 . We first map C1 into P1 by the identity map. If C1 and C2 intersect along the geodesic g, then let P2 be a totally geodesic copy of H2 which intersects P1 along g and makes an angle θ with P1 , and let hθ map C2 isometrically into P2 . The rest of the construction is forced by the fact that we require that hθ be ρθ -equivariant. Every time we extend hθ over a new copy CN of C1 or C2 , we assume that hθ has already been defined on an adjacent copy CA of C2 or C1 which lies in a hyperbolic plane PA . One then finds a hyperbolic plane PN meeting PA along the geodesic hθ (CA ∩ CN ) at an angle θ. We then map CN isometrically into PN . Continuing this process indefinitely produces hθ . Notice that since we have taken care to alway bend in the same direction, that hθ (H2 ) bounds a convex region in H3 , if hθ is an embedding. Let L be the minimum distance between two copies of the pre-image of β ∗ . One may use hyperbolic geometry to show that there exists θL so that any piecewise geodesic in H3 so that each segment has length at least L and two adjacent segments make an angle at most θ, then the piecewise geodesic is a quasi-isometric embedding. It then immediately follows that if |θ| < θL , then hθ is a quasi-isometric embedding, so ρθ is convex cocompact. (With a little more work one can show that hθ is an embedding). Armed with this intuition, we attempt to sketch a proof of the fact that the result of projective bending is always a Benoist representation. Our sketch should probably be viewed as an invitation to work out the proof for yourself and draw the pictures necessary to convince yourself. Proposition 21.1. If ρt : Γ → PSL(n + 1, R) is obtained as a result of projective bending of a discrete, faithful, cocompact representation ρ0 : Γ → PSO(n, 1), then ρt is a Benoist representation. We loosely follow the treatment in Goldman [104]. Goldman works in the case n = 2 and I encourage you to focus on this case first as well. (Our discussion will be somewhat awkward since we have not developed the natural language of developing maps.) We regard Hn as the unit disk Dn in an affine chart. Then, if π : Hn → M is the covering map, π −1 (X) is a discrete collection of totally geodesic hyperplanes in Hn , which appear in our model as a collection of disks in projective hyperplanes which accumulate only at ∂Hn . Each component of Hn −π −1 (X) is (the interior of) a component of either π −1 (M1 ) or π −1 (M2 ), each component has infinitely 74 many boundary components, each of which is a component of π −1 (X), and no two components of π −1 (Mi ) intersect along a component of π −1 (X). We now describe the domain Ωt so that ρt (Γ) divides Ωt . Let C1 and C2 be the components of π −1 (M1 ) and π −1 (M2 ) which intersect along the component P0 of π −1 (X) stabilized by ΓX . Notice that Bt (C2 ) is entirely contained in the crescent R0 = Bt (R0 ) bounded by the tangent planes to Dn at ∂P0 which contains Dn . If we work in the traditional affine chart A = {~x | xn+1 } = 1, then P0 is the unit disk in the x2 -x3 · · · xn−1 -xn -plane and the crescent R0 = {(x1 , x2 , . . . , xn ) | (0, x2 , . . . , xn ) ∈ P0 }. Pictures would help a lot here It follows that C1 ∪ Bt (C2 ) is properly convex since C1 and Bt (C2 ) are convex and any line segment joining a point in C1 to a point in Bt (C2 ) must pass through P0 . Moreover, C1 is preserved by Γ1 and C2 is preserved by Bt Γ2 B−t . The rest of the construction is determined equivariantly. We must show that the procedure generates a properly convex domain tesselated by copies of C1 and C2 . Each component P of ∂C2 has the form γ2 (P0 ) for some γ2 ∈ Γ. The equivariance of our construction