Rokhlin dimension for actions of residually finite groups

Transcription

1 Rokhlin dimension for actions of residually finite groups Dynamics and C*-Algebras: Amenability and Soficity Banff International Research Station Gábor Szabó (joint with Jianchao Wu and Joachim Zacharias) WWU Münster October / 23

2 1 Introduction 2 Rokhlin dimension 3 Box spaces 4 Topological actions 2 / 23

3 Introduction 1 Introduction 2 Rokhlin dimension 3 Box spaces 4 Topological actions 3 / 23

4 Introduction Within the theory of C -algebras, the crossed product construction is long known as a nearly inexhaustible source of interesting examples. In particular, certain crossed products are often natural test cases for the Elliott classification program. Motivated by the increasing importance of nuclear dimension in the Elliott program, the following question constitutes the main theme of this talk: 4 / 23

5 Introduction Within the theory of C -algebras, the crossed product construction is long known as a nearly inexhaustible source of interesting examples. In particular, certain crossed products are often natural test cases for the Elliott classification program. Motivated by the increasing importance of nuclear dimension in the Elliott program, the following question constitutes the main theme of this talk: Question Let A be a C -algebra, G a countable group and (α, w) : G A a cocycle action. How does nuclear dimension behave with respect to passing to the (twisted) crossed product? A A α,w G 4 / 23

6 Introduction Within the theory of C -algebras, the crossed product construction is long known as a nearly inexhaustible source of interesting examples. In particular, certain crossed products are often natural test cases for the Elliott classification program. Motivated by the increasing importance of nuclear dimension in the Elliott program, the following question constitutes the main theme of this talk: Question Let A be a C -algebra, G a countable group and (α, w) : G A a cocycle action. How does nuclear dimension behave with respect to passing to the (twisted) crossed product? A A α,w G Answering this question in full generality seems to be far out of reach at the moment. However, by inventing the concept of Rokhlin dimension, Hirshberg, Winter and Zacharias have paved the way towards very satisfactory partial answers. This notion was initially defined for actions of finite groups and integers, and was also adapted to actions of Z m. 4 / 23

7 Introduction Within the theory of C -algebras, the crossed product construction is long known as a nearly inexhaustible source of interesting examples. In particular, certain crossed products are often natural test cases for the Elliott classification program. Motivated by the increasing importance of nuclear dimension in the Elliott program, the following question constitutes the main theme of this talk: Question Let A be a C -algebra, G a countable group and (α, w) : G A a cocycle action. How does nuclear dimension behave with respect to passing to the (twisted) crossed product? A A α,w G Answering this question in full generality seems to be far out of reach at the moment. However, by inventing the concept of Rokhlin dimension, Hirshberg, Winter and Zacharias have paved the way towards very satisfactory partial answers. This notion was initially defined for actions of finite groups and integers, and was also adapted to actions of Z m. We will discuss a generalization to cocycle actions of residually finite groups: 4 / 23

8 Rokhlin dimension 1 Introduction 2 Rokhlin dimension 3 Box spaces 4 Topological actions 5 / 23

9 Rokhlin dimension Definition Let A be a separable C -algebra and G a countable, discrete group and H G a subgroup with finite index. Let (α, w) : G A be a cocycle action. Let d N be a natural number. 6 / 23

10 Rokhlin dimension Definition Let A be a separable C -algebra and G a countable, discrete group and H G a subgroup with finite index. Let (α, w) : G A be a cocycle action. Let d N be a natural number. Then α has Rokhlin dimension d with respect to H, written dim Rok (α, H) = d, if 6 / 23

11 Rokhlin dimension Definition Let A be a separable C -algebra and G a countable, discrete group and H G a subgroup with finite index. Let (α, w) : G A be a cocycle action. Let d N be a natural number. Then α has Rokhlin dimension d with respect to H, written dim Rok (α, H) = d, if d is the smallest number such that there exist equivariant c.p.c. order zero maps ϕ l : (C(G/H), G-shift) (F (A), α ) (l = 0,..., d) with ϕ 0 (1) + + ϕ d (1) = 1. If no number satisfies this condition, we set dim Rok (α, H) :=. 6 / 23

