Rokhlin dimension for actions of residually finite groups

# Rokhlin dimension for actions of residually finite groups

Gábor Szabó, Jianchao Wu and Joachim Zacharias Fraser Noble Building, Institute of Mathematics, University of Aberdeen,     Aberdeen AB24 3UE, Scotland, UK Department of Mathematics, Pennsylvania State University, -    109 McAllister Building, University Park, PA 16802, USA School of Mathematics and Statistics, University of Glasgow, -    University Gardens, Glasgow Q12 8QW, Scotland, UK
###### Abstract.

We introduce the concept of Rokhlin dimension for actions of residually finite groups on -algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not been considered previously. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product -algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group -algebras have finite nuclear dimension. This extends an analogous result about -actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing -algebra.

###### 2010 Mathematics Subject Classification:
Primary 46L55, 54H20; Secondary 46L35, 20F69, 20F18
Supported by: SFB 878 Groups, Geometry and Actions, GIF Grant 1137-30.6/2011, ERC Advanced Grant ToDyRiC 267079, EPSRC Grant EP/I019227/2 and EPSRC Grant EP/N00874X/1

## Introduction

Since its very beginning, the theory of operator algebras has been strongly influenced and even motivated by ideas of dynamical nature. The study of -dynamical systems, i.e. group actions of locally compact groups on -algebras, is interesting in many ways. An ample reason for this is certainly the crossed product construction, which naturally associates a new -algebra to a -dynamical system, reflecting the structure of the group, the coefficient algebra, and the action. This construction has, by now, proved to be a virtually inexhaustible source of interesting examples of -algebras. As such, it is not surprising that the crossed product construction is harder to understand than most other standard constructions to create new -algebras. Moreover, crossed products are often simple -algebras of importance in the Elliott classification programme. In view of the astonishing progess of -algebra classification over the last few years [GongLinNiu14, ElliottNiu15, ElliottGongLinNiu15, TikuisisWhiteWinter15], which is largely driven by the discovery of various regularity properties (see for example [Winter14Lin, Winter10, Winter12, RobertTikuisis14, MatuiSato12, KirchbergRordam12, TomsWhiteWinter12, Sato13, MatuiSato14UHF, Winter14, SatoWhiteWinter14, BBSTWW]), a natural and important question appears, which can be regarded as the main motivation of this paper:

###### Question.

Given a -dynamical system , under what conditions on does regularity pass from to ?

Of course, the vague term ‘regularity’ leaves some room for interpretation, but this question is interesting for all possible versions of regularity. Within this paper, we will mainly focus on the regularity property of having finite nuclear dimension, which is an approximation property stronger than nuclearity. We will also have a secondary focus on -stability in section 9, for a chosen strongly self-absorbing -algebra .

General results on nuclear dimension of crossed products are hard to come by. What seems to be required is a certain regularity property of the action. As evidenced by recent developments, Rokhlin-type properties appear to be very good candidates for such a regularity property. The idea of defining a Rokhlin property stems from a basic early result in the dynamics of group actions on measure spaces, the so-called Rokhlin Lemma. This result states that a measure-preserving, aperiodic integer action can be approximated by cyclic shifts in a suitable sense. A von Neumann theoretic translation of this approximation was used successfully by Connes as a key technique towards classifying automorphisms on the hyperfinite II-factor up to outer conjugacy, see [Connes75, Connes77]. Since then, lots of generalizations and reinterpretations have emerged within the realm of von Neumann algebras. Preliminary variants of the Rokhlin property emerged in works of Herman-Jones [HermanJones82] and Herman-Ocneanu [HermanOcneanu84] for cyclic group actions on UHF algebras. Eventually, Kishimoto was successful in fleshing out the Rokhlin property for integer actions on -algebras, see for instance [Kishimoto95, Kishimoto96, Kishimoto96_R, Kishimoto98, Kishimoto98II]. Then Izumi started his pioneering work on finite group actions with the Rokhlin property, see [Izumi04, Izumi04II]. For a selection of other papers related to Rokhlin-type properties for -dynamical systems, see [Phillips11, Phillips12, OsakaPhillips12, Santiago14, Nakamura99, Sato10, MatuiSato12_2, MatuiSato14, Kishimoto02, BratteliKishimotoRobinson07, HirshbergWinter07, Gardella14_5]. (This list is not exhaustive)

