Dynamics at infinity. of symmetric spaces. J. Porti, Universitat Autònoma de Barcelona 11th KAIST Geometric Topology Fair Jeonju, Korea, August 2013

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1 Dynamics at infinity of symmetric spaces J. Porti, Universitat Autònoma de Barcelona 11th KAIST Geometric Topology Fair Jeonju, Korea, August 2013

2 Motivation Γ Isom(H n ) discrete subgroup H n = S n 1 ideal boundary and Λ = Γx H n limit set Γ acts properly discontinously on Ω = H n \Λ Γ convex cocompact if C H n Γ-inv. convex, C/Γ compact. If Γ is convex cocompact, then Ω/Γ is compact.

3 Motivation Γ Isom(H n ) discrete subgroup H n = S n 1 ideal boundary and Λ = Γx H n limit set Γ acts properly discontinously on Ω = H n \Λ Γ convex cocompact if C H n Γ-inv. convex, C/Γ compact. If Γ is convex cocompact, then Ω/Γ is compact. Whan can one say for Γ Isom(X), when X is a symmetric space of rank 2?

4 Motivation Γ Isom(H n ) discrete subgroup H n = S n 1 ideal boundary and Λ = Γx H n limit set Γ acts properly discontinously on Ω = H n \Λ Γ convex cocompact if C H n Γ-inv. convex, C/Γ compact. If Γ is convex cocompact, then Ω/Γ is compact. Whan can one say for Γ Isom(X), when X is a symmetric space of rank 2? The action of Isom(X) is not transtive on X. Hence look for domains in Isom(X)-orbits in X Eg: orbits on (SL(n+1,R)/SO(n+1)) are P n, Flag(P n ), etc Understand geometrically Anosov groups à la Labourie and Guichard-Wienhard

5 Motivation Γ Isom(H n ) discrete subgroup H n = S n 1 ideal boundary and Λ = Γx H n limit set Γ acts properly discontinously on Ω = H n \Λ Γ convex cocompact if C H n Γ-inv. convex, C/Γ compact. If Γ is convex cocompact, then Ω/Γ is compact. Whan can one say for Γ Isom(X), when X is a symmetric space of rank 2? I Symmetric spaces of noncompact type II Ideal boundary III Dynamics at infinity (joint with M. Kapovich, B. Leeb),

6 I Symmetric spaces of noncompact type

7 Symmetric spaces: Definition X complete Riemannian manifold. An inversion at x X is an isometry σ x : X X s.t. σ x (x) = x and dσ x = Id Tx X (σ 2 x = Id) X is called a symmetric space if every x X has an inversion If γ : (ε,ε) X is a geodesic with γ(0) = x, then σ x γ(t) = γ( t)

8 Symmetric spaces: Definition X complete Riemannian manifold. An inversion at x X is an isometry σ x : X X s.t. σ x (x) = x and dσ x = Id Tx X (σ 2 x = Id) X is called a symmetric space if every x X has an inversion If γ : (ε,ε) X is a geodesic with γ(0) = x, then σ x γ(t) = γ( t) Proposition: A symmetric space is homogeneous: x 1,x 2 X g Isom(X) such that gx 1 = x 2 Proof: If y midpoint of x 1 and x 2, then σ y (x 1 ) = σ y σ x1 (x 1 ) = x 2 x 2 x y 1

9 Symmetric spaces: Definition X complete Riemannian manifold. An inversion at x X is an isometry σ x : X X s.t. σ x (x) = x and dσ x = Id Tx X (σ 2 x = Id) X is called a symmetric space if every x X has an inversion If γ : (ε,ε) X is a geodesic with γ(0) = x, then σ x γ(t) = γ( t) Proposition: A symmetric space is homogeneous: x 1,x 2 X g Isom(X) such that gx 1 = x 2 Proof: If y midpoint of x 1 and x 2, then σ y (x 1 ) = σ y σ x1 (x 1 ) = x 2 x 2 x y 1 By moving x 2, one can chose g = σ y σ x1 Isom 0 (X) Corollary: X = G/K, for G connected Lie group, K G compact G = Isom 0 (X) and K = G x.

10 Symmetric spaces as homogeneous spaces X is a symmetric space if every x X has an inversion Theorem:X complete Riemannian manifold. The following are equiv: (1) X is a symmetric space (2) X = G/K, G connected Lie group, K G compact, σ : G G automorphism s.t. σ 2 = Id and (G σ ) 0 K G σ.

11 Symmetric spaces as homogeneous spaces X is a symmetric space if every x X has an inversion Theorem:X complete Riemannian manifold. The following are equiv: (1) X is a symmetric space (2) X = G/K, G connected Lie group, K G compact, σ : G G automorphism s.t. σ 2 = Id and (G σ ) 0 K G σ. Classification (E. Cartan 1926) Examples H n = SO(n,1) 0 /SO(n) σ : SO(n,1) 0 SO(n,1) 0 A (A t ) 1 (SO(n,1) 0 ) σ = SO(n) CH n = SU(n,1) 0 /SU(n), S n = SO(n+1)/SO(n)

12 Locally symmetric spaces X is a locally symmetric space if every x X has an inversion defined in a neighborhood of x (eg H n /Γ). Theorem: X complete Riemannian manifold. The following are equiv: (1) X is locally symmetric (2) R = 0 (parallel curvature tensor)

13 Locally symmetric spaces X is a locally symmetric space if every x X has an inversion defined in a neighborhood of x (eg H n /Γ). Theorem: X complete Riemannian manifold. The following are equiv: (1) X is locally symmetric (2) R = 0 (parallel curvature tensor) Lemma: Y(t) vector field along a geodesic γ(t). Then γ Y = 0 iff dσ γ(t) Y(t+s) = Y(t s) (Use uniqueness of parallel transport)

14 Locally symmetric spaces X is a locally symmetric space if every x X has an inversion defined in a neighborhood of x (eg H n /Γ). Theorem: X complete Riemannian manifold. The following are equiv: (1) X is locally symmetric (2) R = 0 (parallel curvature tensor) Lemma: Y(t) vector field along a geodesic γ(t). Then γ Y = 0 iff dσ γ(t) Y(t+s) = Y(t s) (Use uniqueness of parallel transport) (1) (2): γ geodesic p = γ(0), X, Y, Z parallel along γ. Then dσ p (R(X(t),Y(t))Z(t)) = R(dσ p X(t),dσ p Y(t))dσ p Z(t) = R( X(t), Y(t))( Z(t)) = R(X(t),Y(t))Z(t) Hence γ (t)( R(X(t),Y(t))Z(t) ) = 0 R = 0.

