Invariant Random Subgroups

Transcription

1 Invariant Random Subgroups Lewis Bowen Group Theory International Webinar May 2012 Lewis Bowen (Texas A&M) Invariant Random Subgroups 1 / 29

2 Set-up G : a locally compact group. Lewis Bowen (Texas A&M) Invariant Random Subgroups 2 / 29

3 Set-up G : a locally compact group. Sub(G) : the space of closed subgroups. Lewis Bowen (Texas A&M) Invariant Random Subgroups 2 / 29

4 Set-up G : a locally compact group. Sub(G) : the space of closed subgroups. G acts on Sub(G) by conjugation. g H := ghg 1. Lewis Bowen (Texas A&M) Invariant Random Subgroups 2 / 29

5 Set-up G : a locally compact group. Sub(G) : the space of closed subgroups. G acts on Sub(G) by conjugation. g H := ghg 1. M(G) : space of G-invariant Borel probability measures on Sub(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 2 / 29

6 Set-up G : a locally compact group. Sub(G) : the space of closed subgroups. G acts on Sub(G) by conjugation. g H := ghg 1. M(G) : space of G-invariant Borel probability measures on Sub(G). An invariant random subgroup (IRS) is a random subgroup H < G with law in M(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 2 / 29

7 Examples N G δ N M(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

8 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

9 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Let Γ < G be a lattice. a G-invariant prob. meas. λ on G/Γ. Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

10 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Let Γ < G be a lattice. a G-invariant prob. meas. λ on G/Γ. Define Φ : G/Γ Sub(G) by Φ(gΓ) := gγg 1. Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

11 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Let Γ < G be a lattice. a G-invariant prob. meas. λ on G/Γ. Define Φ : G/Γ Sub(G) by Φ(gΓ) := gγg 1. Then Φ λ := µ Γ M(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

12 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Let Γ < G be a lattice. a G-invariant prob. meas. λ on G/Γ. Define Φ : G/Γ Sub(G) by Φ(gΓ) := gγg 1. Then Φ λ := µ Γ M(G). Let G (X, µ) be a probability-measure-preserving Borel action. Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

13 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Let Γ < G be a lattice. a G-invariant prob. meas. λ on G/Γ. Define Φ : G/Γ Sub(G) by Φ(gΓ) := gγg 1. Then Φ λ := µ Γ M(G). Let G (X, µ) be a probability-measure-preserving Borel action. Stab : X Sub(G), Stab(x) := {g G : gx = x}. Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

14 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Let Γ < G be a lattice. a G-invariant prob. meas. λ on G/Γ. Define Φ : G/Γ Sub(G) by Φ(gΓ) := gγg 1. Then Φ λ := µ Γ M(G). Let G (X, µ) be a probability-measure-preserving Borel action. Stab : X Sub(G), Stab(x) := {g G : gx = x}. Stab is G-equivariant Stab µ M(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

15 Examples N G δ N M(G). If N < G has only finitely many conjugates uniform prob. meas. on the conjugates of N is in M(G). Let Γ < G be a lattice. a G-invariant prob. meas. λ on G/Γ. Define Φ : G/Γ Sub(G) by Φ(gΓ) := gγg 1. Then Φ λ := µ Γ M(G). Let G (X, µ) be a probability-measure-preserving Borel action. Stab : X Sub(G), Stab(x) := {g G : gx = x}. Stab is G-equivariant Stab µ M(G). (Abert-Glasner-Virag) every measure in M(G) arises this way. Lewis Bowen (Texas A&M) Invariant Random Subgroups 3 / 29

16 M(G) is a simplex Definition A convex closed metrizable subset K of a locally convex linear space is a simplex if each point in K is the barycenter of a unique probability measure supported on the subset e K of extreme points of K. Lewis Bowen (Texas A&M) Invariant Random Subgroups 4 / 29

17 M(G) is a simplex Definition A convex closed metrizable subset K of a locally convex linear space is a simplex if each point in K is the barycenter of a unique probability measure supported on the subset e K of extreme points of K. If µ 1, µ 2 M(G) and t [0, 1] then tµ 1 + (1 t)µ 2 M(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 4 / 29

18 Research directions in IRS s Problem: classify the ergodic IRS s of a given group or describe M(G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 5 / 29