As powerful as these earlier Rokhlin properties are, most variants share a common disadvantage. As their definition involves Rokhlin towers consisting of projections, they impose strong restrictions on the -algebra in question, ruling out important examples such as the Jiang-Su algebra or any other sufficiently projectionless -algebra. To circumvent this problem, Hirshberg, Winter and the third author introduced the concept of Rokhlin dimension for actions of finite groups and the integers in [HirshbergWinterZacharias14]. See also [Szabo14] for an extension to -actions. The concept of Rokhlin dimension yields a generalization of many of the commonly used Rokhlin-type properties, motivated by the idea of covering dimension. Instead of requiring one (multi-) tower consisting of projections to reflect a shift-type behaviour in the given dynamical system, one allows boundedly many of such towers consisting of positive elements, the bound defining the dimensional value. It turns out that this generalization provides a much more flexible concept, while still being strong enough to be compatible with the regularity property of having finite nuclear dimension. For a selection of other papers that, at least in large part, emerged out of the ideas within [HirshbergWinterZacharias14], see [BarlakEndersMatuiSzaboWinter, Szabo14, Gardella14_3, Gardella14_4, HirshbergPhillips15, Liao15, Liao16, HirshbergWu15, HirshbergSzaboWinterWu16, BrownTikuisisZelenberg16].

The main results in [HirshbergWinterZacharias14] are a bound on the nuclear dimension of crossed products by single automorphisms (i.e. -actions) of finite Rokhlin dimension, a bound for the Rokhlin dimension for minimal actions on finite dimensional compact metric spaces, as well as a permanence result for -stability and a genericity result for finiteness of Rokhlin dimension in the set of all actions on a given algebra. The first two results were generalized by the first author to -actions in [Szabo14], who also gave a better framework for establising a bound for the Rokhlin dimension on compact metric spaces requiring only freeness of the action. This used some crucial topological arguments inspired by [Lindenstrauss95, Gutman14], the result of which can be regarded as a topological version of the classical Rokhlin Lemma. Combined with the nuclear dimension bound, it shows that -algebras associated to such free and minimal dynamical systems have finite nuclear dimension and thus fall within the scope of Elliott’s classification programme. Such nuclear dimension results were first obtained by Toms and Winter in [TomsWinter13] by different methods.

The main purpose of this paper is to generalize the concept of Rokhlin dimension to actions of all countable, discrete and residually finite groups and to extend the results from [HirshbergWinterZacharias14] and [Szabo14] to this setting. Moreover, we enlarge the class of -dynamical systems under consideration to non-unital -algebras, and cocycle actions instead of ordinary actions.

###### Definition A.

Let be a separable -algebra, a countable, residually finite group and a cocycle action. The (full) Rokhlin dimension of is defined as the infimum of natural numbers such that for any finite-index subgroup of , there exist equivariant c.p.c. order zero maps

 φl:(C(G/H),G-shift)⟶(F∞(A),~α∞),for l=0,…,d

with .

Here the left-hand side involves the canonical action of on the algebra of functions over the finite quotient , while on the right-hand side, is Kirchberg’s central sequence algebra and is the action on naturally induced from (see Definition 4.8). The definition can be refined by considering only a sufficiently separating sequence of finite-index subgroups, and the separability of is not an essential prerequisite for the definition (see Definition 4.4). In fact, the whole setting can be generalized to include also uncountable groups. We also give an equivalent characterization involving approximate towers of positive elements on the algebraic level, in the spirit of [HirshbergWinterZacharias14] (see Proposition 4.5).

