Anosov representations-3

ANOSOV REPRESENTATIONS 37 Part 3. Convex cocompact representations in rank one Lie groups Just us kids hangin’ out today Watchin’ our long hair turnin’ gray Not so skinny maybe not so free Not so many as we used to be —————James McMurtry [164] We begin with a quick review of the Teichmüller space of a closed surface. This theory is the motivation for much of what is now known as Higher Teichmüller theory. In particular, Fuchsian representations are the prototypical example of an Anosov representation. The first generalization of the theory of Fuchsian representations is the theory of convex cocompact representations into rank one Lie groups. They are also another class of classical examples of Anosov representations. We will survey this theory with an emphasis on the aspects of the theory which inspire and generalize to the theory of Anosov representations. We will restrict to the case of O0 (n, 1) which is the isometry group of real hyperbolic space Hn . Everything we do in this section generalizes in some form to the other rank one Lie groups, which are “essentially” U(n, 1), the isometry group of complex hyperbolic space CHn , Sp(n, 1), the group of orientation-preserving isometries of quaternionic hyperbolic space, and the isometry group of the Cayley plane. See, for example, Bridson-Haefliger [49, Section II.10] or Parker’s lecture notes [173] (which are available on his webpage). Our working definition of a rank one Lie group is that it is a semi-simple Lie group whose quotient symmetric space is negatively curved. This turns out to be equivalent to the assumption that the quotient symmetric space does not contain an isometrically embedded Euclidean plane. More generally, the rank of a semi-simple Lie group is the maximal dimension of an isometrically embedded copy of Euclidean space in the quotient symmetric space. 9. Teichmüller space: a refresher And it’s here I see pictures and my madness is clear And there’s no longer logic so therefore no fear ————–Ian Hunter [118] If you are not already somewhat familiar with Teichmüller theory, I recommend you immediately put down these notes and go read a more complete treatment of this beautiful subject. Farb and Margalit [96] give a nice treatment from a modern geometrical/topological viewpoint. Bers’ survey paper [27] is a beautiful treatment of the classical complex analytic approach (and contains an oblique, but poignant, commentary on the tension of working in a subject named after an ardent Nazi). Thurston [203, Section 4.6] gives a concise treatment of the FenchelNielson coordinates. Abikoff [2] gives a treatment of the classical theory with an eye towards the modern viewpoint. Recall that the upper half-plane model for the hyperbolic plane is given by H2 = {z ∈ C | Im(z) > 0} with Riemannian metric ds2hyp = 1 dxdy. y2 38 In this metric, the geodesics are lines and semi-circles perpendicular to the real line and the group of Möbius transformations with real co-efficients acts as the group of orientationpreserving isometries of H2 , i.e. Isom+ (H2 ) = PSL(2, R). The hyperbolic plane has constant curvature −1, which is manifested explicitly in the fact that if P is a n-gon in H2 with internal angles {α1 , . . . , αn }, then Area(P ) = π(n − 2) − n X αi . i=1 A complete orientable Riemannian surface X is said to be hyperbolic if it is locally isometric to H2 . In this case, the universal cover X̃ is a simply connected complete Riemannian manifold locally isometric to H2 and hence can be identified with H2 . Therefore, X = H2 /Γ where Γ is a discrete subgroup of Isom+ (H2 ) ∼ = PSL(2, R). Notice that Γ is only well-defined up to conjugacy, since the identification of X̃ with H2 is not canonical. A marked hyperbolic structure on a closed orientable surface S is a pair (X, f ) where f : S → X is a homeomorphism and X is a hyperbolic surface. If X = H2 /Γ, then f∗ : π1 (S) → π1 (X) ∼ = Γ is an isomorphism and hence we obtain a discrete, faithful representation ρ : π1 (S) → PSL(2, R). However, ρ is only well-defined up to conjugation in PGL(2, R) (which we interpret as the full group of isometries of H2 ). In classical Teichmüller theory, both S and X are oriented so one gets a representation which is well-defined up to conjugacy into PSL(2, R). It will be more convenient for us to ignore orientation. One may view this as a sign of the depravity I have fallen into since entering Higher Teichmüller theory. I used to hate the fool in me, but only in the morning Now I tolerate him all day long —————Mike Cooley [90] One may build a hyperbolic surface of genus two, by starting with a regular hyperbolic octagon, all of whose internal angles are π4 and then gluing by the standard gluing pattern. Similarly, one may build a hyperbolic surface of genus g by starting with a regular (4g − 4)-gon π with internal angles 2g . In turn, one can build any hyperbolic surface of genus two from a hyperbolic octagon, all of whose angles add up to 2π. One can see this, by first noticing that the surface is obtained this way topologically, so one has a bouquet of circles on the hyperbolic surface whose complement is an open disk. If we fix the vertex of the bouquet of circles at one point on the surface and pull the edges tight so that they form geodesic arcs, then the complement of the resulting geodesic bouquet of circles is a hyperbolic octagon. One might then try to guess how big the Teichmüller space of marked hyperbolic structures on S is. The space of hyperbolic octagons is 16-dimensional, since an octagon is