19 Research directions in IRS s Problem: classify the ergodic IRS s of a given group or describe M(G). Remark 1. M(G) is compact in the weak* topology. So it can be viewed as a compactification of the set of lattice subgroups. Lewis Bowen (Texas A&M) Invariant Random Subgroups 5 / 29

20 Research directions in IRS s Problem: classify the ergodic IRS s of a given group or describe M(G). Remark 1. M(G) is compact in the weak* topology. So it can be viewed as a compactification of the set of lattice subgroups. Remark 2. If K is an IRS then K \G can be thought of as something like a group. Although it need not be homogeneous, it possesses statistical homogeneity. Lewis Bowen (Texas A&M) Invariant Random Subgroups 5 / 29

21 Higher rank simple Lie groups Theorem (Stuck-Zimmer, 1994) If G is a simple Lie group of real rank 2 and K < G is an ergodic IRS then either K is a lattice a.s. or K = {e}. Lewis Bowen (Texas A&M) Invariant Random Subgroups 6 / 29

22 Higher rank simple Lie groups Theorem (Stuck-Zimmer, 1994) If G is a simple Lie group of real rank 2 and K < G is an ergodic IRS then either K is a lattice a.s. or K = {e}. Let X = G/K. An X-manifold M is a manifold locally modeled on X (i.e., M = X/Γ for some lattice Γ < G). Lewis Bowen (Texas A&M) Invariant Random Subgroups 6 / 29

23 Higher rank simple Lie groups Theorem (Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet) If G is as above, and M i is a sequence of X-manifolds such that lim vol(m i) = +, i lim inf i injrad(m i) > 0 k, b k (M i ) lim i vol(m i ) = β k(x). Lewis Bowen (Texas A&M) Invariant Random Subgroups 7 / 29

24 Higher rank simple Lie groups Theorem (Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet) If G is as above, and M i is a sequence of X-manifolds such that lim vol(m i) = +, i lim inf i injrad(m i) > 0 k, b k (M i ) lim i vol(m i ) = β k(x). Sketch. Let M i = X/Γ i. By Stuck-Zimmer, µ Γi converges in M(G) to δ e. Show that L 2 -betti numbers vary continuously on M(G) using a generalized version of Lück approximation. Lewis Bowen (Texas A&M) Invariant Random Subgroups 7 / 29

25 Random walks Let G = a, b R be a 2-generator group, Lewis Bowen (Texas A&M) Invariant Random Subgroups 8 / 29

26 Random walks Let G = a, b R be a 2-generator group, µ = uniform prob. meas. on {a, b, a 1, b 1 }, Lewis Bowen (Texas A&M) Invariant Random Subgroups 8 / 29

27 Random walks Let G = a, b R be a 2-generator group, µ = uniform prob. meas. on {a, b, a 1, b 1 }, {X i } i=1 = i.i.d. random variables with law µ, Lewis Bowen (Texas A&M) Invariant Random Subgroups 8 / 29

28 Random walks Let G = a, b R be a 2-generator group, µ = uniform prob. meas. on {a, b, a 1, b 1 }, {X i } i=1 = i.i.d. random variables with law µ, Z n = X 1 X n. Lewis Bowen (Texas A&M) Invariant Random Subgroups 8 / 29

29 Random walks Let G = a, b R be a 2-generator group, µ = uniform prob. meas. on {a, b, a 1, b 1 }, {X i } i=1 = i.i.d. random variables with law µ, Z n = X 1 X n. {Z n } is the simple random walk on G with µ-increments. Lewis Bowen (Texas A&M) Invariant Random Subgroups 8 / 29

30 Entropy Let µ n be the law of Z n, Lewis Bowen (Texas A&M) Invariant Random Subgroups 9 / 29

31 Entropy Let µ n be the law of Z n, H(µ n ) := g G µ({g}) log µ({g}); Lewis Bowen (Texas A&M) Invariant Random Subgroups 9 / 29

32 Entropy Let µ n be the law of Z n, H(µ n ) := g G µ({g}) log µ({g}); 1 h µ (G) := lim n n H(µn ). Lewis Bowen (Texas A&M) Invariant Random Subgroups 9 / 29

33 Entropy Let µ n be the law of Z n, H(µ n ) := g G µ({g}) log µ({g}); 1 h µ (G) := lim n n H(µn ). 0 h µ (G) h µ (F 2 ). Lewis Bowen (Texas A&M) Invariant Random Subgroups 9 / 29