The benefit of this generalization is threefold: it not only allows us to study a wider class of examples, but also unifies all the arguments made (so far separately) for finite groups and the integers, and provides a clearer conceptual picture for how the Rokhlin dimension relates to -algebraic regularity properties such as the nuclear dimension. The crucial insight is that the applications of the Rokhlin dimension require a complementary ingredient of coarse geometric nature, namely the box space (in the sense of Roe, see [RoeCG, 11.24]) associated to a residually finite group and some chosen sequence of finite-index subgroups . It turns out that a group yields an interesting Rokhlin dimension theory if it admits a box space with finite asymptotic dimension. This insight culminates in Theorem 5.2, where we establish the following upper bound for the nuclear dimension of (twisted) crossed products:

###### Theorem B.

For every -algebra and every cocycle action , we have the upper bound

 dim+1nuc(A⋊α,wG)≤asdim+1(□σG)⋅dim+1Rok(α,σ)⋅dim+1nuc(A).

In the above inequality, the dimensions appearing on the right-hand side are the asymptotic dimension of the box space for , the Rokhlin dimension of along , and the nuclear dimension of the coefficient algebra , respectively. For notational convenience, each dimension in the formula is increased by , as denoted by the superscripts. By introducing box spaces in this context, this bound is improved and gives a coarse geometric interpretation of the ad-hoc constants appearing in previous such estimates.

The asymptotic dimension and the box space construction have individually played significant roles in coarse geometry and have had remarkable applications to the Baum-Connes conjecture; however, to the best of the authors’ knowledge, this is the first time the effects of combining these two notions come under careful investigation. We conduct such an investigation from a general point of view in Section 2, even before we can generalize the Rokhlin dimension theory. It turns out that some care has to be taken in the choice of the finite-index subgroups featuring in the box space but that good choices always exist (see Definition 2.5 and Corollary 2.10). We also give a key characterization of the asymptotic dimension in terms of partitions of unity in Lemma 2.13 that is geared towards our main application in Theorem 5.2.

A central question in Sections 2 and 3 is what groups admit box spaces with finite asymptotic dimension, a condition that is vital for the applications of our Rokhlin dimension theory. This is certainly satisfied for finite groups and the integers, and was already (implicitly) crucial in the existing theory for these groups. On the other hand, it follows from classical results in coarse geometry that such a condition implies the amenability of the group. We devote the entire Section 3 to proving the following (see Theorem 3.10):

###### Theorem C.

Every finitely generated, virtually nilpotent group admits a box space with finite asymptotic dimension.

This provides a reasonably large class of groups to which Theorems B and F can be applied. Our proof is done by developing a criterion for a discrete subgroup of a locally compact group to have finite dimensional box spaces that involves the existence of a sequence of expanding automorphisms of the group. For nilpotent groups, we then apply this to known embeddings into certain Lie groups. For further examples of groups with finite dimensional box spaces, see a preprint by Finn-Sell and the second author [FinnSellWu14], which was inspired by and improves the results on box spaces in this paper. Among other things, it is shown there that for every virtually polycyclic group, there exists some box space whose asymptotic dimension is at most the Hirsch length of the group.

There are further interesting connections of our work to ongoing developments in dimension theory for topological dynamics. Recently, Guentner, Willett and Yu have invented various stronger versions of amenability of discrete group actions on compact metric spaces, see [GuentnerWillettYu14_1, GuentnerWillettYu15]. In particular, they introduced for such actions a notion that may be termed amenability dimension111Our choice of this terminology “amenability dimension” is based on an early draft of Guentner, Willett and Yu, where they used the term “-amenable” to denote what we call having amenability dimension at most . Since then, they have generalized the notion to what they call “-BLR”, which is closely related to the dynamic asymptotic dimension defined in the same paper (see [GuentnerWillettYu15, Remark 4.14]).. This is a dynamical analogue of asymptotic dimension, the finiteness of which may be regarded as a strong form of amenability of the action. The original motivation to introduce amenability dimension was to facilitate the computation of -theoretic invariants, and in particular to prove the Farrell-Jones conjecture or the Baum-Connes conjecture in certain cases. In fact, their work is in part inspired by the pioneering work of Bartels, Lück and Reich on the Farrell-Jones conjecture for hyperbolic groups in [BartelsLuckReich08]. Somewhat surprisingly, there is a theorem among Guentner-Willett-Yu’s results that showcases the compatibility of amenability dimension with nuclear dimension, in a similar fashion as for Rokhlin dimension. This raises the question to what extent amenability dimension and Rokhlin dimension are related concepts. After examining amenability dimension in rather -algebraic terms, we show that for groups with finite dimensional box spaces, the two dimensional invariants are of the same order of magnitude (see Theorem LABEL:dimrok_amdim):