12 Rokhlin dimension Definition Let A be a separable C -algebra and G a countable, discrete and residually finite group. Let σ = (G n ) n be a residually finite approximation of G, i.e. a decreasing and separating sequence of subgroups with finite index. We define dim Rok (α, σ) = sup dim Rok (α, G n ) n N and dim Rok (α) = sup dim Rok (α, H). H fin G 7 / 23

13 Rokhlin dimension Definition Let A be a separable C -algebra and G a countable, discrete and residually finite group. Let σ = (G n ) n be a residually finite approximation of G, i.e. a decreasing and separating sequence of subgroups with finite index. We define dim Rok (α, σ) = sup dim Rok (α, G n ) n N and dim Rok (α) = sup dim Rok (α, H). H fin G Remark If G is finite or Z m, the second expression agrees with the previously known definition. 7 / 23

14 Rokhlin dimension As hinted before, the following permanence properties served as motivation for introducing Rokhlin dimension: 8 / 23

15 Rokhlin dimension As hinted before, the following permanence properties served as motivation for introducing Rokhlin dimension: Theorem (Hirshberg-Winter-Zacharias) If α : G A is a finite group action on a unital C -algebra, we have dim +1 nuc(a α G) dim +1 Rok (α) dim+1 nuc(a). 8 / 23

16 Rokhlin dimension As hinted before, the following permanence properties served as motivation for introducing Rokhlin dimension: Theorem (Hirshberg-Winter-Zacharias) If α : G A is a finite group action on a unital C -algebra, we have dim +1 nuc(a α G) dim +1 Rok (α) dim+1 nuc(a). Theorem (Hirshberg-Winter-Zacharias) If A is a unital C -algebra and α Aut(A), we have dim +1 nuc(a α Z) 2 dim +1 Rok (α) dim+1 nuc(a). 8 / 23

17 Rokhlin dimension As hinted before, the following permanence properties served as motivation for introducing Rokhlin dimension: Theorem (Hirshberg-Winter-Zacharias) If α : G A is a finite group action on a unital C -algebra, we have dim +1 nuc(a α G) dim +1 Rok (α) dim+1 nuc(a). Theorem (Hirshberg-Winter-Zacharias) If A is a unital C -algebra and α Aut(A), we have Theorem (S) dim +1 nuc(a α Z) 2 dim +1 Rok (α) dim+1 nuc(a). If α : Z m A is an action on a unital C -algebra, we have dim +1 nuc(a α G) 2 m dim +1 Rok (α) dim+1 nuc(a). 8 / 23

18 Rokhlin dimension Here comes the main result of this talk. The following unifies (and in fact improves some of) the previous estimates: 9 / 23

19 Rokhlin dimension Here comes the main result of this talk. The following unifies (and in fact improves some of) the previous estimates: Theorem (S-Wu-Zacharias) Let G be a countable, discrete, residually finite group. Let σ be a residually finite approximation of G. Let A be any C -algebra and (α, w) : G A a cocycle action. Then we have dim +1 nuc(a α,w G) asdim +1 ( σ G) dim +1 Rok (α, σ) dim+1 nuc(a). 9 / 23

20 Rokhlin dimension Here comes the main result of this talk. The following unifies (and in fact improves some of) the previous estimates: Theorem (S-Wu-Zacharias) Let G be a countable, discrete, residually finite group. Let σ be a residually finite approximation of G. Let A be any C -algebra and (α, w) : G A a cocycle action. Then we have dim +1 nuc(a α,w G) asdim +1 ( σ G) dim +1 Rok (α, σ) dim+1 nuc(a). The above constant denotes the asymptotic dimension of the so-called box space of G associated to σ. We shall elaborate in the next part. 9 / 23

