Rokhlin dimension for actions of residually finite groups

Transcription

1 Rokhlin dimension for actions of residually finite groups Workshop on C*-algebras and dynamical systems Fields Institute, Toronto Gábor Szabó (joint work with Jianchao Wu and Joachim Zacharias) WWU Münster June / 19

2 1 Introduction 2 Rokhlin dimension 3 The box space 4 Topological actions 2 / 19

3 Introduction 1 Introduction 2 Rokhlin dimension 3 The box space 4 Topological actions 3 / 19

4 Introduction The structure theory of simple nuclear C -algebras is currently undergoing revolutionary progress, driven by the discovery of regularity properties of various flavours. The regularity property of having finite nuclear dimension is the one we are going to focus on: 4 / 19

5 Introduction The structure theory of simple nuclear C -algebras is currently undergoing revolutionary progress, driven by the discovery of regularity properties of various flavours. The regularity property of having finite nuclear dimension is the one we are going to focus on: Definition (Winter-Zacharias) Let A be a C*-algebra. We say that A has nuclear dimension r, and write dim nuc (A) = r, if r is the smallest natural number with the following property: 4 / 19

6 Introduction The structure theory of simple nuclear C -algebras is currently undergoing revolutionary progress, driven by the discovery of regularity properties of various flavours. The regularity property of having finite nuclear dimension is the one we are going to focus on: Definition (Winter-Zacharias) Let A be a C*-algebra. We say that A has nuclear dimension r, and write dim nuc (A) = r, if r is the smallest natural number with the following property: For all F A and ε > 0, there exists a finite dimensional C -algebra F and a c.p.c. map ψ : A F and c.p.c. order zero maps ϕ (0),..., ϕ (r) : F A such that a r ϕ (l) ψ(a) ε for all a F. l=0 4 / 19

7 Introduction Question How does nuclear dimension behave with respect to passing to (twisted) crossed products? A A α G 5 / 19

8 Introduction Question How does nuclear dimension behave with respect to passing to (twisted) crossed products? A A α 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. 5 / 19

9 Introduction Question How does nuclear dimension behave with respect to passing to (twisted) crossed products? A A α 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. 5 / 19

10 Introduction Question How does nuclear dimension behave with respect to passing to (twisted) crossed products? A A α 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: 5 / 19

11 Rokhlin dimension 1 Introduction 2 Rokhlin dimension 3 The box space 4 Topological actions 6 / 19

12 Rokhlin dimension From now on, all C -algebras are assumed to be separable and unital for convencience. For what follows, neither condition is actually necessary. 7 / 19

13 Rokhlin dimension From now on, all C -algebras are assumed to be separable and unital for convencience. For what follows, neither condition is actually necessary. Definition Let A be a separable, unital C -algebra and G a countable, discrete and residually finite group. Let (α, w) : G A be a cocycle action. Let d N be a natural number. 7 / 19

14 Rokhlin dimension From now on, all C -algebras are assumed to be separable and unital for convencience. For what follows, neither condition is actually necessary. Definition Let A be a separable, unital C -algebra and G a countable, discrete and residually finite group. Let (α, w) : G A be a cocycle action. Let d N be a natural number. Then α has Rokhlin dimension d, written dim Rok (α) = d, if d is the smallest number with the following property: 7 / 19

15 Rokhlin dimension From now on, all C -algebras are assumed to be separable and unital for convencience. For what follows, neither condition is actually necessary. Definition Let A be a separable, unital C -algebra and G a countable, discrete and residually finite group. Let (α, w) : G A be a cocycle action. Let d N be a natural number. Then α has Rokhlin dimension d, written dim Rok (α) = d, if d is the smallest number with the following property: For every subgroup H G with finite index, there exist equivariant c.p.c. order zero maps ϕ l : (C(G/H), G-shift) (A A, α ) (l = 0,..., d) with ϕ 0 (1) + + ϕ d (1) = 1. If no number satisfies this condition, we set dim Rok (α) :=. Remark If G is finite or Z m, this agrees with the previous definition. 7 / 19

16 Rokhlin dimension As mentioned at the beginning, the following permanence properties served as motivation for introducing Rokhlin dimension: 8 / 19

17 Rokhlin dimension As mentioned at the beginning, 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 / 19

18 Rokhlin dimension As mentioned at the beginning, 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 / 19

19 Rokhlin dimension As mentioned at the beginning, 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 / 19

20 Rokhlin dimension Here comes the main result of this talk. The following unifies (and in fact improves some of) the previous estimates: 9 / 19

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 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 ( G) dim +1 Rok (α) dim+1 nuc(a). 9 / 19

22 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 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 ( G) dim +1 Rok (α) dim+1 nuc(a). The above constant denotes the asymptotic dimension of the box space of G. We shall elaborate: 9 / 19

23 The box space 1 Introduction 2 Rokhlin dimension 3 The box space 4 Topological actions 10 / 19

24 The box space Definition (Roe) Let G be a countable, residually finite group. The box space G is the disjoint union of all finite quotients groups of G, equipped with its minimal connected G-invariant coarse structure for the left action of G by translation. 11 / 19