34 Entropy Let µ n be the law of Z n, H(µ n ) := g G µ({g}) log µ({g}); 1 h µ (G) := lim n n H(µn ). 0 h µ (G) h µ (F 2 ). Problem What are all possible values of h µ (G) as G varies over all 2-generator groups? Lewis Bowen (Texas A&M) Invariant Random Subgroups 9 / 29

35 Random walk entropy This problem is related to the structure theory of stationary actions and Furstenberg entropy. Lewis Bowen (Texas A&M) Invariant Random Subgroups 10 / 29

36 Random walk entropy This problem is related to the structure theory of stationary actions and Furstenberg entropy. Theorem is dense in [0, h µ (F 2 )]. {h µ (G) : G a 2-generator group} Lewis Bowen (Texas A&M) Invariant Random Subgroups 10 / 29

37 Random walks on random coset spaces For K < F 2, consider the random walk {KZ n } n=1 on K \F 2. Lewis Bowen (Texas A&M) Invariant Random Subgroups 11 / 29

38 Random walks on random coset spaces For K < F 2, consider the random walk {KZ n } n=1 on K \F 2. Let µ n K ({Kg}) = Prob (KZ n = Kg), Lewis Bowen (Texas A&M) Invariant Random Subgroups 11 / 29

39 Random walks on random coset spaces For K < F 2, consider the random walk {KZ n } n=1 on K \F 2. Let µ n K ({Kg}) = Prob (KZ n = Kg), 1 h µ (λ) := lim n n H(µ n K ) dλ(k ). Lewis Bowen (Texas A&M) Invariant Random Subgroups 11 / 29

40 Random walk entropy Theorem There exists a path-connected subspace N M e (F 2 ) on which the map λ N h µ (λ) is continuous and surjects onto [0, h µ (F 2 )]. Lewis Bowen (Texas A&M) Invariant Random Subgroups 12 / 29

41 Random walk entropy Theorem There exists a path-connected subspace N M e (F 2 ) on which the map λ N h µ (λ) is continuous and surjects onto [0, h µ (F 2 )]. Theorem The finitely-supported measures in N are dense and these correspond to normal subgroups of F 2. Therefore, is dense in [0, h µ (F 2 )]. {h µ (G) : G a 2-generator group} Lewis Bowen (Texas A&M) Invariant Random Subgroups 12 / 29

42 Let K n < F 2 be the group generated by all elements of the form ghg 1 where g a n, b n and either h = a k b r a k for some 1 k n 1 and r Z or h = b k a r b k for some 1 k n 1 and r Z. Lewis Bowen (Texas A&M) Invariant Random Subgroups 13 / 29

43 A covering construction Choose 0 p 1 and choose each loop of K n \F 2 with probability p independently. Lewis Bowen (Texas A&M) Invariant Random Subgroups 14 / 29

44 A covering construction Choose 0 p 1 and choose each loop of K n \F 2 with probability p independently. Consider the resulting 2-complex. Lewis Bowen (Texas A&M) Invariant Random Subgroups 14 / 29

45 A covering construction Choose 0 p 1 and choose each loop of K n \F 2 with probability p independently. Consider the resulting 2-complex. Take its universal cover. Lewis Bowen (Texas A&M) Invariant Random Subgroups 14 / 29

46 A covering construction Choose 0 p 1 and choose each loop of K n \F 2 with probability p independently. Consider the resulting 2-complex. Take its universal cover. This is the Schreier coset graph of an IRS with law λ n,p. Lewis Bowen (Texas A&M) Invariant Random Subgroups 14 / 29

47 A covering construction h µ (λ n,p ) is continuous in p, Lewis Bowen (Texas A&M) Invariant Random Subgroups 15 / 29

48 A covering construction h µ (λ n,p ) is continuous in p, λ n,0 = δ e h µ (λ n,0 ) = h µ (F 2 ), Lewis Bowen (Texas A&M) Invariant Random Subgroups 15 / 29

49 A covering construction h µ (λ n,p ) is continuous in p, λ n,0 = δ e h µ (λ n,0 ) = h µ (F 2 ), lim n h µ (λ n,1 ) = 0. Lewis Bowen (Texas A&M) Invariant Random Subgroups 15 / 29