21 Rokhlin dimension Here comes the main result of this talk. The following unifies (and in fact improves some of) the previous estimates: Theorem (S-Wu-Zacharias) Let G be a countable, discrete, residually finite group. Let σ be a residually finite approximation of G. Let A be any C -algebra and (α, w) : G A a cocycle action. Then we have dim +1 nuc(a α,w G) asdim +1 ( σ G) dim +1 Rok (α, σ) dim+1 nuc(a). The above constant denotes the asymptotic dimension of the so-called box space of G associated to σ. We shall elaborate in the next part. Remark One particular instance of the above inequality is dim +1 nuc(a α,w G) asdim +1 ( s G) dim +1 Rok (α) dim+1 nuc(a), where s G is a so-called standard box space of G. 9 / 23

22 Box spaces 1 Introduction 2 Rokhlin dimension 3 Box spaces 4 Topological actions 10 / 23

23 Box spaces Definition (Roe, Khukhro) Let G be a countable, discrete and residually finite group. Let σ = (G n ) n be a residually finite approximation of G. Equip G with a right-invariant, proper metric d. (e.g. right-invariant word-length metric.) 11 / 23

24 Box spaces Definition (Roe, Khukhro) Let G be a countable, discrete and residually finite group. Let σ = (G n ) n be a residually finite approximation of G. Equip G with a right-invariant, proper metric d. (e.g. right-invariant word-length metric.) The box space σ G of G associated to σ is the disjoint union n N G/G n, endowed with a metric d B such that this metric, when restricted to some G/G n, is induced by d under the quotient map G G/G n, and such that dist db (G/G n, G/G m ) max {diam db (G/G n ), diam db (G/G m )} for all n, m N. 11 / 23

25 Box spaces Remark The previous definition implicitely contains a theorem stating that such a metric space σ G exists, and is independant (up to coarse equivalence) by the choices of either d or d B. 12 / 23

26 Box spaces Remark The previous definition implicitely contains a theorem stating that such a metric space σ G exists, and is independant (up to coarse equivalence) by the choices of either d or d B. Definition Let σ = (G n ) n be a residually finite approximation of G. We call σ dominating, if for all subgroups H G with finite index, there is some n such that G n H. 12 / 23

27 Box spaces Remark The previous definition implicitely contains a theorem stating that such a metric space σ G exists, and is independant (up to coarse equivalence) by the choices of either d or d B. Definition Let σ = (G n ) n be a residually finite approximation of G. We call σ dominating, if for all subgroups H G with finite index, there is some n such that G n H. Definition If σ is dominating, we will call s G = σ G a standard box space. 12 / 23

28 Box spaces It is known that the coarse structure of σ G encodes important features of G. We would like to pick out one particular instance of this: 13 / 23

29 Box spaces It is known that the coarse structure of σ G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) σ G has property A if and only if G is amenable. 13 / 23

30 Box spaces It is known that the coarse structure of σ G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) σ G has property A if and only if G is amenable. Theorem (Higson-Roe) Finite asymptotic dimension implies property A. 13 / 23

31 Box spaces It is known that the coarse structure of σ G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) σ G has property A if and only if G is amenable. Theorem (Higson-Roe) Finite asymptotic dimension implies property A. Theorem (S-Wu-Zacharias) Regardless of any choices, one has asdim( s G) = min σ asdim( σ G). 13 / 23

32 Box spaces It is known that the coarse structure of σ G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) σ G has property A if and only if G is amenable. Theorem (Higson-Roe) Finite asymptotic dimension implies property A. Theorem (S-Wu-Zacharias) Regardless of any choices, one has asdim( s G) = min σ asdim( σ G). So for what kind of groups do we have asdim( s G) <? 13 / 23

33 Box spaces Example The box space of a finite group is always coarsely equivalent to a one-point space, hence it has asymptotic dimension 0. asdim( σ Z m ) = m for any σ. 14 / 23

34 Box spaces Example The box space of a finite group is always coarsely equivalent to a one-point space, hence it has asymptotic dimension 0. asdim( σ Z m ) = m for any σ. Theorem (S-Wu-Zacharias) Finitely generated, virtually nilpotent groups G satisfy asdim( s G) <. 14 / 23

35 Box spaces Example The box space of a finite group is always coarsely equivalent to a one-point space, hence it has asymptotic dimension 0. asdim( σ Z m ) = m for any σ. Theorem (S-Wu-Zacharias) Finitely generated, virtually nilpotent groups G satisfy asdim( s G) <. Remark This result has recently been slightly improved by Wu, in that asdim +1 ( σ G) 3 l Hir(G) for every residually finite approximation σ of G. 14 / 23