15 Locally symmetric spaces X is a locally symmetric space if every x X has an inversion defined in a neighborhood of x (eg H n /Γ). Theorem: X complete Riemannian manifold. The following are equiv: (1) X is locally symmetric (2) R = 0 (parallel curvature tensor) Lemma: Y(t) vector field along a geodesic γ(t). Then γ Y = 0 iff dσ γ(t) Y(t+s) = Y(t s) (Use uniqueness of parallel transport) (2) (1): Define a (local) map exp p (tv) exp p ( tv). This map preserves R by parallelism, hence it is an isometry by Cartan Ambrose Hicks theorem

16 Examples H n = SO(n,1) 0 /SO(n) σ(a) = (A t ) 1 R n = Isom 0 (R n )/SO(n) Isom 0 (R n ) SL(R,n+1) S n = SO(n+1)/SO(n) σ(a) = JAJ, J = H 2 H 2 SL(n,R)/SO(n) σ(a) = (A t ) SL(3,R)/SO(3) = {volume 1 ellipsoids in R 3 }/isometries

17 Decompostition X = G/K g = Lie(G) k = Lie(K) p = T x X, σ k = Id and σ p = Id g = k p, Lemma [k,k] k, [k,p] p, [p,p] k Proof: X,Y p, σ[x,y] = [σx,σy] = [ X, Y] = [X,Y] k. Lemma (R(X,Y)X,Y) = ([X,[Y,X]],Y) X,Y p

18 Decompostition X = G/K g = Lie(G) k = Lie(K) p = T x X, σ k = Id and σ p = Id g = k p, Lemma [k,k] k, [k,p] p, [p,p] k Proof: X,Y p, σ[x,y] = [σx,σy] = [ X, Y] = [X,Y] k. Lemma (R(X,Y)X,Y) = ([X,[Y,X]],Y) X,Y p Theorem X = R r X 1 X n, X i = G i /K i irr, G i semisimple Proof: Decompose p = p 0 p n Ad K -equiv

19 Decompostition X = G/K g = Lie(G) k = Lie(K) p = T x X, σ k = Id and σ p = Id g = k p, Lemma [k,k] k, [k,p] p, [p,p] k Proof: X,Y p, σ[x,y] = [σx,σy] = [ X, Y] = [X,Y] k. Lemma (R(X,Y)X,Y) = ([X,[Y,X]],Y) X,Y p Theorem X = R r X 1 X n, X i = G i /K i irr, G i semisimple Proof: Decompose p = p 0 p n Ad K -equiv Killing form: B : g g R ad G -invariant B(X,Y) = tr(adx ady) B gi 1 unique ad G -invariant B ki 0 negative definite B pi 1 positive definite iff G i noncompact

20 Symmetric spaces of noncompact type X = R r X 1 X n, Def: X of noncompact type if r = 0 and each X i noncompact

21 Symmetric spaces of noncompact type X = R r X 1 X n, Def: X of noncompact type if r = 0 and each X i noncompact X noncompact type X = G/K, K maximal compact subgroup g = k p, k = Lie(K), B k negative definite, B p positive definite

22 Symmetric spaces of noncompact type X = R r X 1 X n, Def: X of noncompact type if r = 0 and each X i noncompact X noncompact type X = G/K, K maximal compact subgroup g = k p, k = Lie(K), B k negative definite, B p positive definite Chose B as metric on p = T x X sec(x,y) = [X,Y] 2 0, X,Y p. Corollary Xnoncompact type, then X is CAT(0). Examples: H n, H 2 H 2, SL(n,R)/SO(n).

23 Flats and rank Def: a flat F X is a totally geodesic subespace F = R K Lemma abelian subalgebras ofp Flats throughx 0 h exp(h)x 0 (use sec(x,y) = [X,Y] 2 0)

24 Flats and rank Def: a flat F X is a totally geodesic subespace F = R K Lemma abelian subalgebras ofp Flats throughx 0 h exp(h)x 0 (use sec(x,y) = [X,Y] 2 0) All maximal abelian algebras in p are conjugate. F 1,F 2 X maximal flats, then g G such that gf 1 = F 2 (if x 0 F 1 F 2 can chose g G x0 )

25 Flats and rank Def: a flat F X is a totally geodesic subespace F = R K Lemma abelian subalgebras ofp Flats throughx 0 h exp(h)x 0 (use sec(x,y) = [X,Y] 2 0) All maximal abelian algebras in p are conjugate. F 1,F 2 X maximal flats, then g G such that gf 1 = F 2 (if x 0 F 1 F 2 can chose g G x0 ) Def: rank(x) = dim(maximal flat) rank(x) = 1 sec(x) < 0 Exs rankh n = 1, rankh n H m = 2, ranksl(n+1)/so(n+1) = n λ 0 λ1 h =... λ 0 + +λ n = 0 λ n

26 Flats and rank Def: a flat F X is a totally geodesic subespace F = R K Lemma abelian subalgebras ofp Flats throughx 0 h exp(h)x 0 (use sec(x,y) = [X,Y] 2 0) All maximal abelian algebras in p are conjugate. F 1,F 2 X maximal flats, then g G such that gf 1 = F 2 (if x 0 F 1 F 2 can chose g G x0 ) Def: rank(x) = dim(maximal flat) rank(x) = 1 sec(x) < 0 Exs rankh n = 1, rankh n H m = 2, ranksl(n+1)/so(n+1) = n λ 0 λ1 h =... λ 0 + +λ n = 0 λ n x,y F, σ y σ x transvection (it preserves F, σ y σ x F translation xy )

27 Regular directions Every geodesic γ : R X is contained in a maximal flat Def: γ is regular if contained in only one maximal flat, otherwise singular. Example: H 2 H 2 (s,t) (γ 1 (s),γ 2 (t)) is a flat (γ i : R H 2 geodesic,) γ 2 regular singular γ 1 singular directions s or t constant (horizontal or vertical)

28 Regular directions Every geodesic γ : R X is contained in a maximal flat Def: γ is regular if contained in only one maximal flat, otherwise singular. Example: {( SL(3)/SO(3) ) } λ0 h = λ1 λ 0 +λ 1 +λ 2 = 0 = R 2, exp(h)x 0 flat through x 0 λ 2 ) λ 0 = λ 1 can conjugate by : it is singular ( Regular directions through 0: λ 0 λ 1 λ 2 λ 0 λ 1 λ 0 = λ 1 λ 1 = λ 2 λ 0 λ 2 λ 0 = λ 2