determined by its vertices. There are 5 constraints, coming from the fact that the internal angles must add up to 2π and that lengths of edges that are paired must agree, giving a 11-dimensional space of allowed octagons, but PSL(2, R) acts as congruences of these octagons, so one really has a 8-dimensional space of octagons. Finally, an octagon gives a hyperbolic surface plus a specified point on that surface, so the space of hyperbolic surfaces should be 6-dimensional. ANOSOV REPRESENTATIONS 39 We will choose to formalize Teichmüller space by using representations. Recall that a marked hyperbolic structure on a closed surface S, gives rise to a (conjugacy class of a) discrete, faithful representation ρ : π1 (S) → PSL(2, R). In turn, a discrete, faithful representation ρ : π1 (S) → PSL(2, R) gives rise to a hyperbolic surface Xρ = H2 /ρ(π1 (S)). Since Xρ is homotopy equivalent to S, it is homeomorphic to S. Moreover, there is homeomorphism hρ : S → Xρ so that (hρ )∗ is conjugate to ρ. (Here, we are using a special property of the topology of closed surfaces. The Nielsen-Baer Theorem, see Farb-Margalit [96, Chapter 8], gives that every homotopy equivalence of a closed orientable surface is homotopic to a homeomorphism.) We then let DF(π1 (S), PSL(2, R)) = {ρ : π1 (S) → PSL(2, R) | ρ discrete, faithful} and the Teichmüller space of S is the quotient T (S) = DF(π1 (S), PSL(2, R))/PGL(2, R) where DF(π1 (S), PSL(2, R)) inherits a topology as a subset of Hom(π1 (S), PSL(2, R)), PGL(2, R) acts by conjugation and T (S) inherits the quotient topology. Notice that the Milnor-Svarc Lemma implies that if ρ ∈ DF(π1 (S), PSL(2, R)), then the orbit map τρ : π1 (S) → H2 is a quasi-isometry and Corollary 3.6 implies that there is a ρ-equivariant homeomorphism ξρ : ∂π1 (S) → ∂H2 ∼ = S1. Alternatively, one may define T (S) to be the space of marked hyperbolic structure on S up to the equivalence (X1 , f1 ) ∼ (X2 , f2 ) if and only if f2 ◦ f1−1 is homotopic to an isometry. One may think of X as hyperbolic clothing for the naked topological surface S and f as instructions for how to wear the clothing. The equivalence relation allows one to adjust the clothing, but not to wear it backwards or to stick your head through the hole designated for the arm. It is a classical theorem, going back to the 19th century, that T (S) is homeomorphic to R6g−6 if g ≥ 2 is the genus of S. (Notice that π1 (S) has a presentation with 2g relations and one relation, one would expect that DF(π1 (S), PSL(2, R)) has dimension (2g)3 − 3 = 6g − 3, so one would predict that Teichmüller space has dimension 6g − 6.) The mapping class group Mod(S) is the group of (isotopy classes of) self-homeomorphisms of S. Fricke showed that the mapping class group acts properly discontinuously, but not freely, on T (S) and its quotient is the moduli space of unmarked hyperbolic structures on S. We will soon prove a very general version of Fricke’s theorem. There are a variety of metrics on Teichmüller space, all of which have their own advantages and disadvantages. The most prominent are the Teichmuller metric, which is complete, but only a Finsler metric and is not non-positively curved, and the Weil-Petersson metric which is Riemannian and negatively curved, but not complete. Teichmüller showed that Teichmüller space has a natural complex structure (I.e. one invariant under the action of the mapping class group) and Ahlfors [5] showed that the Weil-Petersson metric is Kähler. Wolpert (see [219] for a survey) extensively studied the resulting symplectic structure on Teichmüller space. This hopefully provides a quick taste of the bounty of structure associated to Teichmüller space. We now give a quick sketch of the Fenchel-Nielsen coordinates on Teichmüller space. Suppose that X is a closed orientable hyperbolic surface of genus g ≥ 2. Recall that, since X is negatively curved, every homotopically non-trivial closed curve is homotopic to a unique closed geodesic. Morever, if two homotopically non-trivial simple closed curves are disjoint and non-parallel, then their geodesic representatives are also disjoint. Let α = {α1 , . . . , α3g−3 } be a maximal collection of disjoint simple closed curves and let α∗ be their geodesic representatives on X. 40 pictures needed The components of X − α∗ are a collection of 2g − 2 hyperbolic pairs of pants with geodesic boundary. (A topological pair of pants is a disk with two holes.) Therefore, every closed hyperbolic surface may be built from hyperbolic pairs of pants. If P is a hyperbolic pair of pants with geodesic boundary and s1 , s2 and s3 are the shortest paths joining boundary components (called seams), then P − {s1 , s2 , s3 } is a pair of all-right hyperbolic hexagons (i.e. hexagons all of whose interior angles are π2 ). An all-right hexagon is determined by the lengths of any 3 non-consecutive sides. Moreover, any 3 lengths can be achieved. It follows that P is the double of the unique all-right hexagon with alternate sides having lengths agreeing with the lengths of the seams of P . Moreover, we can build a geodesic pair of pants with any collection of boundary lengths and this geodesic pair of pants is entirely determined by its boundary lengths. So the hyperbolic structure on X is determined, up to isometry, by the lengths of the components of α∗ and instructions for gluing the pants together. Since the pants are glued along closed geodesic curves, there is a one-dimensional space of ways to glue them. This suggests more forcefully that the space of hyperbolic structures on X has