50 A covering construction h µ (λ n,p ) is continuous in p, λ n,0 = δ e h µ (λ n,0 ) = h µ (F 2 ), lim n h µ (λ n,1 ) = 0. We can approximate λ n,p be choosing a periodic collection of loops of K n \F 2 and then taking the universal cover of the 2-complex, Lewis Bowen (Texas A&M) Invariant Random Subgroups 15 / 29

51 A covering construction h µ (λ n,p ) is continuous in p, λ n,0 = δ e h µ (λ n,0 ) = h µ (F 2 ), lim n h µ (λ n,1 ) = 0. We can approximate λ n,p be choosing a periodic collection of loops of K n \F 2 and then taking the universal cover of the 2-complex, which gives a Schreier coset graph for a group with only finitely many conjugates. Lewis Bowen (Texas A&M) Invariant Random Subgroups 15 / 29

52 A covering construction h µ (λ n,p ) is continuous in p, λ n,0 = δ e h µ (λ n,0 ) = h µ (F 2 ), lim n h µ (λ n,1 ) = 0. We can approximate λ n,p be choosing a periodic collection of loops of K n \F 2 and then taking the universal cover of the 2-complex, which gives a Schreier coset graph for a group with only finitely many conjugates. Its normal core has entropy approximating λ n,p. Lewis Bowen (Texas A&M) Invariant Random Subgroups 15 / 29

53 Classification Results Theorem (Stuck-Zimmer, 1994) If G is a simple Lie group of real rank 2 and K < G is an ergodic IRS then either K is a lattice a.s. or K = {e}. Lewis Bowen (Texas A&M) Invariant Random Subgroups 16 / 29

54 Classification Results Theorem (Stuck-Zimmer, 1994) If G is a simple Lie group of real rank 2 and K < G is an ergodic IRS then either K is a lattice a.s. or K = {e}. Theorem (Vershik, 2010) There is a nice classification of IRS s of S = n S n. Lewis Bowen (Texas A&M) Invariant Random Subgroups 16 / 29

55 Classification Results Theorem (Stuck-Zimmer, 1994) If G is a simple Lie group of real rank 2 and K < G is an ergodic IRS then either K is a lattice a.s. or K = {e}. Theorem (Vershik, 2010) There is a nice classification of IRS s of S = n S n. Theorem (Bader-Shalom, 2006) If G 1, G 2 are just non-compact infinite property (T) groups then every ergodic IRS K < G 1 G 2 either splits as a product K = H 1 H 2 or K is a lattice subgroup a.s. Lewis Bowen (Texas A&M) Invariant Random Subgroups 16 / 29

56 What sort of simplex is M(G)? A simplex Σ is Poulsen if e Σ is dense in Σ; Bauer if e Σ is closed in Σ. Lewis Bowen (Texas A&M) Invariant Random Subgroups 17 / 29

57 What sort of simplex is M(G)? A simplex Σ is Poulsen if e Σ is dense in Σ; Bauer if e Σ is closed in Σ. Theorem (Lindenstrauss-Olsen-Sternfeld, 1978) There is a unique Poulsen simplex Σ up to affine isomorphism. Moreover, e Σ = L 2. Lewis Bowen (Texas A&M) Invariant Random Subgroups 17 / 29

58 What sort of simplex is M(G)? A simplex Σ is Poulsen if e Σ is dense in Σ; Bauer if e Σ is closed in Σ. Theorem (Lindenstrauss-Olsen-Sternfeld, 1978) There is a unique Poulsen simplex Σ up to affine isomorphism. Moreover, e Σ = L 2. There are uncountably many nonisomorphic Bauer simplices. Lewis Bowen (Texas A&M) Invariant Random Subgroups 17 / 29

59 Simplices M(F r ) is neither. Lewis Bowen (Texas A&M) Invariant Random Subgroups 18 / 29

60 Simplices M(F r ) is neither. Let M fi (F r ) M e (F r ) be those measures coming from finite-index subgroups. Lewis Bowen (Texas A&M) Invariant Random Subgroups 18 / 29

61 Simplices M(F r ) is neither. Let M fi (F r ) M e (F r ) be those measures coming from finite-index subgroups. M fi (F r ) is countable and each µ M fi (F r ) is isolated in M e (F r ). Lewis Bowen (Texas A&M) Invariant Random Subgroups 18 / 29