29 Regular directions Every geodesic γ : R X is contained in a maximal flat Def: γ is regular if contained in only one maximal flat, otherwise singular. Example: {( SL(3)/SO(3) ) } λ0 h = λ1 λ 0 +λ 1 +λ 2 = 0 = R 2, exp(h)x 0 flat through x 0 λ 2 ) λ 0 = λ 1 can conjugate by : it is singular ( Regular directions through 0: λ 0 λ 1 λ 2 λ 0 e λ 2 e λ 0 e λ 1

30 Roots Proposition: F maximal flat, x 0 F, then: 1. {γ F singular geodesic throughx 0 } = α Λ H α, withh α hyperplane,λfinite 2. α Λ, s α G s.t. s α (F) = F, s α F = reflection alongh α

31 Roots Proposition: F maximal flat, x 0 F, then: 1. {γ F singular geodesic throughx 0 } = α Λ H α, withh α hyperplane,λfinite 2. α Λ, s α G s.t. s α (F) = F, s α F = reflection alongh α Proof: h p max abelian F = exp(h)x 0 v h is singular iff w p, w / h, [v,w] = ad v (w) = 0. ad v : g g is symmetric for (, ) = B(,σ ), which is positive definite {ad v,v h} diagonalize simultaneously: g = g 0 α Λg α α : h R is called a root, α( ) = (,e α ), e α h is the root element. Proof of 1: set H α = kerα.

32 Roots Proposition: F maximal flat, x 0 F, then: 1. {γ F singular geodesic throughx 0 } = α Λ H α, withh α hyperplane,λfinite 2. α Λ, s α G s.t. s α (F) = F, s α F = reflection alongh α Proof: h p max abelian F = exp(h)x 0 v h is singular iff w p, w / h, [v,w] = ad v (w) = 0. ad v : g g is symmetric for (, ) = B(,σ ), which is positive definite {ad v,v h} diagonalize simultaneously: g = g 0 α Λg α α : h R is called a root, α( ) = (,e α ), e α h is the root element. Proof of 1: set H α = kerα. Proof of 2: α Λ, write g α = k g α p g α If k +p g α, k k, p p, then [h,p] = α(h)k, [h,k] = α(h)p. Claim If p = 1 then [k,p] = e α, [k,e α ] = e α 2 p, and [k,h α ] = 0 [ 1 e α k,p] = 1 e α e α, [ 1 e α k, 1 e α e α] = p, and [ 1 e α k,h α] = 0

33 Roots Proposition: F maximal flat, x 0 F, then: 1. {γ F singular geodesic throughx 0 } = α Λ H α, withh α hyperplane,λfinite 2. α Λ, s α G s.t. s α (F) = F, s α F = reflection alongh α Proof: h p max abelian F = exp(h)x 0 v h is singular iff w p, w / h, [v,w] = ad v (w) = 0. ad v : g g is symmetric for (, ) = B(,σ ), which is positive definite {ad v,v h} diagonalize simultaneously: g = g 0 α Λg α α : h R is called a root, α( ) = (,e α ), e α h is the root element. Proof of 2: α Λ, write g α = k g α p g α If k +p g α, k k, p p, then [h,p] = α(h)k, [h,k] = α(h)p. Claim If p = 1 then [k,p] = e α, [k,e α ] = e α 2 p, and [k,h α ] = 0 [ 1 e α k,p] = 1 e α e α, [ 1 e α k, 1 e α e α] = p, and [ 1 e α k,h α] = 0 Claim exp(t 1 e α k) rotation in plane e α,p. Take s α := exp( π e α k),

34 Roots Proposition: F maximal flat, x 0 F, then: 1. {γ F singular geodesic throughx 0 } = α Λ H α, withh α hyperplane,λfinite 2. α Λ, s α G s.t. s α (F) = F, s α F = reflection alongh α Proof: h p max abelian F = exp(h)x 0 v h is singular iff w p, w / h, [v,w] = ad v (w) = 0. ad v : g g is symmetric for (, ) = B(,σ ), which is positive definite {ad v,v h} diagonalize simultaneously: g = g 0 α Λg α α : h R is called a root, α( ) = (,e α ), e α h is the root element. Proof of 2: α Λ, write g α = k g α p g α If k +p g α, k k, p p, then [h,p] = α(h)k, [h,k] = α(h)p. Claim If p = 1 then [k,p] = e α, [k,e α ] = e α 2 p, and [k,h α ] = 0 proof: [h,[k,p]] = [k,[h,p]] [p,[k,h]] = 0 [k,p] h by maximality B([k,p],h) = B(p, [k,h]) = B(p,α(h)p) = α(h) = B(e α,h)

35 SL(3, R)/SO(3) h = {( λ0 λ1 λ 2 ) } λ 0 +λ 1 +λ 2 = 0 Singular directions: λ i = λ j λ 1 λ 0 = λ 1 λ 1 = λ 2 λ 0 g = h λ 2 λ 0 = λ 2 0 g λ0 λ 1 g λ0 λ 2 g λ1 λ 0 0 g λ1 λ 2 s λ0 λ 1 = g λ2 λ 0 g λ2 λ 1 0 ( )

36 SL(3, R)/SO(3) h = {( λ0 λ1 λ 2 ) } λ 0 +λ 1 +λ 2 = 0 Singular directions: λ i = λ j λ 1 λ 0 = λ 1 λ 1 = λ 2 λ 0 g = h λ 2 λ 0 = λ 2 0 g λ0 λ 1 g λ0 λ 2 g λ1 λ 0 0 g λ1 λ 2 s λ0 λ 1 = g λ2 λ 0 g λ2 λ 1 0 ( ) W = s λi λ j = Σ 3

37 Weyl group Def: The Weyl group is W = {g K gf = F} = s α α Λ W = normalizer(h)/centralizer(h), F = exp(h)x 0 (F,W) is a (euclidean) Coxeter complex

38 Weyl group Def: The Weyl group is W = {g K gf = F} = s α α Λ W = normalizer(h)/centralizer(h), F = exp(h)x 0 (F,W) is a (euclidean) Coxeter complex Def: A wall is fix(s α ) = exp(ker(α)) A chamber is the closure of a component of F \wall x, y lie in the same chamber iff α(x)α(y) 0 α Λ.