36 Box spaces When a box space σ G of a residually finite group has finite asymptotic dimension, one might be tempted to think that this value encodes the complexity of the group G in some sense with respect to σ. This turns out to be true, in that the value simultaniously keeps track of both large-scale geometry and periodic behavior. 15 / 23

37 Box spaces When a box space σ G of a residually finite group has finite asymptotic dimension, one might be tempted to think that this value encodes the complexity of the group G in some sense with respect to σ. This turns out to be true, in that the value simultaniously keeps track of both large-scale geometry and periodic behavior. The usefulness for our main result comes from the fact that this gives rise to decay functions with similar properties as found for Z. 15 / 23

38 Box spaces When a box space σ G of a residually finite group has finite asymptotic dimension, one might be tempted to think that this value encodes the complexity of the group G in some sense with respect to σ. This turns out to be true, in that the value simultaniously keeps track of both large-scale geometry and periodic behavior. The usefulness for our main result comes from the fact that this gives rise to decay functions with similar properties as found for Z. Lemma (S-Wu-Zacharias) Let G be a residually finite group and σ = (G n ) n a residually finite approximation. Then for every s N, the statement asdim( σ G) s is equivalent to the following condition: 15 / 23

39 Box spaces When a box space σ G of a residually finite group has finite asymptotic dimension, one might be tempted to think that this value encodes the complexity of the group G in some sense with respect to σ. This turns out to be true, in that the value simultaniously keeps track of both large-scale geometry and periodic behavior. The usefulness for our main result comes from the fact that this gives rise to decay functions with similar properties as found for Z. Lemma (S-Wu-Zacharias) Let G be a residually finite group and σ = (G n ) n a residually finite approximation. Then for every s N, the statement asdim( σ G) s is equivalent to the following condition: For every ε > 0 and M G, there exists n and functions µ (l) : G [0, 1] for l = 0,..., s such that supp(µ (l) ) supp(µ (l) )h = for all l and h G n \ {1}. s l=0 h G n µ (l) (gh) = 1 for all g G. Each µ (l) is (M, ε)-flat with respect to left-translation, i.e. µ (l) µ (l) (g ) ε for all l and g M. 15 / 23

40 Box spaces Theorem (S-Wu-Zacharias) For all (α, w) : G A and σ, we have dim +1 nuc(a α,w G) asdim +1 ( σ G) dim +1 }{{} Rok (α, σ) dim +1 }{{} nuc(a). }{{} s+1 d+1 r+1 16 / 23

41 Box spaces Theorem (S-Wu-Zacharias) For all (α, w) : G A and σ, we have dim +1 nuc(a α,w G) asdim +1 ( σ G) dim +1 }{{} Rok (α, σ) dim +1 }{{} nuc(a). }{{} s+1 d+1 r+1 Proof (rough sketch of main theorem for actions) Let F A α,w G and ε > 0. We may pretend that F consists of au g for certain a A and g M G. Choose n N and decay functions µ (0),..., µ (s) according to the Lemma. Define G (l) = supp(µ (l) ). Choose Rokhlin towers ϕ 0,..., ϕ d : (C(G/G n ), G-shift) (F (A), α ). 16 / 23

42 Box spaces Theorem (S-Wu-Zacharias) For all (α, w) : G A and σ, we have dim +1 nuc(a α,w G) asdim +1 ( σ G) dim +1 }{{} Rok (α, σ) dim +1 }{{} nuc(a). }{{} s+1 d+1 r+1 Proof (rough sketch of main theorem for actions) Let F A α,w G and ε > 0. We may pretend that F consists of au g for certain a A and g M G. Choose n N and decay functions µ (0),..., µ (s) according to the Lemma. Define G (l) = supp(µ (l) ). Choose Rokhlin towers ϕ 0,..., ϕ d : (C(G/G n ), G-shift) (F (A), α ). Consider A α G (A α G) s l=0 θ l M G (0)(A) M G (s)(a) s d l=0 j=0 σ l,j 16 / 23