###### Theorem D.

Let be a -algebra, a residually finite group admitting a box space with finite asymptotic dimension, and a cocycle action. Then the Rokhlin dimension of is finite if and only if its amenability dimension is finite.

As a major application of this result together with techniques developed in [Szabo14] and some geometric group theory, we extend the finiteness results for Rokhlin dimension to free actions of infinite, finitely generated, nilpotent groups on finite dimensional compact metric spaces, see Corollary LABEL:free_nilpotent_dimnuc. When combined with the astounding recent progress in the Elliott programme by many hands [GongLinNiu14, ElliottNiu15, ElliottGongLinNiu15, TikuisisWhiteWinter15], this shows that a large class of simple transformation group -algebras associated to actions of nilpotent groups is classifiable (see Theorem LABEL:crossed_product_classifiable):

###### Theorem E.

Let be a finitely generated, infinite, nilpotent group and a compact metric space of finite covering dimension. Let be a free and minimal action. Then the transformation group -algebra is a simple ASH algebra of topological dimension at most .

This considerably extends results which previously have only been known for and -actions. We point out that some time after the initial preprint version of this paper was published, Bartels [Bartels16] independently proved a similar finiteness result for more general actions of virtually nilpotent groups, by using slightly different methods. As a consequence, Theorem C can be extended from nilpotent groups to virtually nilpotent groups (cf. Theorem LABEL:thm:Bartels-extension).

In the case of noncommutative coefficient algebras, we also explore generalizations of some other results from [HirshbergWinterZacharias14]. Firstly, we investigate the notion of Rokhlin dimension with commuting towers and its relation to another type of important -algebraic regularity property — -stability for a strongly self-absorbing -algebra . We show that for groups with finite dimensional box spaces, finite Rokhlin dimension with commuting towers allows -stability to pass from the coefficient -algebra to the (twisted) crossed product (see Theorem LABEL:dimrokc_sigma_D):

###### Theorem F.

Let be a separable, -stable -algebra, let be a countable, residually finite group admitting a box space with finite asymptotic dimension, and let be a cocycle action having finite Rokhlin dimension with commuting towers. Then the twisted crossed product is -stable.

Secondly, we examine the genericity of cocycle actions with small Rokhlin dimension on -stable or UHF-stable -algebras (see Theorem LABEL:generic_dimrok_UHF and Theorem LABEL:generic_dimrok_Z):

###### Theorem G.

Let be a discrete, finitely generated and residually finite group and let be a separable -algebra. Then among all cocycle actions ,

1. those with Rokhlin dimension are generic if tensorially absorbs the universal UHF algebra;

2. those with Rokhlin dimension at most are generic if tensorially absorbs the Jiang-Su algebra.