39 Weyl group Def: The Weyl group is W = {g K gf = F} = s α α Λ W = normalizer(h)/centralizer(h), F = exp(h)x 0 (F,W) is a (euclidean) Coxeter complex Def: A wall is fix(s α ) = exp(ker(α)) A chamber is the closure of a component of F \wall x, y lie in the same chamber iff α(x)α(y) 0 α Λ. Example: SL(3, R)/SO(3) λ 0 = λ 1 λ 1 λ 0 e 01 e 12 e 02 λ 1 = λ 2 e 02 e 01 e 12 λ 2 λ 0 = λ 2 Choice of positive chamber: {x R 2 (x,e 10 ) > 0, (x,e 12 ) > 0} {e 01,e 12 } primitive root system (type A 2 )

40 Weyl group Def: The Weyl group is W = {g K gf = F} = s α α Λ W = normalizer(h)/centralizer(h), F = exp(h)x 0 (F,W) is a (euclidean) Coxeter complex Def: A wall is fix(s α ) = exp(ker(α)) A chamber is the closure of a component of F \wall x, y lie in the same chamber iff α(x)α(y) 0 α Λ. Irreducible Dynkin diagrams

41 II Ideal boundary

42 Recall... X = G/K symmetric space of nocompact type K < G maximal compact, sec 0 F = exp(h)x 0 X maximal flat (dimf = rankx) {singular geodesics inf troughx0 } = α Λ where wall α = kerα : h R, Λ (finite) set of roots. Weyl group = s α reflection through wall α G x0 = K (F,W) is a Euclidean Coxeter complex. wall α

43 Recall... X = G/K symmetric space of nocompact type K < G maximal compact, sec 0 F = exp(h)x 0 X maximal flat (dimf = rankx) {singular geodesics inf troughx0 } = α Λ where wall α = kerα : h R, Λ (finite) set of roots. wall α Weyl group = s α reflection through wall α G x0 = K (F,W) is a Euclidean Coxeter complex. Ex: H 2 H 2 (s,t) (γ 1 (s),γ 2 (t)) is a flat (γ i : R H 2 geodesic,) t Singular directions: s = 0 or t = 0 Weyl group: Z/2Z Z/2Z s

44 Recall... X = G/K symmetric space of nocompact type K < G maximal compact, sec 0 F = exp(h)x 0 X maximal flat (dimf = rankx) {singular geodesics inf troughx0 } = α Λ where wall α = kerα : h R, Λ (finite) set of roots. wall α Weyl group = s α reflection through wall α G x0 = K (F,W) is a Euclidean Coxeter {( complex. ) } λ0 Ex: SL(3,R)/SO(3): h = λ1 λ 0 +λ 1 +λ 2 = 0 λ 2 λ 1 λ 0 = λ 1 Singular directions: λ i = λ j Weyl group: Σ 3 = D 3 λ 1 = λ 2 λ 2 λ 0 λ 0 = λ 2

45 X = G/K symmetric space of nocompact type K < G maximal compact, sec 0 Ideal boundary

46 Ideal boundary X = G/K symmetric space of nocompact type K < G maximal compact, sec 0 X= {r : [0,+ ) X geodesic}/ = asymptote classes of geodesic rays r 1 r 2 if d(r 1 (t),r 2 (t)) C

47 Ideal boundary X = G/K symmetric space of nocompact type K < G maximal compact, sec 0 X= {r : [0,+ ) X geodesic}/ = asymptote classes of geodesic rays r 1 r 2 if d(r 1 (t),r 2 (t)) C U(T x0 (X)) X = S dimx 1 v {exp x0 (tv) t 0} is a homeomorphism. G acts continuously on X X = ball.

48 Ideal boundary X = G/K symmetric space of nocompact type K < G maximal compact, sec 0 X= {r : [0,+ ) X geodesic}/ = asymptote classes of geodesic rays r 1 r 2 if d(r 1 (t),r 2 (t)) C U(T x0 (X)) X = S dimx 1 v {exp x0 (tv) t 0} is a homeomorphism. G acts continuously on X X = ball. The action of G on X is not transitive. For ξ X let G ξ = {g G gξ = ξ}. Lemma G ξ acts transitively on X.

49 Proof that G ξ acts transitively on X. p,q X p q r 1 r 2 X ξ t d(r 1 (t),r 2 (t)) is convex and bounded, hence monotonic φ t i = σ r i (t)σ ri (0) G transvection along r i,

50 Proof that G ξ acts transitively on X. p,q X p q r 1 r 2 X ξ t d(r 1 (t),r 2 (t)) is convex and bounded, hence monotonic φ t i = σ r i (t)σ ri (0) G transvection along r i, φ t 1(p) = r 1 (t) at distance C from r 2 (t), φ t 2 φt 1(p) lies at distance C from r 2 (0) = q, at dist. 2C from p, Hence φ t k 2 φ t k 1 φ for some sequence t k, φ (ξ) = ξ

51 Proof that G ξ acts transitively on X. p,q X p q r 1 r 2 X ξ t d(r 1 (t),r 2 (t)) is convex and bounded, hence monotonic φ t i = σ r i (t)σ ri (0) G transvection along r i, φ t 1(p) = r 1 (t) at distance C from r 2 (t), φ t 2 φt 1(p) lies at distance C from r 2 (0) = q, at dist. 2C from p, Hence φ t k 2 φ t k 1 φ for some sequence t k, φ (ξ) = ξ φ (r 1 (t)) is a ray at ctnt distance from r 2 (t): they bound a flat strip φ r 1 φ (p) ξ q r 2

52 Proof that G ξ acts transitively on X. t d(r 1 (t),r 2 (t)) is convex and bounded, hence monotonic φ t i = σ r i (t)σ ri (0) G transvection along r i, φ t 1(p) = r 1 (t) at distance C from r 2 (t), φ t 2 φt 1(p) lies at distance C from r 2 (0) = q, at dist. 2C from p, Hence φ t k 2 φ t k 1 φ for some sequence t k, φ (ξ) = ξ φ (r 1 (t)) is a ray at ctnt distance from r 2 (t): they bound a flat strip φ r 1 φ (p) ξ q φ r 1 and r 2 lie in a 2-flat F = R 2 τ transvection in F st τ(φ (p)) = q, hence τ(φ (ξ)) = τ(ξ) = ξ r 2