dimension 6g − 6. More formally, we get a map L : T (S) → R3g−3 where +  3g−3 L(X, f ) = `X (f (αi )∗ ) . i=1 At each element of α we can define a twist coordinate in R which records how the geodesic pairs of pants are glued along f (αi )∗ , so we obtain Θ : T (S) → R3g−3 . (I will wave my hands about this in class, but you can read about it carefully elsewhere.) One can then see that (L, Θ) : T (S) → R3g−3 × R3g−3 ∼ = R6g−6 + is a homeomorphism. For a careful discussion of twist coordinates see, for example, Thurston [203, Section 4.6], Farb-Margalit [96, Section 10.6] or Martelli [159, Chapter 7]. One of the crucial properties of Teichmüller space is that it is an entire component of the representation variety. In some circles, a Higher Teichmüller space is defined to be a component of the character variety consisting of Anosov representations, but I would argue that this definition is too restrictive. Theorem 9.1. If S is a closed oriented surface of genus g ≥ 2, then T (S) is a component of X(π1 (S), PSL(2, R)) = Hom(π1 (S), PSL(2, R))/PGL(2, R)). We will sketch a simple hands-on proof in our situation. Proof. It suffices to prove that T (S) is open and closed in X(π1 (S), PSL(2, R)), since we have already sketched a proof that it is connected. Suppose that a sequence {[ρn ]} ⊂ T (S) converges to [ρ] ∈ X(π1 (S), PSL(2, R)). Then, one may find representatives, ρn ∈ Hom(π1 (S), PSL(2, R)) which converge to a representative ρ of [ρ]. Corollary 6.2 implies that ρ is also discrete and faithful, so [ρ] ∈ T (S). Therefore, T (S) is closed. In the proof of open-ness we will restrict to the case where S has genus 2, but the proof generalizes. Suppose [ρ] ∈ T (S) and ρ : π1 (S) → PSL(2, R) is a representative. We have seen that Xρ may be obtained by gluing up a hyperbolic octagon. So there is an octagon D which is a fundamental domain for the action of ρ(Γ) on H2 . Then we can find a generating set {a1 , a2 , a3 , a4 } for π1 (S) so that the edges of D occur in the order E1 , E2 , ρ(a1 )(E1 ), ρ(a2 )(E2 ), E3 , E4 , ρ(a3 )(E3 ), ρ(a4 )(E4 ). ANOSOV REPRESENTATIONS 41 Let Li be the bi-infinite geodesic containing Ei . Then, we can choose a neighborhood U of ρ in Hom(π1 (S), PSL(2, R)), so that if σ ∈ U , then L1 , L2 , σ(a1 )(L1 ), σ(a2 )(L2 ), L3 , L4 , σ(a3 )(L3 ), σ(a4 )(L4 ) cut out an octagon Dσ close to (and combinatorially equivalent to) D. One can then show that Dσ glues up to give a hyperbolic surface of genus two, and so is a fundamental domain for the action of σ(π1 (S)) on H2 . It follows that U ⊂ DF (π1 (S), PSL(2, R)) and that the projection [U ] of U is an open neighborhood of ρ in X(π1 (S), PSL(2, R)) which is contained in T (S). (Notice that the pre-image of [U ] is the PGL(2, R) orbit of U which is open in Hom(π1 (S), PSL(2, R)) and contained in DF(π1 (S), PSL(2, R)).) It follows that T (S) is open in X(π1 (S), PSL(2, R)) which completes the proof.  We will later see several facts which allow one to give a more general and more conceptual proof that T (S) is open. We will see (Theorem 11.4) that every representation into O0 (n, 1) whose orbit map is a quasi-isometric embedding into Hn has a neighborhood consisting of representations whose orbit maps are quasi-isometric embeddings. Since a representation of a torsion-free group whose orbit map is a quasi-isometric embedding is discrete and faithful and O0 (2, 1) ∼ = PSL(2, R), this gives a proof of open-ness. 10. Hyperbolic geometry in dimension n > 2 You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life....I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind..... I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time.... For God’s sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life. —–Farkas Bolyai [165] (advising his son not to work on hyperbolic geometry).1 One may again define (real) hyperbolic space of dimension n as the unique simply connected manifold of dimension n with constant sectional curvature −1. There is again an upper half space model for hyperbolic n-space Hn = {~x ∈ Rn | xn > 0} with line element q 1 dx21 + · · · + dx2n . ds = xn It is easy to see that the subset Y = {~x ∈ Hn | x2 = · · · = xn−1 = 0} is a totally geodesic copy of H2 sitting within Hn (since reflection in the x1 -xn -plane fixes Y ). One can then check that inversions in hemispheres orthogonal to {xn = 0} and reflections in hyperplanes orthogonal to {xn = 0} give isometries of Hn . Since, the group generated by these inversions and reflections 1I found this quote, which I had only seen portions of before, in a delightful blog post by Evelyn Lamb. https://blogs.scientificamerican.com/roots-of-unity/hyperbolic-quotes-about-hyperbolic-geometry/ 42 acts transitively on the orthonormal frame bundle of Hn , we see that this group is the full isometry group of Hn . Therefore, Hn has constant sectional curvature, and since it contains a totally geodesic copy of H2 , the constant is −1. Moreover, one sees that the geodesics are semi-circles and lines orthogonal to {xn = 0}. In the case n = 3, we may identify this group with the group of conformal automorphisms b = C ∪ {∞} = ∂H3 . So, in this case we can identify Isom+ (H3 ) with of the Riemann sphere C PSL(2, C). However, in general, this group doesn’t have a nice presentation as a matrix group from this viewpoint. By introducing the hyperboloid model for Hn , we can identify Isom+ (Hn ) with PSO(n, 1) (or with SO0 (n, 1) if one prefers). Let B(~x, ~y ) = x1 y1 + x2 y2 + · · · + xn yn − xn+1 yn+1 be the bilinear form associated to the quadratic form