forces us to attach Bt γ2 Bt−1 (C1 ) to C1 ∪ Bt (C2 ) along Bt (γ2 (P0 )). Notice that Bt γ2 Bt−1 (C1 ) is contained in the crescent RPt spanned by the tangent lines to Bt (Dn ) at Bt (∂P ) since C1 ⊂ RP0 and Bt γ2 Bt−1 (RP0 ) = RPt . Let [ X1 = C1 ∪ Bt (C2 ) Bt γ2 Bt−1 (C1 ) γ2 ∈Γ2 be the result of attaching copies of C1 along all components of Bt (∂C2 ). Notice that if P = γ2 (P0 ) is a component of ∂C2 , then Bt (γ2 (C1 )) is a properly convex subset of RPt , X1 is contained in RPT and P separates Bt γ2 Bt−1 (C2 ) from the remainder of X1 . Therefore, any line segment joining a point in Bt γ2 Bt−1 (C2 ) to a point in the remainder of X1 must pass through P . So, the line segment joining any two points in X1 may be divided up into segments joining copies of P which lie on the boundary of some translate of C1 and C2 , and hence lies in X1 . So, X1 is convex. Notice that if QPt denote the portion of RPt which lies on the same side of P as Bt (C2 ), then QPt ⊂ R0 . Moreover, if we choose some γ̂2 ∈ Γ2 − {id} and let P̂2 = γ̂2 (P0 ), then X1 − C1 is contained in the portion of RP̂t which lies on the same side of P0 as C2 , which 2 is bounded. Therefore, X1 is a bounded subset of A and hence properly convex. Similarly, each component P of ∂C1 has the form γ1 (P0 ) for some γ1 ∈ Γ. The equivariance of our construction forces us to attach γ1 (Bt (C1 )) to C1 ∪ Bt (C2 ) along γ1 (P0 ). Notice that each γ1 (Bt (C1 )) is contained in the crescent RPt = RP0 spanned by the tangent lines to Dn at ∂P . Let [ [ X2 = C1 ∪ Bt (C2 ) Bt γ2 Bt−1 (C1 )) γ1 (Bt (C1 )) γ2 ∈Γ2 γ1 ∈Γ1 be the result of attaching copies of C2 along all components of ∂C1 . We check, just as before, that X2 is convex and if we choose γ̂1 ∈ Γ1 − {id} and let P̂1 = γ̂1 (P0 ), then X2 is contained in RP̂t ∩ RP̂t , which is a bounded subset of A. Therefore X2 is properly convex. 1 2 Notice that Ωt is constructed by successively adding translates of C1 and C2 . Each time t we attach a new copy of Ci along α(P0 ), for some α ∈ ρt (Γ), then there is a crescent Rα(P , 0) bounded by tangent planes to α(∂Ω) passing through α(∂P0 ), which contains the previous t copies of C1 and C2 and is separated in Rα(P from the previous copies by α(P0 ). Therefore, 0) ANOSOV REPRESENTATIONS 75 the domain remains convex after adding the new copy of Ci . One also checks, via the nesting of the crescents, that Ωt is contained in RP̂t ∪ RP̂t , which is a bounded subset of A. Thus, Ωt 1 2 is properly convex. Since Ωt is divided by the hyperbolic group ρt (Γ) it is strictly convex, so ρt is a Benoist representation. Remarks: (1) Johnson and Millson [124] introduced projective bending, which generalizes the bending construction of Thurston [202] in the hyperbolic setting (see also Apanasov [9] and Kouroniotis [142]). Goldman [104], in dimension n = 2, and Benoist [22] were the first to notice that projective bending always produces Benoist representations. (2) Goldman used a more general version of Proposition 21.1, when n = 2, in his construction of a parametrization of the space of Benoist representations of π1 (S) into PSL(3, R). This space is homeomorphic to R16g−16 and his coordinates are natural generalizations of the FenchelNielsen coordinates. In this setting, Goldman refers to projective bending as bulging. Kapovich [129] and Marquis [157] produced much more general versions which work in all dimensions. (3) Johnson and Millson [124] used projective bending to show that if Γ is a cocompact lattice in PSO(n, 1), then Hom(Γ, PSL(n + 1, R) can have singularites at the identity when n ≥ 3. The singularities they discover arise when H3 /Γ contains intersecting totally geodesic codimension