53 Proof that G ξ acts transitively on X. t d(r 1 (t),r 2 (t)) is convex and bounded, hence monotonic φ t i = σ r i (t)σ ri (0) G transvection along r i, φ t 1(p) = r 1 (t) at distance C from r 2 (t), φ t 2 φt 1(p) lies at distance C from r 2 (0) = q, at dist. 2C from p, Hence φ t k 2 φ t k 1 φ for some sequence t k, φ (ξ) = ξ φ (r 1 (t)) is a ray at ctnt distance from r 2 (t): they bound a flat strip φ r 1 φ (p) ξ q φ r 1 and r 2 lie in a 2-flat F = R 2 τ transvection in F st τ(φ (p)) = q, hence τ(φ (ξ)) = τ(ξ) = ξ Corollary For r 1 r 2, r 1 is regular iff r 2 is regular. Hence we talk about regular/singular ideal points (ideal chambers later) r 2

54 Example H 2 H 2. F = S 0 S 0 = S 1 (H 2 H 2 ) = S 1 S 1 = S 3 S 0 = R S 1 = H 2 S 1 chamber chamber (length π 2 ) vertex (ins 0 ) S 1

55 Example H 2 H 2. F = S 0 S 0 = S 1 (H 2 H 2 ) = S 1 S 1 = S 3 S 0 = R S 1 = H 2 S 1 chamber chamber (length π 2 ) vertex (ins 0 ) S 1 G = PSL(2,R) PSL(2,R) Regular orbits = S 1 S 1, singular orbits = S 1. Stabilizers of ideal points: G ξ = ( 0 ) ( 0 ) for ξ regular, G ξ = ( ) ( 0 ) for ξ singular. Orbits: G/G ξ

56 Example X = SL(3,R)/SO(3). F = S 1 : chamber (length π 3 ) vertex X = S 4 ξ X ( ξ regular, G ξ ) = 0, Gξ = G/G ξ 0 0 = flag manifold of P 2 ( ξ singular, G ξ ) = 0 0 or G ξ = ( 0 0, Gξ = G/G ξ = P 2 ), Gξ = G/G ξ = (P 2 ) {(p,l) P 2 (P 2 ) p l}

57 Stabilizer of ideal points ξ X. h h, ξ = lim t + exp(th)x 0 X, G ξ = {g G gξ = ξ} Lemma Lie(G ξ ) = h g α α(h) 0

58 Stabilizer of ideal points ξ X. h h, ξ = lim t + exp(th)x 0 X, G ξ = {g G gξ = ξ} Lemma Lie(G ξ ) = h g α α(h) 0 Proof (sketch): [g α,g β ] g α+β h α(h) 0 g α is a subalgebra. exph translates e th x 0 to a parallel line because [h,h] = 0.

59 Stabilizer of ideal points ξ X. h h, ξ = lim t + exp(th)x 0 X, G ξ = {g G gξ = ξ} Lemma Lie(G ξ ) = h g α α(h) 0 Proof (sketch): [g α,g β ] g α+β h α(h) 0 g α is a subalgebra. exph translates e th x 0 to a parallel line because [h,h] = 0. For v g α, [h,v] = α(h)v: d(e v e th x 0,e th x 0 ) = d(e th e v e th x 0,x 0 ) = d(e exp( tα(h))v x 0,x 0 ) which is bounded iff α(h) 0 (it converges to 0 iff α(h) > 0).

60 Stabilizer of ideal points ξ X. h h, ξ = lim t + exp(th)x 0 X, G ξ = {g G gξ = ξ} Lemma Lie(G ξ ) = h g α α(h) 0 Proof (sketch): [g α,g β ] g α+β h α(h) 0 g α is a subalgebra. exph translates e th x 0 to a parallel line because [h,h] = 0. For v g α, [h,v] = α(h)v: d(e v e th x 0,e th x 0 ) = d(e th e v e th x 0,x 0 ) = d(e exp( tα(h))v x 0,x 0 ) which is bounded iff α(h) 0 (it converges to 0 iff α(h) > 0). Remark: the distance converges to 0 for regular directions. Corollary: Define ideal chambers/simplices as G ξ being constant. P = G ξ parabolic subgroup B = G chamber = G ξ, ξ regular, Borel subgrup (max solvable/min parab)

61 Properties of flats in X. (1) x X, σ chamber in X,!F X flat s.t. x F, σ F (2) Given an ideal chamber σ X, X = {F flat σ F} (3) F 1, F 2 maximal flats asymptotic to σ, then ξ int(σ) there exist rays r i F i asymptotic to ξ, s.t. d(r 1 (t),r 2 (t)) 0 (4) σ,σ X chambers, F X flat such that σ,σ F

62 Properties of flats in X. (1) x X, σ chamber in X,!F X flat s.t. x F, σ F (2) Given an ideal chamber σ X, X = {F flat σ F} (3) F 1, F 2 maximal flats asymptotic to σ, then ξ int(σ) there exist rays r i F i asymptotic to ξ, s.t. d(r 1 (t),r 2 (t)) 0 (4) σ,σ X chambers, F X flat such that σ,σ F Existence (1) and (2): G ξ transitive on X. uniqueness of (1): uniqueness of a geodesic from x 0 to ξ int(σ) (3): estimate previous slide (4) Intersection Borel subalgebras contains a Cartan subalgebra (or open/close argument)

63 Ideal chambers G acts transitively on the set of ideal chambers (transitivity on flats + Weyl group) Def: The Fürstenberg boundary of X is F X = {ideal chambers} = G/G σ = {regulargorbit} F (H 2 H 2 ) = S 1 S 1 S 1 chamber p S 1 zz F SL(n+1,R)/SO(n+1) = Flag manifold ofp n

64 Tits building F = R r maximal flat, ( F,W) is a spherical Coxeter complex A chamber is a simplex of maximal dimension Theorem: X is a Tits building with apartments ( F,W) of dim r 1. Namely, X is a simplical complex with subcomplexes F s.t.: 1. It is thick: every simplex of dim < r 1 belongs to at least 3 chambers 2. Every apartment is thin (a wall belongs to only 2 chambers). 3. ξ 1,ξ 2 X there is an apartment A s.t. ξ 1,ξ 2 A 4. ξ 1,ξ 2 A A X then φ : A = A, φ(ξ i ) = ξ i