Q(~x) = B(~x, ~x) = x21 + · · · + x2n − x2n+1 of signature (n, 1). Consider the hyperboloid H n with two sheets in Rn+1 given by H n = {~x ∈ Rn+1 |Q(~x, ~x) = −1}. Let O(n, 1) be the set of matrices A ∈ GL(n + 1, R) preserving the indefinite quadratic form Q. Explicitly, O(n, 1) = {A ∈ GL(n + 1, R) | AT JA = J} where J is the diagonal matrix with entries (1, 1, . . . , 1, −1). Let’s check that this is the correct formula. If ~x, ~y ∈ Rn+1 and A ∈ GL(n + 1, R), then B(~x, ~y ) = ~xT J~y and B(A~x, A~y ) = (A~x)T JA~y = ~xT AT JA~y . Therefore, AT JA = J if and only if B(A~x, A~y ) = B(~x, ~y ) for all ~x, ~y ∈ Rn+1 . Notice that if U ∈ O(n) ⊂ GL(n, R), then   U 0 ∈ O(n, 1) 0 1 so O(n) may be identified with a subgroup of O(n, 1) which acts transitively on the set of orthonormal frames for Ten+1 H n . One may also check that   ~0 ~0 In−1 0 · · · 0 cosh t sinh t  ∈ O(n, 1) 0 · · · 0 sinh t cosh t for all t ∈ R, where In−1 is the identity matrix in SL(n − 1, R) and ~0 is the trivial vector in Rn−1 . Therefore, O(n, 1) acts transitively on the intersection of the upper sheet of H n with the xn -xn+1 -plane. Moreover J ∈ O(n, 1) takes en+1 to −en+1 . Combining these observations we see that O(n, 1) acts transitively on the orthonormal frame bundle of T H n . Notice that Ten+1 H n = {~x ∈ Rn+1 | xn+1 = 0}, so B restricts to a positive definite bilinear form on Ten+1 H n . Since O(n, 1) preserves B and acts transivitely on H n , we see that B restricts to a positive definite bilinear form on T~x H n for all ~x ∈ H n . Therefore, B induces a Riemannian metric on H n . ANOSOV REPRESENTATIONS 43 Since O(n, 1) acts as a group of isometries of H n and acts transitively on the space of orthonormal frames in T H n , H n has constant sectional curvature and O(n, 1)/O(n) = Hn . There is a nice distance formula for points on the upper sheet of H n given by d(~x, ~y ) = arccosh (−B(~x, ~y )) = arccosh(−x1 y1 − · · · − xn yn + xn+1 yn+1 ). Notice that reflection in the xn -xn+1 -plane lies in O(n, 1) so is an isometry of H n . Therefore the intersection Z of the xn -xn+1 -plane with H n is a geodesic in H n . One then checks that this formula works for ~x = 0 and ~y = (0, · · · , 0, sinh t, cosh t) and then observes that the formula is invariant under O(n, 1) and that any two points may be moved into this position by an element of O(n, 1). Notice that since O(n, 1) acts transitively on the orthonormal frame bundle of T H n , the Riemannian metric on H n has constant sectional curvature. In order to evaluate the curvature we can examine the right-angled triangle with vertices en+1 , (0, · · · , 0, sinh t, cosh t) and (0, · · · , 0, sinh t, 0, cosh t), which has two sides of length |t| and one side of length arccosh(cosh2 (t)). It is immediately clear that this formula is inconsistent with positive or zero curvature, and one may compare with H2 to check that it is consistent with sectional curvature −1, so in fact H n must have constant sectional curvature exactly −1. We can thus identify Hn with the top sheet of H n (i.e. the sheet containing en+1 ). The isometry group of Hn is the component O0 (n, 1) of isometries of H n which preserve the top sheet. Equivalently, we can identify Isom(Hn ) with PO(n, 1) and identify Isom+ (Hn ) with PSO(n, 1). Notice that there is an isomorphism τ3 : PSL(2, R) → PSO(2, 1) ⊂ PSL(3, R) which is also known as the irreducible representation. There is also an isomorphism between PSL(2, C) and PSO(3, 1). Let K be the copy of O(n) sitting inside of O0 (n, 1) and fixing the xn+1 -axis. Every element of K has the form   U 0 0 1 where U ∈ O(n) ⊂ GL(n, R). Then K is the stabilizer of the point en+1 within O0 (n, 1). Since O0 (n, 1) acts transitively on Hn , we may identify Hn = O0 (n, 1)/K where, if B ∈ O0 (n, 1), the coset BK is identified with B(en+1 ) ∈ Hn . Let A be the subgroup of all elements of O0 (n, 1) of the form   ~0 ~0 In−1 At = 0 · · · 0 cosh t sinh t  ∈ O(n, 1) 0 · · · 0 sinh t cosh t for some t ∈ R. Notice that   " √1 cosh t sinh t = √12 sinh t cosh t 2 − √12 √1 2 # et 0 0 e−t " √1 2 − √12 √1 2 √1 2 # so the eigenvalues and singular values of At agree and are equal to {et , 1, . . . , 1, e−t } (in that order if t ≥ 0). 44 If T ∈ O0 (n, 1), then we may choose L ∈ K so that L−1 T (en+1 ) = (0, · · · , 0, sinh t, cosh t) where dHn (en+1 , T (en+1 )) = t ≥ 0. Then A−t L−1 T (en+1 ) = en+1 so there exists K ∈ K so that A−t L−1 T = K. So, to conclude, if T ∈ O(n, 1), then there exists K, L ∈ K and t ≥ 0 so that T = LAt K and dHn (en+1 , T (en+1 )) = t ≥ 0. This is the Cartan decomposition of O0 (n, 1), also known as the KAKdecomposition. It turns out that A is a maximal abelian subgroup and K is a maximal compact subgroup of O0 (n, 1). Since the singular values, but not necessarily the eigenvalues, of T agree with those of A. We see that σ1 (T ) = et , σ2 (T ) = · · · = σn (T ) = 1 and σn+1 (T ) = e−t . It follows that, if T ∈ O0 (n, 1), then dHn (en+1 , T (en+1 )) = log σ1 (T ) = log σ1 (T ) 1 σ1 (T ) = log . σ2 (T ) 2 σn+1 (T ) One can read more about the hyperboloid model for Hn in Thurston [203, Chapter 2] or Martelli [159, Chapter 2]. 