The paper is organized as follows. In Section 1, we fix some notations and review the existing Rokhlin dimension theories for finite groups and . Section 2 develops the theory for the asymptotic dimension of box spaces from a general point of view, while Section 3 focuses on its finiteness for finitely generated virtually nilpotent groups. Section 4 discusses the definitions and basic properties of the Rokhlin dimension for residually finite groups. Section 5 elaborates its application to the nuclear dimension of crossed products. Sections LABEL:sec:dimrok-amdim and LABEL:sec:dimBLR study the amenability dimension and relate it to the Rokhlin dimension, while Section LABEL:sec:nilpotent-dimnuc applies our theory to free minimal actions of nilpotent groups on finite dimensional metric spaces. Section LABEL:sec:commuting-towers investigates Rokhlin dimenison with commuting towers and its application to -stability. Section LABEL:sec:genericity establishes the genericity of finite Rokhlin dimension for actions on -stable algebras.

Acknowledgements. The authors would like to express their gratitude to Rufus Willett and Guoliang Yu for some very helpful discussions. Moreover, the authors are grateful to Siegfried Echterhoff, Stuart White and Wilhelm Winter for pointing out a number of inaccuracies in a previous preprint version of this paper.

## 1. Preliminaries

###### Notation 1.1.

Unless specified otherwise, we will stick to the following notations throughout the paper.

• denotes a -algebra.

• denotes a group action on a -algebra or a compact metric space.

• denotes a countable, discrete group.

• If is some set and is a finite subset, we write .

• For and in some normed space, we write as a shortcut for .

• Assume that “” is one of the notions of dimension that appear in this paper, and an object on which this dimension can be evaluated. Then we sometimes use the convenient notation .

First we recall the existing notion of Rokhlin dimension for finite group actions and -actions on unital -algebras:

###### Definition 1.2 (cf. [HirshbergWinterZacharias14, 1.1]).

Let be a unital -algebra, a finite group and let be an action. We say that has Rokhlin dimension , and write , if is the smallest natural number with the following property:

For all , there exist positive contractions in satisfying the following properties:

1. ;

2. for all and in ;

3. for all and ;

4. for all and .

If there is no such , we write .

###### Notation.

When considering actions of in this section, we denote

 Bn={0,…,n−1}m⊂Zm.
###### Definition 1.3 (cf. [Szabo14, 1.6]).

Let be a unital -algebra, and an action. We say that has Rokhlin dimension , and write , if is the smallest natural number with the following property:

For all , there exist positive contractions in satisfying the following properties:

1. ;

2. for all and in ;

3. for all and ;
(Note that denotes , whenever .)

4. for all and .

If there is no such , we write .

Using central sequences, there is an elegant reformulation of these definitions. We will omit the easy proofs for the sake of brevity222However, a more general statement is proved in detail in 4.5.. Let be a free filter on . As usual, let denote the -sequence algebra for a -algebra . itself embeds into as (representatives of) constant sequences. The relative commutant of inside is called the central sequence algebra of and is denoted . If is a group action of a discrete group on , component-wise application of yields well-defined actions on both and , which we will both denote by . In what follows, we will mostly work with being the filter of all cofinite subsets in , and in this case we will insert the symbol in these notations.

###### Lemma 1.4.

Let be a separable, unital -algebra.

1. Let be an action of a finite group. Let be a natural number. Then if and only if there are equivariant c.p.c. order zero maps

 φl:(C(G),G-shift)⟶(A∞∩A′,α∞)(l=0,…,d)

such that .

2. Let be an action. Let be a natural number. Then if and only if for all , there are equivariant c.p.c. order zero maps

 −−−φl:(C(Zm/nZm),Zm-shift)⟶(A∞∩A′,α∞)(l=0,…,d)

such that .

###### Remark 1.5.

The above perspective 1.4 about the existing notions of finite Rokhlin dimension makes the similarities a lot more apparent. Regarding (2), it is not immediately clear why one would have to use the sequence of subgroups , instead of any other suitable separating sequence of finite-index subgroups. When we define Rokhlin dimension for residually finite groups in the upcoming sections, we will introduce a little more flexibility concerning this point.

We note that for integer actions, there have originally been several versions of Rokhlin dimension, see [HirshbergWinterZacharias14, 2.3]. The above definition for -actions [Szabo14, 1.6] extends the single tower version for -actions, see [HirshbergWinterZacharias14, 2.3c), 2.9].