65 Examples of Tits buildings (H 2 H 2 ) = S 1 S 1 = S 3 S 1 chamber S 1 ( F,W) = (S 1,Z/2Z Z/2Z) type A 1 A 1

66 Examples of Tits buildings (H 2 H 2 ) = S 1 S 1 = S 3 ( F,W) = (S 1,Z/2Z Z/2Z) A 1 A 1 S 1 chamber S 1 H 2 H 2 flat η 1 η 1 η 2 η 2

67 Examples of Tits buildings (H 2 H 2 ) = S 1 S 1 = S 3 ( F,W) = (S 1,Z/2Z Z/2Z) A 1 A 1 S 1 chamber S 1 H 2 H 2 flats γ 1 η 1 γ 1 γ 2 γ 2 η 1 η 2 η 2

68 Examples of Tits buildings SL(n+1,R)/SO(n+1) = S n(n+3)/2 1, ( F,W) = (S n 1,Σ n ), type A n. A 2 A 3

69 Tits distance Def: The Tits distance between ξ 1,ξ 2 X is d T (ξ 1,ξ 2 ) = sup x X x (ξ 1,ξ 2 ) ξ 1 x X ξ 2 Flat case: F = R r = S r 1 with the standard metric. d T is realized by apartments If X is CAT( 1) then d T (ξ 1,ξ 2 ) = π.

70 III Dynamics at infinity (joint with M. Kapovich and B. Leeb)

71 Examples of Tits building structures at X (H 2 H 2 ) = S 1 S 1 = S 3 ( F,W) = (S 1,Z/2Z Z/2Z) A 1 A 1 S 1 chamber S 1 H 2 H 2 flat η 1 η 1 η 2 η 2

72 Examples of Tits building structures at X (H 2 H 2 ) = S 1 S 1 = S 3 ( F,W) = (S 1,Z/2Z Z/2Z) A 1 A 1 S 1 chamber S 1 H 2 H 2 flats γ 1 η 1 γ 1 γ 2 γ 2 η 1 η 2 η 2

73 Examples of Tits building structures at X SL(n+1,R)/SO(n+1) = S n(n+3)/2 1, ( F,W) = (S n 1,Σ n )

74 Examples of Tits building structures at X SL(n+1,R)/SO(n+1) = S n(n+3)/2 1, ( F,W) = (S n 1,Σ n ) ( SL(3,R)/SO(3) ) = S 4, W = Σ 3 = D 3 F X = regular orbit = Flag(P 2 ) = {(p,l) p l} P 2 (P 2 ) Two singular SL(3,R)-orbits: P 2 and (P 2 ) ( SL(3,R)/SO(3) ) = Flag(P 2 ) (0,1) P 2 (P 2 )

75 Regular subgroups X = G/K symmetric space of noncompact type. Γ < G discrete subgroup, γ n Γ sequence. Assume xγ n x regular x V(x,σ n ) γ n x σ n X chamber V(x,σ n ) = {rays fromxtoσ n }. (σ n depends on γ n and x) Def: Γ regular if γ n Γ, γ n, then d(γ n x, V(x,σ n ))

76 Regular subgroups X = G/K symmetric space of noncompact type. Γ < G discrete subgroup, γ n Γ sequence. Assume xγ n x regular x V(x,σ n ) γ n x σ n X chamber V(x,σ n ) = {rays fromxtoσ n }. (σ n depends on γ n and x) Def: Γ regular if γ n Γ, γ n, then d(γ n x, V(x,σ n )) If Λ Γ = X Γx is regular Γ regular, but. There is a well defined notion of limit chamber σ n σ and chamber limit set: Λ Ch (Γ) F X Aim: find disc. domains in G-orbits in X depending on Λ Ch (Γ)

77 Relative position (a mod,σ mod ) = reference model apartment and chamber Chart α : (a mod,σ mod ) = ( F,σ) Def: For fixed σ X, the position relative σ is: pos(,σ) : X a mod ξ pos(ξ,σ) = α 1 (ξ)

78 Relative position (a mod,σ mod ) = reference model apartment and chamber Chart α : (a mod,σ mod ) = ( F,σ) Def: For fixed σ X, the position relative σ is: pos(,σ) : X a mod ξ pos(ξ,σ) = α 1 (ξ) G orbit W orbit

79 Relative position (a mod,σ mod ) = reference model apartment and chamber Chart α : (a mod,σ mod ) = ( F,σ) Def: For fixed σ X, the position relative σ is: pos(,σ) : X a mod G orbit ξ pos(ξ,σ) = α 1 (ξ) W orbit Example: (SL(3,R)/SO(3)) F = S 1 A reference chamber is equiv. to a flag (p,l) P 2 (P 2 ), p l l ref. chamber p

80 Example of relative position Example: (SL(3,R)/SO(3)) F = S 1 A reference chamber is equiv. to a flag (p,l) P 2 (P 2 ), p l There are two W -orbits of singular points (corresponding to pts /lines) {p p l, p p} l ref. chamber {l p l } p {p p l} {l p l, l l}

81 Example of relative position Example: (SL(3,R)/SO(3)) F = S 1 A reference chamber is equiv. to a flag (p,l) P 2 (P 2 ), p l There are two W -orbits of singular points (corresponding to pts /lines) {p p l, p p} l ref. chamber {l p l } p {p p l} {l p l, l l} {ξ G orbit pos(ξ,σ) = ctnt} Schubert cell (maybe not closed) {ξ G orbit pos(ξ,σ) = ctnt} Schubert cycle Want to remove certain Schubert cycles relative to chambers in Λ Ch (Γ)

82 Partial ordering in Weyl orbits Wη 0 a mod Weyl orbit, η 1,η 2 Wη 0 Def: η 1 η 2 if {ξ pos(ξ,σ) = η 1 } {ξ pos(ξ,σ) = η 2 }

83 Partial ordering in Weyl orbits Wη 0 a mod Weyl orbit, η 1,η 2 Wη 0 Def: η 1 η 2 if {ξ pos(ξ,σ) = η 1 } {ξ pos(ξ,σ) = η 2 } Example: {p p l, p p} l ref. chamber {l p l } p {p p l} {l p l, l l}

84 Partial ordering in Weyl orbits Wη 0 a mod Weyl orbit, η 1,η 2 Wη 0 Def: η 1 η 2 if {ξ pos(ξ,σ) = η 1 } {ξ pos(ξ,σ) = η 2 } Example: {p p l, p p} l ref. chamber {l p l } p {p p l} {l p l, l l} If η 0 is regular, then Wη 0 = W and is the Bruhat ordering on W Longest element w 0 W : σ mod and w 0 σ mod are antipodal (opposite)