11. Convex cocompact representations into Isom(Hn ) We’ve been doing this longer than you’ve been alive Propelled by some mysterious drive And they still let me do it as weird as that seems And I do it most nights and then again in my dreams –Ken Bethea, Murry Hammond, Rhett Miller, and Philip Peoples [171] Given a representation ρ : Γ → O0 (n, 1) of a finitely generated group Γ and x0 ∈ Hn , there is an orbit map τρ : Γ → Hn given by τρ (γ) = ρ(γ)(x0 ) for all γ ∈ Γ. We say that ρ is convex cocompact if and only if Γ is finitely generated and τρ is a quasi-isometric embedding. One may check that this definition does not depend on the choice of (finite) generating set for Γ or the choice of the basepoint x0 for the orbit map. It is immediate from this definition that ρ(Γ) is discrete and ρ is almost faithful, i.e. the kernel of ρ is finite. If Γ is torsion-free, then ρ must be faithful. Since Γ is quasi-isometric to its Cayley graph, Corollary 2.5 implies that Γ is a Gromov hyperbolic group. Theorem 3.5 implies that there exists a continuous, injective map ξρ : ∂Γ → ∂Hn so that if {γn } ⊂ Γ converges to z ∈ ∂Γ, then {τρ (γn ))} = {ρ(γn )(x0 )} converges to ξρ (z). Since τρ is ρ-equivariant, it follows that ξρ is also ρ-equivariant. Since ρ(γ) fixes ξρ (γ + ) and ξρ (γ + ) = lim γ n (x0 ), it is easy to check that ξρ is dynamics-preserving, i.e. that ξρ (γ + ) is the attracting fixed point of ρ(γ). We will almost always assume that Γ is non-elementary, i.e. that Γ does not contain a finite index cyclic group. We collect all these observations in the following result. Theorem 11.1. Suppose that Γ is a finitely generated group and ρ : Γ → O0 (n, 1) is a convex cocompact representation. (1) ρ is discrete and almost faithful. (2) Γ is Gromov hyperbolic. ANOSOV REPRESENTATIONS 45 (3) There exists a continuous, dynamics-preserving, injective ρ-equivariant map ξρ : ∂Γ → ∂Hn . Notice that, by the equivariance of the orbit map, the orbit map τρ (with respect to some point x0 ∈ Hn ) of a representation ρ : Γ → O0 (n, 1) is a (K, C)-quasi-isometric embedding if and only if 1 d(1, γ) − C ≤ d(x0 , ρ(γ)(x0 )) ≤ Kd(1, γ) + C K (T ) for all γ ∈ Γ. Therefore, if we choose x0 = en+1 and recall that dHn (en+1 , T (en+1 )) = log σσ21 (T ) for all T ∈ O(n, 1), we obtain the following more Lie-theoretic characterization of convex cocompact representations into O0 (n, 1). (This should remind you of our definition of Anosov representations in Chapter 1.) Lemma 11.2. A representation ρ : Γ → O0 (n, 1) is convex cocompact if and only if there exists K and C so that σ1 (ρ(γ)) 1 d(1, γ) − C ≤ log ≤ Kd(1, γ) + C K σ2 (ρ(γ)) for all γ ∈ Γ. If ρ is discrete and almost faithful, we define its limit set Λ(ρ) ⊂ ∂Hn to be the set of accumulation points of an orbit, i.e. Λ(ρ) = ρ(Γ)(x0 ) − ρ(Γ)(x0 ) ⊂ ∂Hn for some x0 ∈ Hn . One may check that Λ(ρ) does not depend on the choice of x0 . In the upper half-space model, ∂Hn is identified with Rn−1 ∪ {∞} where Rn−1 is identified with the hyperplane {xn = 0}. In the hyperboloid model we let L be the “light cone,” i.e. L = {~x | x21 + · · · + x2n − x2n+1 = 0} and then ∂Hn = P(L) where a sequence {xn } in Hn converges to [z] ∈ ∂Hn if and only if the ←→ ← → sequence of lines {0xn } converges to 0z . The limit set may be defined more dynamically as the minimal Γ-invariant subset of ∂Hn . The name convex cocompact arises because if ρ is convex cocompact, then it acts cocompactly on the convex hull CH(Λ(ρ)) of its limit set. The quotient C(Nρ ) = CH(Λ(ρ))/ρ(Γ) is called the convex core of Nρ = Hn /ρ(Γ)). Proposition 11.3. A discrete faithful representation ρ : Γ → PO(n, 1) is convex cocompact if and only if there exists a convex subset Ω of Hn so that ρ(Γ) preserves and acts cocompactly on Ω. Moreover, if ρ is convex cocompact, ξρ (∂Γ) = Λ(ρ) and ρ(Γ) acts cocompactly on CH(Λ(ρ)). Proof. If ρ(Γ) preserves and acts cocompactly on Ω, then τρ gives a quasi-isometry from Γ to Ω (by the Milnor-Svarc Lemma). Since Ω is convex, it isometrically embeds in Hn . Therefore, τρ is a quasi-isometric embedding into Hn , so ρ is convex cocompact. Now suppose ρ is convex cocompact, so τρ is a quasi-isometric embedding into Hn . We choose Ω = CH(Λ(ρ)), which may be formed as the union of all ideal polyhedral n-simplices in Hn with endpoints in the limit set. We may assume that x0 has been chosen to lie in Ω, which implies, since Ω is Γ-invariant, that τρ (Γ) ⊂ Ω. Notice that ρ(Γ) acts cocompactly on Ω if and only if τρ (Γ) is coarsely dense in Ω, i.e. there exists A > 0 so that if x ∈ Ω, then there exists γ ∈ Γ so that d(x, ρ(γ)(x0 )) ≤ A. 46 Recall that τρ extends to a (K, C)-quasi-isometric embedding τ̂ρ : CΓ → Hn where CΓ is the Cayley graph of Γ. Notice that the image of every edge of CΓ has uniformly bounded length, so it suffices to show that τ̂ρ (CΓ ) is coarsely dense in Ω. The Fellow Traveller Property implies that there exists R so that if α : [a, b] → Hn is a (K, C)-quasi-isometric embedding, then α([a.b]) is a Hausdorff distance at most R apart from the geodesic α(a)α(b) joining α(a) to α(b). If z 6= w ∈ Λ(ρ), then there exists {xn } and {yn } in τρ (Γ) so that xn → z and yn → w. Then xn yn lies in the (closed) neighborhood NR (τ̂ρ (CΓ )) of τ̂ρ (CΓ ) of radius R, for all n, and xn yn → zw, so zw ⊂ NR (τ̂ρ (CΓ )). Notice that there exists Bn so that if T is an ideal polyhedral n-simplex in Hn , then every point in T lies within Bn of an edge of T . Therefore, every point in Ω lies with R + Bn of a point in τ̂ρ (Γ), so ρ(Γ) preserves and acts cocompactly on the convex set Ω. Finally, we check that ξρ (∂Γ) = Λ(ρ). If z ∈ ∂Γ, then there exists a sequence {γn } ⊂ Γ which converges to c. Then, ξρ (z) = lim γn (x0 ) lies in the limit set by definition. On the other hand, if w ∈ Λ(ρ), then there exists a sequence {ρ(γn )} ⊂ ρ(Γ), so that lim ρ(γn (x0 ) = w. If {γnk } is a convergent subsequence of {γn } with lim γnk = z, then ξρ (z) = w. Therefore, ξρ (∂Γ) = Λ(ρ).  