one submanifolds. (4) One may use a similar construction to define the twist flow on Teichmüller space. Suppose that C is a separating curve on a closed surface S determining the decomposition π1 (S) =  λρ 0 Γ1 ∗ΓC Γ2 and that β generates ΓC . If ρ ∈ T (S), we may assume that ρ(β) = where 0 λρ λρ > 1. We then define the time t twist of ρ about C as ρt where ρt |C1 = ρ and ρt = Bt ρB−t where # " t 0 e log λρ . Bt = −t 0 e log λρ 22. The Hilbert geodesic flow We recall, from section 14, that the Hilbert metric on a properly convex domain is induced by a Finsler norm FΩH . If ~v ∈ Tx Ω. let p+ = p+ (x, , ~v ) and p− = p− (x, ~v ) are the endpoints of the line segment {x + t~v | t ∈ R} ∩ Ω̄, oriented so that ~v points toward p+ . Then   |~v | 1 1 H FΩ (x, ~v ) = + 2 |x − p− | |x − p+ | If Ω is strictly convex, then ∂Ω is C 1 , so FΩH is also C 1 . We can then consider the unit tangent bundle T 1 Ω associated to the Hilbert metric on Ω and define a geodesic flow {φt }t∈R on T 1 Ω. Explicitly, if cx,~v : R → Ω is a unit speed geodesic so that cx,~v (0) = x and c0x,~v (0) = ~v , then φt (x, ~v ) = (cx,~v (t), c0x,~v (t)). One may calculate that φt (x, ~v ) = x+ et/2 − 1 |x − p+ |et/2 + |x − p− | ! ~v , ! ! et/2 |x − p+ |et/2 + |x − p− |
2 ~v . 76 We will assume for the remainder of the section that Ω is strictly convex and ∂Ω is C 1 . We define the stable leaf through (x, ~v ) by n o −−−−−−−−−−→ → ∩ ← F s (x, ~v ) = (y, w) ~ ∈ T 1 Ω | p+ (y, w) ~ = p+ (x, ~v ) and ← xy p− (x, ~v )p− (y, w) ~ ⊂ Tp+ (x,~v) Ω ∪{(x, ~v )}. The key observation is then: Lemma 22.1. (Benoist [20, Lemma 3.4]) If Ω is strictly convex and ∂Ω is C 1 , then F s (x, ~v ) = {(y, w) ~ ∈ T 1 Ω | lim dH ~ π(φt (x, ~v )) = 0} Ω (π(φt (y, w), t→∞ where π : T 1Ω → Ω is the projection onto the first factor. Proof. We first notice that limt→∞ π(φt (x, ~v )) = p+ (x, ~v ) and limt→∞ π(φt (y, w)) ~ = p+ (y, w), ~ + + H ~ π(φt (x, ~v )) = ∞. so if p (y, w) ~ 6= p (x, ~v ), then limt→∞ dΩ (π(φt (y, w), −−−−−−−−−−→ →∩← If p+ (y, w) ~ = p+ (x, ~v ), then let q = ← xy p− (x, ~v )p− (y, w). ~ Notice that q exists, since ←−−−−−−−− −−−−→ ← → both xy and p (x, ~v )p (y, w) ~ lie in the projective plane spanned by x, y and p+ (x, ~v ). Let xt = π(φt (x, ~v )) and yt = π(φt (y, w)) ~ Then, since [p− (x, ~v ), x, xt , p+ (x, ~v )] = [p− (y, ~v ), y, yt , p+ (x, ~v )] = e2t , we see, by considering 4 lines through q as in the figure, that −−−−→ →∩← yt = ← qx yp+ (x, ~v ). t (See BenositGeodesicFlowFigure1 on the Google Drive.) Suppose that wz =← x→ y ∩ Ω̄ t t t t and assume that the points occur in the order wt , xt , yt , zt . Our result is then equivalent to the claim that lim dH ~ π(φt (x, ~v )) = 0 if and only if if q ∈ Tp+ (x,v) ∂Ω. Ω (π(φt (y, w), t→∞ We first check that if q does not lie in Tp+ (x,v) ∂Ω, then lim t→∞ |wt − yt | > 1, |wt − xt | which implies that [wt , xt , yt , zt ] does not converge to 1, so dH ~ π(φt (x, ~v )) does not Ω (π(φt (y, w), converge to 0 as t → ∞ (and in fact is bounded below as t → ∞.) We normalize so that in our affine chart p+ (x, ~v ) = ~0, x = (1, 0, . . . , 0) and y lies in the x1 -x2 plane, so we may assume that n = 2. We may also assume that Tp+ (x,~v) ∂Ω is the x1 -axis. Let θ be the angle between the line joining p+ (x, v) to q and the line Tp+ (x,~v) ∂Ω and let η be the angle between p+ (x, ~v )x and p+ (x, ~v )y. (See BenoistGeodesicFlowFigure2 in the Google Drive.) We will assume that q and y lie on the same side of the x2 -axis. If δ + t = |xt − p+ (x, ~v )| and θt is the angle between ← x→ tq and Tp+ (x,v) ∂Ω, |wt − xt | ∼ δt sin θt and |xt − yt | = δt sin η , cos(θt + η) ANOSOV REPRESENTATIONS 77 so, since limt→∞ θt = θ and limt→∞ δt , |wt − xt | sin η sin θ lim =1+ > 1. t→∞ |wt − yt | cos(θ + η) On the other hand, if q ∈ Tp+ (x,v) ∂Ω, we may similarly check that limt→∞ dH ~ π(φt (x, ~v )) = 0, Ω (π(φt (y, w), which completes the proof.  