85 Partial ordering in Weyl orbits Wη 0 a mod Weyl orbit, η 1,η 2 Wη 0 Def: η 1 η 2 if {ξ pos(ξ,σ) = η 1 } {ξ pos(ξ,σ) = η 2 } Example: {p p l, p p} l ref. chamber {l p l } p {p p l} {l p l, l l} If η 0 is regular, then Wη 0 = W and is the Bruhat ordering on W Longest element w 0 W : σ mod and w 0 σ mod are antipodal (opposite) Def: S Wη 0 closed if η 1 S and η 2 η 1 imply η 2 S {ξ Gξ 0 pos(ξ,σ) S} is closed if S is closed

86 Thickenings of limit set Γ G discrete and regular. S Wη 0 a mod closed (for ) Λ Ch (Γ) F X Def: S-Thickening of Λ Ch (Γ): Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S}

87 Thickenings of limit set Γ G discrete and regular. S Wη 0 a mod closed (for ) Λ Ch (Γ) F X Def: S-Thickening of Λ Ch (Γ): Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S} Theorem: Γ G discrete and regular. If S is fat then Gξ 0 \Th S (Λ Ch (Γ)) is a domain of discontinuity for Γ S fat: Wη 0 = S w 0 S, where w 0 W is the longest element in W (we need to remove a sufficiently large set to have proper discontinuity)

88 Chamber conical and antipodal Def: A chamber σ Λ Ch (Γ) is conical if N R (V(x,σ)) Γx is infinite, for some R > 0, some (any) x X. x V(x,σ) γx σ X Def: Γ is antipodal if σ 1 σ 2 Λ Ch (Γ) are antipodal/opposite (pos(σ 1,σ 2 ) = w 0 σ mod, w 0 σ mod and σ mod antipodal in a mod ). σ mod w 0 σ mod Def: Γ < G discrete is RCA if regular, chamber conical and antipodal

89 Chamber conical and antipodal Def: A chamber σ Λ Ch (Γ) is conical if N R (V(x,σ)) Γx is infinite, for some R > 0, some (any) x X. x V(x,σ) γx σ X Def: Γ is antipodal if σ 1 σ 2 Λ Ch (Γ) are antipodal/opposite (pos(σ 1,σ 2 ) = w 0 σ mod, w 0 σ mod and σ mod antipodal in a mod ). Def: Γ < G discrete is RCA if regular, chamber conical and antipodal Theorem: Γ G RCA. If S Wη 0 closed and slim, then ( Gξ0 \Th S (Λ Ch (Γ)) ) /Γ is compact S slim: S w 0 S =.

90 Chamber conical and antipodal Def: A chamber σ Λ Ch (Γ) is conical if N R (V(x,σ)) Γx is infinite, for some R > 0, some (any) x X. x V(x,σ) γx σ X Def: Γ is antipodal if σ 1 σ 2 Λ Ch (Γ) are antipodal/opposite (pos(σ 1,σ 2 ) = w 0 σ mod, w 0 σ mod and σ mod antipodal in a mod ). Def: Γ < G discrete is RCA if regular, chamber conical and antipodal Theorem: Γ G RCA. If S Wη 0 closed and slim, then ( Gξ0 \Th S (Λ Ch (Γ)) ) /Γ is compact S slim: S w 0 S =. fat: S w 0 S = Wη 0. balanced = slim + fat Corollary: Γ G RCA. If S Wη 0 closed and balanced, then Gξ 0 \Th S (Λ Ch (Γ)) is a cocompact domain of discont.

91 Examples of thickenings X = SL(n+1,R)/SO(n+1) Wη 0 a mod rel positions of points in P n e.g. G ξ = σ mod = {F 0 F 1 F n 1 } reference chamber S k : points in F k fat ifk (n 1)/2 S k = slim ifk (n 1)/2 {( )} 0 0 0

92 Examples of thickenings X = SL(n+1,R)/SO(n+1) Wη 0 a mod rel positions of points in P n e.g. G ξ = σ mod = {F 0 F 1 F n 1 } reference chamber S k : points in F k fat ifk (n 1)/2 S k = slim ifk (n 1)/2 {( )} Wη 0 a mod orbit of rel )} positions of {(p,h) P n (P n ) p H} e.g. G ξ = {( S = {(p,h) p F i H for somei} is balanced

93 Examples of thickenings X = SL(n+1,R)/SO(n+1) Wη 0 a mod rel positions of points in P n e.g. G ξ = σ mod = {F 0 F 1 F n 1 } reference chamber S k : points in F k fat ifk (n 1)/2 S k = slim ifk (n 1)/2 {( )} Wη 0 a mod orbit of rel )} positions of {(p,h) P n (P n ) p H} e.g. G ξ = {( S = {(p,h) p F i H for somei} is balanced Also for X = SL(n+1,C)/SU(n+1)

94 Example: Fuchsian subgroups Γ Isom + (H 2 ), H 2

95 Example: Fuchsian subgroups Γ Isom + (H 2 ), Λ 0 = Γx H 2 H 2 Λ 0

96 Example: Fuchsian subgroups Γ Isom + (H 2 ), Λ 0 = Γx H 2 H 2 Γ Isom + (H 2 ) = SO(2,1) SL(3,R) Q = H 2 = {[x 0 : x 1 : x 2 ] P 2 x 2 0 = x 2 1 +x 2 2} conic Γ is regular: Λ Ch = {(p,l) p Λ 0,l = T p Q} Ω = P 2 \{x x line tangent toqatλ 0 } is a domain of discontinuity for Γ

97 Example: Fuchsian subgroups Γ Isom + (H 2 ), Λ 0 = Γx H 2 H 2 Γ Isom + (H 2 ) = SO(2,1) SL(3,R) Q = H 2 = {[x 0 : x 1 : x 2 ] P 2 x 2 0 = x 2 1 +x 2 2} conic Γ is regular: Λ Ch = {(p,l) p Λ 0,l = T p Q} Ω = P 2 \{x x line tangent toqatλ 0 } is a domain of discontinuity for Γ Ω = Flag(P 2 )\{(p,l) p l, p Λ 0 orltangent toqatλ 0 } is a domain of disc. for Γ, cocompact when Γ cvx cocompact in H 2

98 Example of RCA: Veronese embedding Γ 0 < SL(2,C) discrete, H 3 /Γ 0 hyp. orbifold, Λ 0 P 1 (C) limit set Γ = Sym N (Γ 0 ) SL(N +1,C), Γ X = SL(N +1,C)/SU(N +1) Sym N : SL(2,C) SL(N +1,C) irreducible C N+1 = {p(x,y) C[x,y] homogeneous & deg(p(x,y)) = N} If C 2 = v 1,v 2, then Sym N (C 2 ) = v N 1,v N 1 1 v 2,,v N 2. Γ is regular, and it is RCA if Γ 0 convex cocompact.