Let CC(Γ, O0 (n, 1)) ⊂ Hom(Γ, O0 (n, 1)) d be the set of convex cocompact representations and let CC(Γ, O(n, 1)) be its image in the quotient space X(Γ, O0 (n, 1)) = Hom(Γ, O0 (n, 1))/O0 (n, 1). It is a crucial property of convex cocompact representations, known as stability, that CC(Γ, O0 (n, 1)) is open in Hom(Γ, O0 (n, 1)). Informally, if you wiggle a convex cocompact representation a little bit it remains convex cocompact. Theorem 11.4. If Γ is a finitely generated group and ρ : Γ → O0 (n, 1) is convex cocompact, then there exists a neighborhood U of ρ in Hom(Γ, PSL(2, R)) such that if σ ∈ U , then σ is convex cocompact. Moreover, we may choose U so that there exists K̂ and Ĉ so that if σ ∈ U , then τσ is a (K̂, Ĉ)-quasi-isometric embedding (with respect to a fixed generating set for Γ and some fixed basepoint x0 ∈ Hn ). Theorem 11.4 was establshed by Marden [154, Theorem 10.1] when n = 3 and by Thurston [202, Proposition 8.3.3] in the general case, see also Bowditch [38, Theorem 1.5] or CanaryEpstein-Green [56, Section I.2.5]. Proof. Suppose that the orbit map τρ is a (K, C)-quasi-isometric embedding with respect to a finite generating set S and x0 ∈ Hn . The local-to-global principle, Theorem 8.1, implies that there exists A, K̂, and Ĉ so that if f : J → Hn (where J is an interval in R) is a (K + C + 1, C + 1)-quasi-isometry on all segments of length at most A, then f is a (K̂, Ĉ)quasi-isometry. Let U be an open neighborhood of ρ in Hom(Γ, PSL(2, R)) so that if σ ∈ U , γ ∈ Γ and dS (1, γ) ≤ A + 1, then d(ρ(γ(x0 )), σ(γ)(x0 )) < 1. (We may do so since there are only finitely many elements of γ within A + 1 of id.) ANOSOV REPRESENTATIONS 47 If σ ∈ U , let τσ be the orbit map of σ. We see that if dS (1, γ) ≤ A + 1, then 1 dS (id, γ) − C − 1 ≤ d(τρ (id), τρ (γ)) ≤ KdS (id, γ) + C + 1 K so by the equivariance of τρ 1 dS (α, β) − C − 1 ≤ d(τρ (α), τρ (β)) ≤ KdS (α, β) + C + 1 K whenever dS (α, β) ≤ A + 1. Notice that if we let CΓ be the Cayley graph of Γ with respect to S, then we may extend τσ to a map with domain CΓ , by simply mapping all edges to geodesic segments in H2 . The resulting map is a (K + C + 1, C + 1)-quasi-isometry on all geodesic segments in CΓ of length at most A. Therefore, τσ is a (K̂, Ĉ)-quasi-isometric embedding on all geodesic segments in CΓ , which implies that τσ is a (K̂, Ĉ)-quasi-isometric embedding.  Since the set CC(π1 (S), O0 (n, 1)) is invariant under conjugation, we immediately see that both CC(π1 (S), O0 (n, 1)) and its quotient are open. Corollary 11.5. If Γ is a finitely generated group, then CC(Γ, O0 (n, 1)) is open in Hom(Γ, O0 (n, 1)) d and CC(Γ, O0 (n, 1)) is open in X(Γ, O0 (n, 1)). Similarly, let DF(Γ, O0 (n, 1)) ⊂ Hom(Γ, O0 (n, 1)) be the set of discrete, almost faithful, representations and let AH(Γ, PSL(2, R)) be its image in X(Γ, O0 (n, 1)). Corollary 6.2 implies immediately that DF(Γ, O0 (n, 1)) is closed in Hom(Γ, O0 (n, 1)). Corollary 11.6. If Γ is a finitely generated group which is not virtually cyclic, then DF(Γ, O0 (n, 1)) is closed in Hom(Γ, O0 (n, 1)) and AH(Γ, O0 (n, 1)) is closed in X(Γ, O0 (n, 1)). If Γ is the fundamental group of a closed hyperbolic n-manifold N , one may again use the Milnor-Svarc Lemma to show that ρ : Γ → PO(n, 1) is convex cocompact if and only if ρ d is discrete and faithful. Therefore, as in the case of Teichmüller space, CC(Γ, O0 (n, 1)) = AH(Γ, O0 (n, 1)) is a component of X(Γ, O0 (n, 1)). However, if n ≥ 3, then Mostow’s Rigidity d Theorem (see Mostow [169]) implies that CC(Γ, PO(n, 1)) is exactly one point, so we will not be interested in this situation. However, in general, CC(Γ, O0 (n, 1)) is not closed in Hom(Γ, O0 (n, 1)) and AH(Γ, O0 (n, 1)) is not open in X(Γ, O0 (n, 1)). We will give examples when Γ = F2 is the free group on two generators and n = 2. It will be more convenient to work in PSL(2, R) = SO(2, 1). We first describe the classical Schottky construction of convex cocompact representations of free groups. If {C1 , C2 , . . . , C2n−1 , C2n } is a family of disjoint geodesics in H2 bounding disjoint (closed) half-spaces {D1 , D2 , . . . , D2n−1 , D2n } (whose closures are disjoint in H2 ∪ ∂H2 ), then we may construct a convex cocompact representation ρ : Fn → PSL(2, R) by letting ρ(ai ) be a Möbius transformation taking D2i−1 to H2 − int(D2i ) for all i, where Fn =< a1 , . . . , an >. If S 2 P = H − int(Di ), then one may form a complete hyperbolic surface from P by identifying C2i−1 to C2i by ρ(ai ) for all i. Covering space theory then allows us to conclude that P is a fundamental domain for the action of ρ(Fn ) and that orbits of P tesselate H2 . (One may also verify these facts, using the Ping Pong Lemma, see Section [204].) 48 If we choose x0 ∈ int(P ) and let δ = min{d(Ci , Cj ) | i 6= j}, then one may easily check that d(x0 , γ(x0 )) ≥ δd(1, γ). On the other hand, if K = max{d(x0 , ρ(ai )(x0 ))}, then d(x0 , γ(x0 )) ≤ Kd(1, γ). picture needed Therefore, τρ is a quasi-isometric embedding, so ρ is convex cocompact. Notice that, in this case, one may easily see that all representations near to ρ are also convex cocompact, since wiggling the representation, just amounts to wiggling the Ci . We now observe that not all discrete, faithful representations of F2 are convex cocompact. Suppose that C1 is the x-axis, C2 is the line Re(z) = 1, C3 is a semi-circle based  at 1/4 with 1 1 radius 1/8 and C4 is a semi-circle based at 3/4 with radius 1/8. Let ρ0 (a1 ) = and let 0 1 ρ0 (a2 ) be a Möbius transformation taking the half-space “below” C3 to the half-space “above” C4 and preserving the height of points