43 Box spaces Proof (rough sketch continued) Consider A α G (A α G) s l=0 θ l M G (0)(A) M G (s)(a) s d l=0 j=0 σ l,j 17 / 23

44 Box spaces Proof (rough sketch continued) Consider A α G (A α G) s l=0 θ l M G (0)(A) M G (s)(a) s d l=0 j=0 σ l,j Here, the map θ l : A α G M G (l)(a) is the c.p.c. cutdown with decay factor µ (l). 17 / 23

45 Box spaces Proof (rough sketch continued) Consider A α G (A α G) s l=0 θ l M G (0)(A) M G (s)(a) s d l=0 j=0 σ l,j Here, the map θ l : A α G M G (l)(a) is the c.p.c. cutdown with decay factor µ (l). Each σ l,j : M G (l)(a) (A α G) is given by σ l,j (e g,h a) = u g ϕ j (χ { 1 G })au h. 17 / 23

46 Box spaces Proof (rough sketch continued) Consider A α G (A α G) s l=0 θ l M G (0)(A) M G (s)(a) s d l=0 j=0 σ l,j Here, the map θ l : A α G M G (l)(a) is the c.p.c. cutdown with decay factor µ (l). Each σ l,j : M G (l)(a) (A α G) is given by By lengthy computation, check that Each σ l,j is order zero. σ l,j (e g,h a) = u g ϕ j (χ { 1 G })au h. If ε > 0 is chosen to be sufficiently small at the beginning, then the above diagram is an arbitrarily good c.p. approximation of F. 17 / 23

47 Box spaces Proof (rough sketch continued) Consider A α G (A α G) s l=0 θ l M G (0)(A) M G (s)(a) s d l=0 j=0 σ l,j Here, the map θ l : A α G M G (l)(a) is the c.p.c. cutdown with decay factor µ (l). Each σ l,j : M G (l)(a) (A α G) is given by By lengthy computation, check that Each σ l,j is order zero. σ l,j (e g,h a) = u g ϕ j (χ { 1 G })au h. If ε > 0 is chosen to be sufficiently small at the beginning, then the above diagram is an arbitrarily good c.p. approximation of F. We thus obtain dim +1 nuc(a α G) (s + 1)(d + 1)(r + 1). 17 / 23

48 Topological actions 1 Introduction 2 Rokhlin dimension 3 Box spaces 4 Topological actions 18 / 23

49 Topological actions Let G be a countable, discrete group and d N. Let d G l 1 (G) be the set of all probability measures of G supported on at most d + 1 points. Let G = d N dg be the set of all finitely supported probability measures. 19 / 23

50 Topological actions Let G be a countable, discrete group and d N. Let d G l 1 (G) be the set of all probability measures of G supported on at most d + 1 points. Let G = d N dg be the set of all finitely supported probability measures. Note that there is a canonical G-action β on each of these spaces defined by β g (µ)(a) = µ(g 1 A) for all g G. 19 / 23

51 Topological actions Let G be a countable, discrete group and d N. Let d G l 1 (G) be the set of all probability measures of G supported on at most d + 1 points. Let G = d N dg be the set of all finitely supported probability measures. Note that there is a canonical G-action β on each of these spaces defined by β g (µ)(a) = µ(g 1 A) for all g G. Definition (one of many equivalent ones) Let α : G X be an action on a compact metric space. Then α is amenable if there exist approximately equivariant maps (X, α) ( G, β). 19 / 23

52 Topological actions Let G be a countable, discrete group and d N. Let d G l 1 (G) be the set of all probability measures of G supported on at most d + 1 points. Let G = d N dg be the set of all finitely supported probability measures. Note that there is a canonical G-action β on each of these spaces defined by β g (µ)(a) = µ(g 1 A) for all g G. Definition (one of many equivalent ones) Let α : G X be an action on a compact metric space. Then α is amenable if there exist approximately equivariant maps (X, α) ( G, β). That is, there exists a net of continuous maps f λ : X G such that f λ (α g (x)) β g (f λ (x)) 1 0 as λ for all x X and g G. 19 / 23