25 The box space Definition (Roe) Let G be a countable, residually finite group. The box space G is the disjoint union of all finite quotients groups of G, equipped with its minimal connected G-invariant coarse structure for the left action of G by translation. Remark (Roe-Khukhro) Take a decreasing sequence of normal subgroups G n G with finite index, such that every finite index subgroup H G contains G n for some n. Let S G be a finite generating set, and equip G with the associated right-invariant word-length metric. 11 / 19

26 The box space Definition (Roe) Let G be a countable, residually finite group. The box space G is the disjoint union of all finite quotients groups of G, equipped with its minimal connected G-invariant coarse structure for the left action of G by translation. Remark (Roe-Khukhro) Take a decreasing sequence of normal subgroups G n G with finite index, such that every finite index subgroup H G contains G n for some n. Let S G be a finite generating set, and equip G with the associated right-invariant word-length metric. Then the box space G can be realized (up to coarse equivalence) as the disjoint union n N G/G n, endowed with a metric dist such that this metric, when restricted to some G/G n, is induced by the image of S in G/G n, and such that for all n, m N. dist(g/g n, G/G m ) max {diam(g/g n ), diam(g/g m )} 11 / 19

27 The box space It is known that the coarse structure of G encodes important features of G. We would like to pick out one particular instance of this: 12 / 19

28 The box space It is known that the coarse structure of G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) G has property A if and only if G is amenable. 12 / 19

29 The box space It is known that the coarse structure of G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) G has property A if and only if G is amenable. Theorem (Higson-Roe) Finite asymptotic dimension implies property A. 12 / 19

30 The box space It is known that the coarse structure of G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) G has property A if and only if G is amenable. Theorem (Higson-Roe) Finite asymptotic dimension implies property A. So for what kind of groups do we have asdim( G) <? 12 / 19

31 The box space It is known that the coarse structure of G encodes important features of G. We would like to pick out one particular instance of this: Theorem (Guentner) G has property A if and only if G is amenable. Theorem (Higson-Roe) Finite asymptotic dimension implies property A. So for what kind of groups do we have asdim( G) <? Example The box space of a finite group is a one-point set, hence it has asymptotic dimension 0. asdim( Z m ) = m. Probably all finitely generated, virtually nilpotent groups G satisfy asdim( G) <. (Details still need to be checked!) 12 / 19

32 The box space When the box space of a residually finite group has finite asymptotic dimension, one might be tempted to think that this value encodes the geometric complexity of the group in some sense. 13 / 19

33 The box space When the box space of a residually finite group has finite asymptotic dimension, one might be tempted to think that this value encodes the geometric complexity of the group in some sense. This turns out to be true, and that is what makes the proof of the main result possible. Unfortunately, there is not enough time to get into details. Instead, we would like to look at the case of topological actions. 13 / 19

34 Topological actions 1 Introduction 2 Rokhlin dimension 3 The box space 4 Topological actions 14 / 19

35 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 of G. 15 / 19

36 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 of G. 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. 15 / 19

37 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 of G. 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 versions) 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). 15 / 19

38 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 of G. 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 versions) 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. 15 / 19

39 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 almost equivariant maps (X, α) ( d G, β). 16 / 19

40 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 almost 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) :=. 16 / 19

41 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 almost 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) :=. They mainly use this to describe some sufficient criterions of the Baum-Connes conjecture or the Farrell-Jones conjecture for a group. 16 / 19

42 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 almost 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) :=. They mainly use this to describe some sufficient criterions of the Baum-Connes conjecture or the Farrell-Jones conjecture for a group. But they also prove the following 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). 16 / 19

43 Topological actions Let X be a compact metric space, G a countable, residually finite group, and α : G X an action. Let ᾱ : G C(X) denote the induced C -algebraic action. 17 / 19

44 Topological actions Let X be a compact metric space, G a countable, residually finite group, and α : G X an action. Let ᾱ : G C(X) denote the induced C -algebraic action. Question Is there a connection between dim am (α) and dim Rok (ᾱ)? 17 / 19

45 Topological actions Let X be a compact metric space, G a countable, residually finite group, and α : G X an action. Let ᾱ : G C(X) denote 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) If α : G X is free, one has the following estimates: dim +1 Rok (ᾱ) dim+1 am(α) asdim +1 ( G) dim +1 Rok (ᾱ). 17 / 19

46 Topological actions Let X be a compact metric space, G a countable, residually finite group, and α : G X an action. Let ᾱ : G C(X) denote 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) If α : G X is free, one has the following estimates: dim +1 Rok (ᾱ) dim+1 am(α) asdim +1 ( G) dim +1 Rok (ᾱ). In particular, if asdim( G) <, then α has finite amenability dimension if and only if ᾱ has finite Rokhlin dimension. 17 / 19

47 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) <. In particular, dim nuc (C(X) α Z m ) <. 18 / 19

48 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) <. In particular, dim nuc (C(X) α Z m ) <. Using the methods that were developed for this (e.g. marker property), and also using some additional ingredients, 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) <. In particular, dim nuc (C(X) α G) <. 18 / 19

49 Thank you for your attention! 19 / 19