We call F s (x, ~v ) the stable leaf, since it is easy to check that the collection of stable leaves gives a foliation of T 1 Ω by leaves of dimension n − 1 which is invariant under the geodesic flow, called the stable foliation. Consider the involution ι : T 1 Ω → T 1 Ω given by ι(x, v) = (x, −v). Then ι ◦ φt = φ−t ◦ ι for all t. We may then define the unstable leaf F u (x, ~v ) through (x, ~v ) by setting F u (x, ~v ) = s ι(Fι(x,~ ~ ∈ F u (x, ~v ) if and only if v ) ). Then, by definition and Lemma 22.1, we see that (y, w) limt→−∞ dH ~ π(φt (x, ~v )) = 0. Ω (π(φt (y, w), If Γ is torsion-free and ρ : Γ → PGL(n + 1, R) is a Benoist representation dividing a strictly convex domain Ω, then the geodesic flow on Ω descends to the geodesic flow {φ̂t } on T 1 Mρ where Mρ = Ω/ρ(Γ). Let π̂ : T 1 Mρ → Mρ be the obvious projection map. The stable and unstable foliations descend to foliations Fbs and Fbs of T 1 (Mρ ), so that x and y lie in the same leaf of Fbs if and only if limt→∞ d(π̂(φ̂t (x)), π̂(φ̂(y))) = 0 and x and y lie in the same leaf of Fbu if and only if limt→−∞ d(π̂(φ̂t (x)), π̂(φ̂t (y))) = 0. Moreover, these foliations determine a splitting T (T 1 Mρ ) = Eρs ⊕ Fρ ⊕ Eρu where Eρs |z is the tangent plane to the leaf of Fbs containing z, Fρ |z is the tangent line to the flow line through z, and Eρs |z is the tangent plane to the leaf of Fbs containing z. This splitting lifts to a splitting T (T 1 Ω) = Ẽρs ⊕ F̃ρ ⊕ Ẽρu . We say that a flow {φt } on a closed manifold N is Anosov if there exists a flow-invariant splitting T N = E+ ⊕ F ⊕ E− such that F is tangent to the flow line at each point and there exists constants C and a so that if v + ∈ E + or v − ∈ E − and t > 0, then ||Dφ−t (v + )|| ≤ Ce−at and ||Dφt (v − )|| ≤ Ce−at . (Notice that if a flow is Anosov with respect to one continuous family of norms on T N , then it is Anosov with respect to any other, since N is compact.) We place an equivariant continuous Finsler metric on T 1 Ω. If w ∈ T(x,~v) Ω, we decompose w = wu + wf + ws and let where wu ∈ Ẽρu , wf ∈ F̃ρ , and wu ∈ Ẽρs  1 2 ||w|| = ||Dπ(ws )||2 + ||Dπ(wf )||2 + ||Dπ(wu )||2 . (Notice that this norm is ρ(Γ)-equivariant by construction.) We now check that if us ∈ Eρs , then limt→−∞ ||Dφt (us )|| = 0. (The moral of the story is that the map φt takes the horoball π(F s (x, ~v )) to the horoball π(F s (φt (x, ~v ))) and contracts uniformly.) Let us = (x, ~u). We first normalize so that in our affine chart p+ (x, ~v ) = ~0, 78 x = (1, 0, . . . , 0), p− (x, ~v ) = (2, 0, . . . , 0), and Tp+ (x,~v) ∂Ω and Tp− (x,~v) ∂Ω are both parallel to the x2 -x3 · · · -xn plane, ~u also lies in the x2 -x3 · · · -xn plane, and we may normalize so that ~u = ce2 for some c 6= 0. One may then calculate that Dφt (us ) = |xt |us where π(φt (x, ~v )) = xt and so   1 c|xt | 1 s H ||Dφt (u )|| = FΩ (xt , |xt |~u) = + 2 |xt − wt | |xt − zt | where wt zt is the intersection of the line through xt in the direction ~u = ce2 with Ω̄. One then easily checks that limt→−∞ ||Dφt (us )|| = 0. (See Benoist [20, Section 3.2.6] or Crampon [75, Corollary 4.5] for details.) Similarly, if wu ∈ Eρu , then limt→∞ ||Dφt (wu )|| = 0. Since T 1 Mρ is closed, there exists t0 > 0 such that for all z ∈ T 1 Mρ , there exists tz ∈ [0, t0 ] such that if v u ∈ Eρu , then ||Dφt (v u )|| ≤ 12 ||v u || and if v s ∈ Eρs , then ||Dφt (v s )|| ≤ 12 ||v s ||. ||Dφt (v)|| 2 | v ∈ T (T 1 Mρ ), |t| ≤ t0 , ||v|| 6= 0} Therefore, we may choose a = log t0 and C = sup{ ||v|| and check that the splitting T (T 1 Mρ ) = Eρs ⊕ F ⊕ Eρu is an Anosov splitting with constants C and a. Theorem 22.2. (Benoist [20]) If Γ is torsion-free and ρ : Γ → PL(n + 1, R) is a Benoist representation, then its geodesic flow is Anosov. This generalizes the classical fact that the geodesic flow of a closed hyperbolic manifold is Anosov. Another natural part of the picture here, which we will not elaborate on, is the Busemann function of Ω (see Crampon [75, Section 4.1]). Suppose that Ω is strictly convex and ∂Ω is C1 . → ∩ ∂Ω and q be the point of intersection of T ∂Ω and T ∂Ω. If z ∈ ∂Ω and x, y ∈ Ω, let w = − xy w z →∩− → where the sign is positive if y Then Bz (x, y) is the (signed) distance between y and r = − yz qx lies between r and z and negative otherwise. If (x, v) ∈ T 1 (Ω), then we define the generalized horoball H(x,~v) = {y ∈ Ω | Bp+ (x,~v) (x, y) = 0} and notice that F u (x, ~v ) = {(y, w) ~ | y ∈ H(x,~v) and p+ (x, ~v ) = p+ (y, w)} ~ and F s (x, ~v ) = {(y, w) ~ | y ∈ H(x,−~v) and p− (x, ~v ) = p− (y, w)}. ~ Moreover, one may check that Bz (x, y) = lim d(x, zn ) − d(y, zn ) zn →z where {zn } is any sequence in Ω converging to z. Remark: Benoist [20] further proves that if Γ divides a properly convex domain Ω, then Ω is strictly convex if and only if the geodesic flow on Ω/Γ is Anosov. 23. Further topics: Convex projective manifolds I went walking in the wasted city Started thinking about entropy Smelled the wind from the ruined river Went home to watch TV ANOSOV REPRESENTATIONS 79 —————Warren Zevon [228] The geodesic flow and entropy. Benoist [20] further proves that if Ω is strictly convex and divisible, then ∂Ω is C 1+α for some α > 0. Recall that a function f : U → R is C 1+α on an open subset U of Rd if it is C 1 and there exists C > 0 so that |f (y) − f (x) − Dfx (y − x)| ≤ C|x − y|1+α for all x, y ∈ U . We say that ∂Ω is C 1+α if it is locally the graph of a C 1+α function. It follows that if ρ is a Benoist representation, then the geodesic flow on T 1 Mρ is C 1+α for some α > 0. Since {(γ + , γ − ) |γ ∈ Γ infinite order} is dense in ∂Γ × ∂Γ whenever Γ is a hyperbolic group (see Benakli-Kapovich [126]), {(ξρ (γ + ), ξρ (γ − ))} is dense in ∂Ω × ∂Ω if ρ is a Benoist representation dividing Ω. It follows that closed geodesics are dense in T 1 Mρ . Anosov [8] showed, as a consequence of the Anosov Closing Lemma, that an Anosov flow is topologically transitive if periodic orbits are dense. Recall that a flow is topologically transitive, if U and V are two open subsets of T 1 Ω, then φt (U )∩V is non-empty for some t. Benoist [20] further shows that the geodesic flow is topologically mixing, i.e. if U and V are open subsets of T 1 Mρ , then there exists T so that if t ≥ T , then φt (U ) ∩ V is non-empty. (Benoist uses his classification of the Zariski closures of Benoist representations and the main result of Benoist [17] to show that the subgroup of R>0 generated by lengths of closed geodesics is dense. The fact that T 1 Mρ is topologically mixing then follows from standard results, see for example Hasselblatt-Katok [112, Exercise 18.3.4].) Once we know that the geodesic flow is C 1+α and topologically transitive, we may start applying the powerful methods of the Thermodynamic formalism. In particular, one may define the topological entropy of a Benoist representation to be the exponential growth rate of the number of closed orbits of the geodesic flow on T 1 Mρ with period at most T , i.e. log #{[γ] ∈ [Γ] |`ρ (γ) ≤ T } T →∞ T where [Γ] is the collection of conjugacy classes of elements of Γ. Moreover, Benoist [20] shows that h(ρ)T (#{[γ] ∈ [Γ] |`ρ (γ) ≤ T }) lim = 1. T →∞ eh(ρ)T (See Sambarino [185, 186] for generalizations of Benoist’s result.) Crampon [74] proved the following remarkable rigidity result. h(ρ) = lim Theorem 23.1. (Crampon [74]) If ρ : Γ → PGL(n + 1, R) is a Benoist representation, then h(ρ) ≤ n − 1 with equality if and only if ρ(Γ) is conjugate into PO(n, 1). Crampon [74] also proves that h(ρ) is the exponential growth rate of the volume of balls of radius T on the universal cover of Mρ , so the same result holds for volume growth entropy. Potrie and Sambarino [179] give a very different proof of Theorem 23.1. Crampon [75] notes that entropy varies continuously on Ben(Γ, n). Pollicott and Sharp [177] show that entropy varies analytically on Ben(Γ, 2). Bridgeman, Canary, Labourie and Sambarino [45] show that entropy varies analytically over the smooth points of Ben(Γ, n) for all n. (The key new tool here is a proof that the limit map varies analytically over smooth points 80 of Ben(Γ, n).) If n = 2, Nie [170] and Zhang [230] show that the entropy achieves every value between 0 and 1 on Ben(Γ, 2). Kim [139] and Cooper-Delp [68] showed that if ρ, σ ∈ Ben(Γ, n), then ρ is conjugate to either σ or σ ∗ if and only if `σ (γ) = `ρ (γ) for all γ ∈ Γ. Bridgeman, Canary, Labourie and Sambarino [45] show that ρ is conjugate to σ if and only if |λ1 (σ(γ))| = |λ1 (ρ(γ))| for all γ ∈ Γ. Bridgeman, Canary and Labourie [47] show that if n = 2 and Γ is a closed surface group of genus at least 3, then it suffices to consider elements representing simple closed curves in both of the previous results. We will discuss rigidity properties of Benoist and Anosov representations more fully in Section 47. A characterization of Benoist representations. Canary and Tsouvalas [60] give a characterization of Benoist representations when n ≥ 3. Recall that Γ has cohomological dimension m, if m is the minimal dimension so that if R is any ZΓ-module, then H r (Γ, R) = 0 if r > m. For example, if X is a closed n-manifold with contractible universal cover, then π1 (M ) has cohomological dimension n. Theorem 23.2. (Canary-Tsouvalas [60]) Suppose that n ≥ 3, Γ is a torsion-free hyperbolic group of cohomological dimension n, and ρ : Γ → PSL(n + 1, R) is a representation. If there exists a continuous, non-constant ρ-equivariant map ξ : ∂Γ → RPn , then ρ is a Benoist representation. The direct product of a Fuchsian representation into SL(2, R) and a trivial one-dimensional representation, provides a counterexample to this characterization when n = 2. We may combine Theorem 23.2 with Theorems 35.1 and 36.1 to obtain the following partial converses to Corollaries 19.2 and 19.3. Corollary 23.3. Suppose that Γ is a torsion-free hyperbolic group of cohomological dimension n and ρ : Γ → SL(n + 1, R) is a representation. If either (1) There exists J and B so that   1 |λ1 (ρ(γ))| ≥ ||γ|| − B J||γ|| + B ≥ log |λn+1 (ρ(γ))| J for all γ ∈ Γ, or (2) There exists L and D so that  Ld(1, γ) ≥ log σ1 (ρ(γ)) σn+1 (ρ(γ))  ≥ 1 d(1, γ) − D L for all γ ∈ Γ, then ρ is a Benoist representation. Dynamics on Benoist components. Fricke’s Theorem generalizes to the setting of Benoist components. Theorem 23.4. If Ben(Γ, n) is non-empty, then Out(Γ) acts properly discontinuously on Ben(Γ, n)/PGL(n + 1, R). Theorem 23.4 is a special case of a result of Guichard and Wienhard [109, Theorem 5.4] (see also [55, Theorem 6.2]) concerning actions of outer automorphism groups on deformation ANOSOV REPRESENTATIONS 81 spaces of Anosov representations. We will discuss Guichard and Wienhard’s result more fully in Section 5, see Theorem 28.2. If n ≥ 3, then Out(Γ) will be finite, so this result is only relevant when n = 2, where it was first proved by Goldman [104], see