62 Simplices M(F r ) is neither. Let M fi (F r ) M e (F r ) be those measures coming from finite-index subgroups. M fi (F r ) is countable and each µ M fi (F r ) is isolated in M e (F r ). Let M ie (F r ) := M e (F r ) \ M fi (F r ) and M i (F r ) = Hull(M ie (F r )). Lewis Bowen (Texas A&M) Invariant Random Subgroups 18 / 29

63 Simplices M(F r ) is neither. Let M fi (F r ) M e (F r ) be those measures coming from finite-index subgroups. M fi (F r ) is countable and each µ M fi (F r ) is isolated in M e (F r ). Let M ie (F r ) := M e (F r ) \ M fi (F r ) and M i (F r ) = Hull(M ie (F r )). Theorem M i (F r ) is a Poulsen simplex. So M ie (F r ) = L 2. Lewis Bowen (Texas A&M) Invariant Random Subgroups 18 / 29

64 Surgery Given two Schreier coset graphs K 1 \F r, K 2 \F r, we can connect them together by replacing a vertex of each with 2 vertices and adding some edges. Lewis Bowen (Texas A&M) Invariant Random Subgroups 19 / 29

65 Ergodic measures are dense Let η M i (F r ). For p (0, 1) we will construct η p M ie (F r ) such that lim p 0 η p = η. Lewis Bowen (Texas A&M) Invariant Random Subgroups 20 / 29

66 Building an ergodic approximation Let K < F r be random with law η. Lewis Bowen (Texas A&M) Invariant Random Subgroups 21 / 29

67 Building an ergodic approximation Color each vertex of K \F r red with prob. p independently. Lewis Bowen (Texas A&M) Invariant Random Subgroups 22 / 29

68 Building an ergodic approximation At a red vertex, choose a random subgroup L < F r with law η independent of K and attach its Schreier coset graph by surgery to K \F r. Lewis Bowen (Texas A&M) Invariant Random Subgroups 23 / 29

69 Building an ergodic approximation At a red vertex, choose a random subgroup J < F r with law η independent of K and other subgroups and attach its Schreier coset graph by surgery to K \F r. Lewis Bowen (Texas A&M) Invariant Random Subgroups 24 / 29

70 Building an ergodic approximation Lewis Bowen (Texas A&M) Invariant Random Subgroups 25 / 29

71 Building an ergodic approximation Lewis Bowen (Texas A&M) Invariant Random Subgroups 26 / 29

72 Building an ergodic approximation Lewis Bowen (Texas A&M) Invariant Random Subgroups 27 / 29

73 Building an ergodic approximation Lewis Bowen (Texas A&M) Invariant Random Subgroups 28 / 29

74 Building an ergodic approximation This is the Schreier coset graph of a random subgroup K < F 2. Let η p be the law of this subgroup. Show: η p is ergodic and lim p 0 η p = η. Lewis Bowen (Texas A&M) Invariant Random Subgroups 28 / 29

75 Further results and questions (Abert-Glasner-Weiss) If K < G is an ergodic IRS then ρ(k \G) = ρ(g) K is amenable a.s. Lewis Bowen (Texas A&M) Invariant Random Subgroups 29 / 29

76 Further results and questions (Abert-Glasner-Weiss) If K < G is an ergodic IRS then ρ(k \G) = ρ(g) K is amenable a.s. (B.) Any ergodic aperiodic probability-measure-preserving equivalence relation (X, µ, E) with cost(e) < r is isomorphic to (Sub(F r ), λ, E Fr ) for some λ M(F r ). Lewis Bowen (Texas A&M) Invariant Random Subgroups 29 / 29

77 Further results and questions (Abert-Glasner-Weiss) If K < G is an ergodic IRS then ρ(k \G) = ρ(g) K is amenable a.s. (B.) Any ergodic aperiodic probability-measure-preserving equivalence relation (X, µ, E) with cost(e) < r is isomorphic to (Sub(F r ), λ, E Fr ) for some λ M(F r ). (Bartholdi-Grigorchuk) There is a finitely generated group G with an ergodic IRS K so that the Schreier coset graph K \G has polynomial growth of irrational degree almost surely. Lewis Bowen (Texas A&M) Invariant Random Subgroups 29 / 29