As we will also treat cocycle actions in this paper, we remind the reader of some basic definitions:

###### Definition 1.6 (cf. [BusbySmith70, 2.1] or [PackerRaeburn89, 2.1]).

Let be a -algebra and a discrete group. A cocycle action is a map together with a map satisfying

 αr∘αs=Ad(w(r,s))∘αrs

and

 αr(w(s,t))⋅w(r,st)=w(r,s)⋅w(rs,t)

for all .

One may always assume and for all . If is trivial, then this just recovers the definition of an ordinary action .

###### Definition 1.7 (cf. [PackerRaeburn89]).

Let be a -algebra and a discrete group. Let be a cocycle action. Then one defines the maximal twisted crossed product to be the universal -algebra with the property that it contains a copy of , there is a map satisfying

 ugau∗g=αg(a)anduguh=w(g,h)⋅ugh

for all and .

The reduced twisted crossed product is defined as the -algebra inside generated by the following two representations:

Let be faithfully represented on a Hilbert space. Then consider

 ι:A→B(ℓ2(G)⊗H)viaι(a)(δt⊗ξ)=δt⊗αt−1(a)ξ

and

 λ:G→U(B(ℓ2(G)⊗H))byλs(δt⊗ξ)=δst⊗w(t−1s−1,s)ξ.

If is amenable, then the maximal and reduced twisted crossed products always coincide.

###### Remark 1.8.

In a more symbolic notation using -valued infinite matrices (that we will use later) one can also write

 aus=∑t∈Get,s−1t⊗αt−1(a)w(t−1,s)∈A⋊r,α,wG

for all .

## 2. Box spaces and asymptotic dimension

###### Definition 2.1.

Let be a countable, discrete group. Let be a decreasing sequence of subgroups with finite index, i.e.  and for all . We say that the sequence is separating if . is (by definition) residually finite if and only if such a sequence exists.

###### Remark 2.2.

A well-known fact in group theory is that for any finite-index subgroup , there is a normal finite-index subgroup such that . It follows that a group is residually finite if and only if it has a separating decreasing sequence of normal finite-index subgroups.

Before we come to the next definition, we need to recall some facts. If is a metric space and we have a discrete group acting on from the right, then the orbit space is defined as the quotient of by identifying any two points in the same orbit. Let be the corresponding quotient map. One can define a quotient pseudometric by

 π∗(d)(π(x1),π(x2))=inf{d(x1⋅g1,x2⋅g2)∣g1,g2∈G}.

In good cases, this will be a metric inducing the quotient topology, such as for geometric actions, i.e. isometric, properly discontinuous and cobounded actions, see [DrutuKapovich, 3.20].

Recall also the definition of a coarse disjoint union of finite metric spaces [NowakYuLSG, 1.4.12]. Moreover, recall that a metric on a discrete set is called proper, if balls are finite sets.

###### Definition 2.3 (following [RoeCG, 11.24] and [Khukhro12]).

Let be a countable, discrete, residually finite group and let be a decreasing sequence of subgroups of . Let us equip with a proper, right-invariant metric . For each , let be the quotient map and the quotient metric on . Then the box space associated to is defined as the coarse disjoint union of the sequence of finite metric spaces . More precisely, it is a coarse space whose underlying set is the disjoint union , and whose coarse structure is determined by a metric such that for all , the metric restricts to on the subset , and such that we have

 distdB(G/Gn,G/Gm)→∞

as and .

###### Remark 2.4.

Box spaces associated to a family of normal finite-index subgroups have seen a fair amount of study. It is known to coarse geometers that any countable, discrete group admits a proper right-invariant metric, and that the choice of the proper right-invariant metric on and the metric does not affect the coarse equivalence class of the resulting box space. This justifies the absence of and in the notation . For a rigorous and comprehensive proof of this, we refer the reader to [Szabo_Diss, Section 1.2, Proposition 1.3.3]. We remark that even though in the statement of [Szabo_Diss, Proposition 1.3.3], the sequence is assumed to be separating and made up of normal subgroups, the actual proof does not make use of these conditions.