on C3 . Let P be the (closure of the) region between C1 and C2 and above C3 and C4 . Consider the hyperbolic surface X obtained from identifying C1 with C2 by ρ0 (a1 ) and identifying C3 with C4 by ρ0 (a2 ) and the sequence of regions Xn given by the quotient of {z ∈ P |e−n ≤ Im(z) ≤ en }. Notice that Xn contains the ball of radius n about the quotient of i + 1/2 and exhausts X. It follows that X is complete. Covering space theory, then guarantees that ρ0 : F2 → PSL(2, R) is discrete and faithful and that P is a fundamental domain for the action of ρ0 (F2 ) on H2 . However, τρ is not a quasi-isometric embedding, since if we choose x0 = 1 + 1/2, then √ n + n2 + 4 n d(x0 , ρ0 (a )(x0 )) = 2 log ∼ 2 log n 2 so ρ0 is not convex cocompact. Notice that it is important to be careful in checking completeness. Suppose that we choose C1 , C2 , C3 and C4 as for ρ0 but then let ρ̂0 (a1 ) to be given by z 7→ 21 z + 1 and ρ̂0 (a2 ) = ρ0 (a2 ). The region Xn is not preserved by the gluings ρ̂(ai ), the quotient of P is not complete and, in fact, ρ̂0 is convex cocompact. One may see that the quotient of P is not complete, by considering the path in the quotient which is the union of horizontal segments in P of height 2n for all n. This path has finite length but leaves every compact subset P of the quotient of P . Notice that the lines {ρ̂(an1 )(C0 )} accumulates at the line Re(z) = ni=0 21n = 2, so the translates of P do not tesselate H2 . A fundamental domain for the action of ρ̂(F2 ) is given by looking at the region below the circle of radius 2 about z = 2 and above the circle of radius 1 about z = 2 and above C3 and C4 . One may then use this picture, just as above, to show that ρ̂0 is convex cocompact. We now observe that ρ is a limit of a sequence {ρn } of representations whose image is not discrete and faithful. For all n ≥ 2, let ρn (a) be an element of PSL(n, R) which fixes ni + 12 and takes i to i + 1 and let ρn (b) = ρ(b). It is then easy to check that {ρn } converges to ρ and that ρn (F2 ) is either indiscrete or not faithful (since either ρn (a) has finite order, or < ρn (a) > is indiscrete). Similarly, we choose ρ̂n (a) ∈ PSL(2, R) to take the interior of the circle R−n of radius n about −n to the exterior of the circle Rn of radius n about n + 1, so that the “height” (i.e. the imaginary component) of points on R−n is preserved and let ρ̂n (b) = ρ(b). Then ρ̂n is convex cocompact for all n and lim ρ̂n = ρ. We summarize in the following proposition: ANOSOV REPRESENTATIONS 49 Proposition 11.7. If F2 is the free group with 2 generators, then CC(F2 , PSL(2, R)) is open, but not closed, in Hom(F2 , PSL(2, R)). Moreover, DF (F2 , PSL(2, R)) is closed, but not open, in Hom(F2 , PSL(2, R)). 12. Fricke’s Theorem Fricke proved that the mapping class group Mod(S) of a closed surface acts properly discontinuously, but not freely, on its Teichmüller space. In this section, we will generalize this by d showing that Out(Γ) acts properly discontinuously on CC(Γ, O0 (n, 1)). Recall that Mod(S) of a closed, orientable surface S is the group of (isotopy classes of) orientation-preserving homeomorphisms of S. It is classical that two homeomorphisms of S are isotopic if and only if they are homotopic and that every homotopy equivalence of S is homotopic to a homeomorphism. Therefore, Mod(S) is identified with an index two subgroup of Out(π1 (S)). Recall that if Γ is a group, then Out(Γ) = Aut(Γ)/Inn(Γ) and Inn(Γ) is the group of inner automorphisms of Γ. (All these facts are covered carefully in Farb and Margalit’s A primer on mapping class groups.) The mapping class group Mod(S) acts naturally on T (S), by φ([ρ]) = [ρ ◦ (φ∗ )−1 ]. Theorem 12.1. (Fricke’s Theorem) If S is a closed oriented surface of genus g ≥ 2, then Mod(S) acts properly discontinuously on T (S). The quotient M(S) = T (S)/Mod(S) is the moduli space of isometry classes of (unmarked) hyperbolic structures on S. By the Uniformization Theorem, it may be identified with the space of conformal (or complex) structures on S. In algebraic geometry, it occurs as the space of smooth algebraic curves of genus g. It admits a natural geometric compactification which has the structure of a projective algebraic variety. Since Mod(S) does not act freely, moduli space has the structure of an orbifold rather than of a manifold. We begin with an elementary observation about translation lengths. Recall that if Γ is a group, then ||γ|| denotes the minimal translation length of the action of γ on Γ. Equivalently, ||γ|| is the minimal length of a word conjugate to γ. We let `X denote the translation length of the action of Γ on X, i.e. `X (γ) = inf dX (x, γ(x)). x∈X The following fact is really part of the discussion surrounding the Milnor-Svarc Lemma. Lemma 12.2. If Γ acts properly discontinuously, cocompactly and by isometries on proper length space X, then there exists J, B > 0 so that 1 ||γ|| − B ≤ `X (γ) ≤ J||γ|| + B J for all γ ∈ Γ. Moreover, if the orbit map τ : Γ → X is a (K, C)-quasi-isometric embedding, for some choice of x0 ∈ X, then we may choose J = K and B = 3C. One may easily check that if X = Hn and A ∈ O0 (n, 1), then `Hn (A) = 2 log λ1 (A) = 2 log λ1 (A) λ1 (A) = log . λ2 (A) λn+1 (A) 50 Proof. Fix x0 ∈ X. The Milnor-Svarc Lemma guarantees that there exists K and C so that the orbit map τ : Γ → X, given by γ 7→ γ(x0 ) is a (K, C)-quasi-isometry. Let α ∈ Γ be an element so that d(α, γα) = d(id, α−1 γα) = ||γ||. Then, d(x0 , α−1 γα(x0 )) ≤ K||γ|| + C so `X (α−1 γα) = `X (γ) ≤ K||γ|| + C. On the other hand, if x ∈ X, then there