53 Topological actions Definition (Guentner-Willett-Yu) Let α : G X be an action on a compact metric space and d N. α is said to have amenability dimension at most d, written dim am (α) d, if there exist approximately equivariant maps (X, α) ( d G, β). 20 / 23

54 Topological actions Definition (Guentner-Willett-Yu) Let α : G X be an action on a compact metric space and d N. α is said to have amenability dimension at most d, written dim am (α) d, if there exist approximately equivariant maps (X, α) ( d G, β). The amenability dimension dim am (α) is defined to be the smallest such d, if it exists. Otherwise dim am (α) :=. 20 / 23

55 Topological actions Definition (Guentner-Willett-Yu) Let α : G X be an action on a compact metric space and d N. α is said to have amenability dimension at most d, written dim am (α) d, if there exist approximately equivariant maps (X, α) ( d G, β). The amenability dimension dim am (α) is defined to be the smallest such d, if it exists. Otherwise dim am (α) :=. Theorem (Guentner-Willett-Yu) For a free action α : G X, one has the estimate dim +1 nuc(c(x) α G) dim +1 am(α) dim +1 (X). 20 / 23

56 Topological actions For a topological dynamical system (X, α, G), denote by ᾱ : G C(X) the induced C -algebraic action. 21 / 23

57 Topological actions For a topological dynamical system (X, α, G), denote by ᾱ : G C(X) the induced C -algebraic action. Question Is there a connection between dim am (α) and dim Rok (ᾱ)? 21 / 23

58 Topological actions For a topological dynamical system (X, α, G), denote by ᾱ : G C(X) the induced C -algebraic action. Question Is there a connection between dim am (α) and dim Rok (ᾱ)? This can be answered as follows: Theorem (S-Wu-Zacharias) Let σ be a residually finite approximation of G. If α : G X is free, one has the following estimates: dim +1 Rok (ᾱ) dim+1 am(α) asdim +1 ( σ G) dim +1 Rok (ᾱ, σ). 21 / 23

59 Topological actions For a topological dynamical system (X, α, G), denote by ᾱ : G C(X) the induced C -algebraic action. Question Is there a connection between dim am (α) and dim Rok (ᾱ)? This can be answered as follows: Theorem (S-Wu-Zacharias) Let σ be a residually finite approximation of G. If α : G X is free, one has the following estimates: dim +1 Rok (ᾱ) dim+1 am(α) asdim +1 ( σ G) dim +1 Rok (ᾱ, σ). In particular, if asdim( s G) <, then α has finite amenability dimension if and only if ᾱ has finite Rokhlin dimension. 21 / 23

60 Topological actions Last year, the following result was obtained: Theorem (S) If α : Z m X is a free action on a compact metric space of finite covering dimension, then dim Rok (ᾱ) <. Hence dim nuc (C(X) α Z m ) <. 22 / 23

61 Topological actions Last year, the following result was obtained: Theorem (S) If α : Z m X is a free action on a compact metric space of finite covering dimension, then dim Rok (ᾱ) <. Hence dim nuc (C(X) α Z m ) <. Combining methods developed for the above result (e.g. marker property) detailed study of amenability dimension some geometric group theory involving nilpotent groups this extends to the following setting: 22 / 23

62 Topological actions Last year, the following result was obtained: Theorem (S) If α : Z m X is a free action on a compact metric space of finite covering dimension, then dim Rok (ᾱ) <. Hence dim nuc (C(X) α Z m ) <. Combining methods developed for the above result (e.g. marker property) detailed study of amenability dimension some geometric group theory involving nilpotent groups this extends to the following setting: Theorem (S-Wu-Zacharias) Let G be a finitely generated, nilpotent group. If α : G X is a free action on a compact metric space of finite covering dimension, then both dim am (α) and dim Rok (ᾱ) are finite. In particular, dim nuc (C(X) α G) has finite nuclear dimension. (In fact at most 3 l Hir(G) dim +1 (X) 2 1.) 22 / 23

63 Thank you for your attention! 23 / 23