also Labourie [145, Theorem 1.0.2]. Finite volume strictly convex projective manifolds. Cooper-Long -Tillmann [70] made an extensive study of finite volume convex projective manifolds. In particular, they extend Benoist’s characterization of strictly convex closed projective manifolds, see Theorem 16.1, to this setting. Theorem 23.5. (Cooper-Long-Tillmann [70, Theorem 0.15]) Suppose that Ω ⊂ RPn is properly convex, Γ ⊂ Aut(Γ) is discrete and torsion-free, and N = Ω/Γ has finite volume and is homeomorphic to the interior of a compact manifold M Then the following are equivalent: (1) Ω is strictly convex. (2) ∂Ω is C 1 (3) Γ is hyperbolic relative to the subgroups associated to boundary components of M . Among the structural tools they develop in this setting are an analogue of the Margulis Lemma ([70, Theorem 0.1], see also Crampon-Marquis [76]), a thick-thin decomposition ([70, Theorem 0.2]), see Choi [65] for the surface case) and a proof that cusps of finite volume strictly convex manifolds are projectively equivalent to cusps of finite volume hyperbolic manifolds ([70, Theorem 0.5]). In a sequel paper [71], they prove an analogue of Koszul’s open-ness theorem for strictly convex projective manifolds of finite volume. Marquis [155, 156] had earlier studied deformation spaces of finite area properly convex surfaces. Barthelmé, Marquis and Zimmer proved a generalization of Crampon’s rigidity theorem for finite volume properly convex projective manifolds. We recall that the volume growth entropy of a properly convex manifold N = Ω/Γ is given by
log vol(B(x0 , T )) hvol (N ) = lim T →∞ T where vol(B(x0 , T )) is the (Hilbert) volume of the (metric) ball of radius T about a fixed point x0 ∈ Ω. Theorem 23.6. (Barthelmé-Marquis-Zimmer [13]) Suppose that Ω ⊂ RPn is properly convex, Γ ⊂ Aut(Γ) is discrete and torsion-free, and N = Ω/Γ has finite volume. Then hvol (N ) ≤ n − 1 with equality if and only if Γ is conjugate into O0 (n, 1). Manning [151] showed that if N is a compact Riemannian manifold with non-positive sectional curvature, then the topological entropy of its geodesic flow and its volume growth entropy coincide. Crampon [74] established the same result in the setting of closed strictly convex projective manifolds and Crampon-Marquis [78] extended it to the setting of finitely volume strictly convex projective manifolds. It is unknown whether it extends to finite volume properly convex projective manifolds. Adeboye, Bray and Constantine [4, 43] have also studied the relationship between volume and entropy for finite volume striclty convex projective manifolds. 82 Closed properly convex projective manifolds. Benoist actually develops much of his theory in the setting of discrete cocompact actions of groups of projective automorphisms of a properly convex domain Ω. A key role in this enlarged theory are properly embedded triangles, i.e. Euclidean triangles whose edges are in ∂Ω and whose interior lies in Ω. In dimension 3, Benoist [23] showed that the JSJ decomposition of a properly convex projective manifold is realized by a disjoint collection of properly embedded triangles. Benoist also exhibited examples but it is a subject of current interest to ascertain which irreducible closed 3-manifolds admit properly convex projective structures, see, for example, Ballas-Danciger-Lee [10]. Bobb [30] and Islam-Zimmer [120, 121] have successfully generalized portions of Benoist’s 3-dimensional results to higher dimensions. Bray [41, 42] has studied dynamics on closed properly convex projective 3-manifolds, proving that their geodesic flows are topologically mixing and are ergodic with respect to their Bowen-Margulis measures.