99 Example of RCA: Veronese embedding Γ 0 < SL(2,C) discrete, H 3 /Γ 0 hyp. orbifold, Λ 0 P 1 (C) limit set Γ = Sym N (Γ 0 ) SL(N +1,C), Γ X = SL(N +1,C)/SU(N +1) Sym N : SL(2,C) SL(N +1,C) irreducible C N+1 = {p(x,y) C[x,y] homogeneous & deg(p(x,y)) = N} If C 2 = v 1,v 2, then Sym N (C 2 ) = v N 1,v N 1 1 v 2,,v N 2. Γ is regular, and it is RCA if Γ 0 convex cocompact. Veronese embedding φ : P 1 P N = P({p C[X,Y]homogeneous & degp = N}) [a : b] [(ax +by) N ] Λ ch (Γ) = { flags osculating toφ(λ 0 ) φ(p 1 )}

100 Example of RCA: Veronese embedding Γ 0 < SL(2,C) discrete, H 3 /Γ 0 hyp. orbifold, Λ 0 P 1 (C) limit set Γ = Sym N (Γ 0 ) SL(N +1,C), Γ X = SL(N +1,C)/SU(N +1) Γ is regular, and it is RCA if Γ 0 convex cocompact. Λ ch (Γ) = { flags osculating toφ(λ 0 ) φ(p 1 )} Theorem: For N > 2, Γ acts properly on P N \Osc [N/2] φ(λ 0 ) and cocompactly on P N \Osc [(N 1)/2] φ(λ 0 ) if Γ 0 convex cocompact Osc k = k-th osculating manifold (k = 1 is the tangent bundle, k = 2 planes approaching second order, etc) For N 1, Γ acts properly (cocompactly when Γ 0 cvx cocompact) on {(p,h) P N (P N ) p H, F i osc toφ(γ 0 ), p F i H}

101 Weak RCA subgroups One can weaken the regularity assumption working with simplices instead of chambers. eg limit sets are not full flags but only partial flags Can talk about weak-rca subgroups Theorem: Γ is weak-rca iff it is anosov (Anosov in the sense of Labourie and Guichard-Wienhard) Benoist: strictly convex projective groups are weak-rca Ω P n limit set in {(p,h) P n (P n ) p H}: Λ = {(p,h) p Ω,H tangent to Ω}

102 Sketch of proof of discontinuity Theorem: Γ G discrete and regular. If S is fat then Ω = Gξ 0 \Th S (Λ Ch (Γ)) is a domain of discontinuity for Γ Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S}

103 Sketch of proof of discontinuity Theorem: Γ G discrete and regular. If S is fat then Ω = Gξ 0 \Th S (Λ Ch (Γ)) is a domain of discontinuity for Γ Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S} Sketch of proof: Take ξ n Gξ 0, ξ n ξ, γ n Γ, γ n, and γ n ξ n ξ. Want to see that if ξ lies in Ω, then ξ doesn t

104 Sketch of proof of discontinuity Theorem: Γ G discrete and regular. If S is fat then Ω = Gξ 0 \Th S (Λ Ch (Γ)) is a domain of discontinuity for Γ Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S} Sketch of proof: Take ξ n Gξ 0, ξ n ξ, γ n Γ, γ n, and γ n ξ n ξ. Want to see that if ξ lies in Ω, then ξ doesn t Step 1: γ ±1 n σ ± Λ Ch (Γ) Step 2: If ˆσ F X is opposite to σ, then γ nˆσ σ + Assume ξ Ω, hence pos(ξ,σ ) / S pos(ξ n,σ ) / S pos(ξ n,ˆσ) S (S fat) pos(γ n ξ n,γ nˆσ) S pos(ξ,σ + ) S (S closed) Thus ξ / Ω.

105 Sketch of proof of cocompactness Theorem: Γ G RCA. If S Wη 0 closed and slim, then ( Gξ0 \Th S (Λ Ch (Γ)) ) /Γ is compact Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S}

106 Sketch of proof of cocompactness Theorem: Γ G RCA. If S Wη 0 closed and slim, then ( Gξ0 \Th S (Λ Ch (Γ)) ) /Γ is compact Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S} Sketch of proof: Step 1. Γ is expansive (à la Sullivan) on Λ Ch (Γ) F X by conicallity: For all σ Λ Ch (Γ) there are γ Γ, σ U F X, C > 1 st: d(γσ 1,γσ 2 ) > Cd(σ 1,σ 2 ) σ 1,σ 2 U Step 2. There is a fibration π : Th S (Λ Ch (Γ)) Λ Ch (Γ) (that extends in a neighbhood in Gξ 0 ) by slimness and antipodality

107 Sketch of proof of cocompactness Theorem: Γ G RCA. If S Wη 0 closed and slim, then ( Gξ0 \Th S (Λ Ch (Γ)) ) /Γ is compact Th S (Λ Ch (Γ)) = σ Λ Ch (Γ) {ξ Gξ 0 pos(ξ,σ) S} Sketch of proof: Step 1. Γ is expansive (à la Sullivan) on Λ Ch (Γ) F X by conicallity: For all σ Λ Ch (Γ) there are γ Γ, σ U F X, C > 1 st: d(γσ 1,γσ 2 ) > Cd(σ 1,σ 2 ) σ 1,σ 2 U Step 2. There is a fibration π : Th S (Λ Ch (Γ)) Λ Ch (Γ) (that extends in a neighbhood in Gξ 0 ) by slimness and antipodality Then d(γπ 1 (σ 1 ),γπ 1 (σ 2 )) > Cd(π 1 (σ 1 ),π 1 (σ 2 )) σ 1,σ 2 U Which implies cocompactness (find a fundamental domain uniformly away from Th S (Λ Ch (Γ)))

108 Thank you for your attention