exists α ∈ Γ so that d(α(x0 ), x) ≤ C. Moreover, d(α(x0 ), γα(x0 ))) ≥ 1 1 d(α, γα) − C ≥ ||γ|| − C K K and d(γα(x0 ), γ(x)) ≤ C so d(x, γ(x)) ≥ K||γ|| − 3C. Therefore, `X (γ) ≥ K||γ|| − 3C and we may take J = K and B = 3C  We will also use the following group-theoretic fact, first discussed in Section 8. Proposition 8.3. If Γ is a torsion-free hyperbolic group, then there exists a finite collection B of elements of π1 (S), so that for any K {φ ∈ Out(Γ) | ||φ(b)|| ≤ K for all b ∈ B} is finite. If Γ = π1 (S), then this is equivalent to the claim that there is a finite collection F of curves on S, so that any self-homeomorphism of S is determined, up to homotopy, by the homotopy classes of the images of curves in F . This is not hard to check using surface topology. We are now ready to establish our generalization of Fricke’s Theorem. d Theorem 12.3. If Γ is a finitely generated torsion-free group and CC(Γ, O0 (n, 1)) is nond empty, then Out(Γ) acts properly discontinuously on CC(Γ, O0 (n, 1)). d Proof. To warm up, we first show that the action of Out(Γ) on CC(Γ, O0 (n, 1)) has discrete d orbits. If ρ ∈ CC(Γ, O0 (n, 1)), then Lemma 12.2 implies that there exists J and B so that 1 ||g|| − B ≤ `(ρ(g)) ≤ J||g|| + B J Let B be the finite collection of elements of Γ provided by Proposition 8.3. If {φn } is a sequence of distinct elements of Out(Γ), then, up to subsequence, there exists b ∈ B so that −1 ||φ−1 n (b)|| → ∞. Therefore, by the above inequality, `(ρ(φn (β)) → ∞, which implies that ANOSOV REPRESENTATIONS 51 d φn (ρ) = ρ ◦ φ−1 n → ∞ in X(Γ, O0 (n, 1)). Therefore, the action of Out(Γ) on CC(Γ, O0 (n, 1)) has discrete orbits. d Let R be a compact subset of CC(Γ, O0 (n, 1)). Theorem 11.4 implies that if ρ ∈ R, then there exist KU , CU and an open neighborhood U of ρ so that if σ ∈ U , then τσ is a (KU , CU )quasi-isometric embeddding (for some choice of x0 ∈ Hn ). Since R is compact, it can be covered by finitely many such neighborhoods. So, there exist K and C so that if ρ ∈ R, then τρ is a (K, C)-quasi-isometric embedding (for some choice of basepoint). Lemma 12.2 then implies that if J = K and B = 3C, then 1 ||γ|| − B ≤ `(ρ(γ)) ≤ J||γ|| + B J for all γ ∈ Γ and all ρ ∈ R. So if {φn } is a sequence of distinct elements of Out(Γ), then, by Proposition 8.3, there exists b ∈ B so that, after perhaps passing to a subsequence, ||φ−1 n (b)|| → ∞. So, if ρ ∈ R, then `(ρ(φ−1 n (β)) ≥ 1 −1 ||φ (b)|| − B → ∞ J n so {φn (R)} exits every compact subset of X(π1 (F ), PSL(2, R)). Therefore, Out(Γ) acts properly d discontinuously on CC(Γ, O0 (n, 1)).  Remarks: 1) The mapping class group Mod(S) does not act properly discontinuously on AH(π1 (S), PO(3, 1)). We say that [φ] ∈ Mod(S) is pseudo-Anosov if whenever α is a homotopically non-trivial simple closed curve α on S and n ∈ N, then φn (α) is not (freely) homotopic to α. Then φ has infinite order in Mod(S) and fixes a point in AH(π1 (S), PO(3, 1)). Thurston proved that the mapping torus Mφ = S × [0, 1]/(x, 0) ∼ (φ(x), 0) admits a hyperbolic metric. So there exists a convex cocompact representation ρ̂φ : π1 (Mφ ) → PO(3, 1). The fixed point is then given by [ρφ ] where ρφ = ρ̂φ |π1 (S) . 2) Goldman [102] proved that X(π1 (S), PSL(2, R)) has 2g − 1 components (indexed by the absolute value of the Euler number of representations in the component). Goldman [105] conjectured that if the genus of S is at least 3, then the mapping class group acts ergodically on all components other than the component which is Teichmüller space. If S has genus 2, Marché and Wolff [152, 153] proved that the mapping class group acts ergodically on the component of X(π1 (S), PSL(2, R)) consisting of representations whose Euler number has modulus 1 and the the component consisting of representations of modulus 0 splits into two open sets which are preserved and acted on ergodically by the mapping class group. In 2014, Souto announced a proof of Goldman’s conjecture for components consisting of representations with Euler number 0. 3) Goldman also conjectured that the mapping class group acts ergodically on X0 (π1 (S), PSL(2, C)) − CC(π1 (S), PSL(2, C)) where X0 (π1 (S), PSL(2, C)) is the component of X0 (π1 (S), PSL(2, C)) containing CC(π1 (S), PSL(2, C)). Minsky [167], Canary-Storm [58] and Lee [148] showed that for many 3-manifolds M , the domain of discontinuity for the action of Out(π1 (M ), OSL(2, C)) on X(π1 (S), PSL(2, C) is strictly larger than CC(π1 (M ), PSL(2, C)), see [55] for a more detailed survey of these results. 52 13. Further topics: Hyperbolic 3-manifolds I left the four lane highway took a blacktop seven miles Down by the old country school I went to as a child Two miles down a gravel road I could see the proud old home A tribute to a way of life that’s almost come and gone. The roots of my raising run deep I come back for the strength that I need And hope comes no matter how far down I sink The roots of my raising run deep. —————–Tommy Collins [110] One may view the theory of Kleinian groups as the lowest of all Higher Teichmüller theories, or perhaps as “only somewhat higher Teichmüller theory.” However, this theory has progressed rather dramatically, beginning with Thurston’s groundbreaking work in the 1970s and 1980s. We will review some of this work in the hopes that it could provide inspiration for future directions in “truly” higher Teichmüller theory. (This hope was one of the motivations for my attempts to begin working in the field, although the main motivation was my enjoyment talking math with the collaborators I found there.)