Although for us, is not required to consist of normal subgroups, one needs a condition that rules out certain pathological examples where the coarse structure of the box space behaves badly with regard to that of itself.

###### Definition 2.5 (cf. [FinnSellWu14, 3.1]).

A decreasing sequence of finite-index subgroups of is called semi-conjugacy-separating if for any nontrivial element , there is such that , where is the conjugacy class of .

###### Notation 2.6.

For the sake of brevity, we shall call a semi-conjugacy-separating decreasing sequence of finite-index subgroups of a regular residually finite approximation of or simply a regular approximation .

###### Lemma 2.7.

Let be a group and a decreasing sequence of finite-index subgroups of . Then the following are equivalent:

1. is semi-conjugacy-separating.

2. .

3. For any finite subset , there is such that the quotient map is injective on for any .

###### Proof.

(1) (2): Condition (2) can be rewritten as: for any nontrivial element , there is such that for any , . But the statement at the end is equivalent to , which is in turn equivalent to . Thus the entire statement is equivalent to (1).

(2) (3): Given any finite subset and with , then , so by (2) we can find such that for all . With decreasing and finite, we can find which works for all pairs of different elements in simultanously, thus verifying (3).

(3) (2): With and as in condition (3), it follows that for all , which implies condition (2). ∎

###### Remark 2.8.

Although we are only interested in decreasing sequences, Definition 2.5 and Lemma 2.7 work equally well for any collection of finite-index subgroups of that is closed under forming intersections.

After we fix a proper, right-invariant metric on and replace by arbitrarily large balls around , the last condition in the above Lemma says that the family of maps achieves arbitrarily large injectivity radii. This suggests the importance of this condition regarding the coarse geometric information of the box space and the group. On the other hand, condition (2) directly implies the following.

###### Corollary 2.9.

Every separating family of finite-index normal subgroups is semi-conjugacy-separating.

Since a group is residually finite if and only it has a separating family of finite-index normal subgroups, we also have:

###### Corollary 2.10.

A group is residually finite if and only if it has a semi-conjugacy-separating family of finite-index subgroups.

The condition of having large injectivity radii is relevant to the coarse geometry of the box spaces, because here it actually implies the generally stronger condition of having large isometry radii, as shown in the next Lemma:

###### Lemma 2.11.

Let be a metric space with an isometric right-action of a group , let be the quotient space by the group action with the induced pseudo-metric, and let be the corresponding projection. Let and be such that is injective when restricted to . Then is an isometry from onto .

###### Proof.

For any and with , we have

 dX(x1⋅h1,x2⋅h1)=dX(x1,x2)≤2R,

while because , being different from but having the same image under as , must lie outside . It follows that

 dY(π(x1),π(x2))=infh1,h2∈HdX(x1⋅h1,x2⋅h2)=dX(x1,x2).

On the other hand, for any , we may pick some . As

 R≥dX(π(x),y)=infh,h′∈HdX(x⋅h,x′⋅h′)=infh,h′∈HdX(x,x′⋅h′h−1),

we see that and thus is onto . This completes the proof. ∎

Our main use of this Lemma will be to perform lifts of uniformly bounded covers, a crucial step in the proof of the upcoming main Lemma of this section.

###### Remark 2.12.

It is well-known that the box spaces of , as coarse metric spaces, encode important properties of . For instance, has property A if and only if is amenable (see [RoeCG, 11.39] and [NowakYuLSG, 4.4.6])333Although these authors only studied box spaces associated to families of normal subgroups, we remark that the proof of this result requires only a condition of large injectivity radii, as formulated in the previous Lemmas, whence the proofs also work for semi-conjugacy-separating families.. For us, the finiteness of asymptotic dimension is the more relevant condition. Let us briefly recall the relevant definition: the asymptotic dimension of a metric space is defined to be the smallest non-negative integer such that for every there exists a uniformly bounded cover of with Lebesgue number such that each consists of pairwise disjoint sets (one may even assume -disjointness, meaning that any two different sets in have distance bounded below by ). Finiteness of asymptotic dimension always implies property A, see [NowakYuLSG, 4.3.6]; hence in the case of box spaces, it also implies the amenability of . When a box space has finite asymptotic dimension, one might be tempted to think that this value encodes the complexity of the group in some sense. This is demonstrated in the main technical result of this section:

###### Lemma 2.13.

Let be a residually finite group. Consider the action of on itself by right multiplication and fix a proper, right-invariant metric. Let be a decreasing sequence of finite-index subgroups of . Then the following conditions are equivalent for all :

1. is semi-conjugacy-separating and the box space has asymptotic dimension at most .

2. For any , there exists and a uniformly bounded cover of with Lebesgue number at least , such that each is -invariant (with regard to the right multiplication) and has mutually disjoint members.

3. For every and , there exists and finitely supported functions for satisfying the following properties:

1. For every , one has

 supp(μ(l))∩supp(μ(l))h=∅for all h∈Gn∖{1}.
2. For every , one has

 s∑l=0∑h∈Gnμ(l)(gh)=1.
3. For every and , one has

 ∥μ(l)−μ(l)(g⋅__)∥∞≤ε.

In the style of [HirshbergWinterZacharias14, Section 2], we will refer to a system of functions satisfying the properties in part (3) of Lemma 2.13 as a system of decay functions.

###### Remark.

Compared with the definition of asymptotic dimension, condition (2) can be viewed as a periodic variant of asymptotic dimension. On the other hand, condition (3) reflects the standard fact that covers with large Lebesgue number give rise to very flat partitions of unity, but again in a periodic way.

###### Proof of 2.13.

For the entire proof, let us fix the following notation: For each , denote by the quotient map. Let be a proper, right-invariant metric on and let be an induced metric on as specified in Definition 2.3.

Given , there is a cover of with Lebesgue number at least , such that the diameters of members of are uniformly bounded by some and each has mutually disjoint members. Let denote the induced finite cover on the finite subspace for some .

Applying Lemma 2.7(3), we may choose such that for any , is injective when restricted to . It then follows from Lemma 2.11 that restricts to an isometry between and for any . Applying Lemma 2.7(2), let us also assume that

 B3R′(1)∩⋃k∈GkGnk−1={1}.

By uniform boundedness, each is contained in for some , and thus we may find that is mapped isometrically onto under . Hence can be written as a disjoint union . Define

 U(l):={UV⋅h | V∈V′(l), h∈Gn}

for all . Then each is -invariant (with regard to multiplication from the right), uniformly bounded and has mutually disjoint members. Moreover, we claim that covers with Lebesgue number at least . Indeed, Let be a subset with diameter less than . Since the cover of has Lebesgue number at least and is contractive, there is some such that , whence

 X⊂π−1n(πn(X))⊂π−1n(V)=⨆h∈GnUV⋅h.

By construction, the set has diameter at most , and the subgroup satisfies the equation

 B3R′(1)∩⋃k∈GkGnk−1={1}.

Thus, for in and , we have

 d(g1h1,g2h2)=d(g1h1h−12g−11,g2g−11)>d(g1h1h−12g−11,1G)−2R≥R.

So we get . But this implies that must be entirely contained in for some , which shows the claim.

Let and be given. Choose big enough so that is contained in . By assumption, there exists and a uniformly bounded cover of with Lebesgue number at least , such that each is -invariant and has mutually disjoint members. For each , upon combining several members of into one, we may assume that is of the form for some (necessarily finite) set such that for any . Now define

 μ(l):G→[0,1], g↦distd(g,G∖U(l))∑V∈Udistd(g,G∖V).

It is easy to see that and forms a partition of unity for . This proves properties (a) and (b). By [NowakYuLSG, 4.3.5], each is Lipschitz with constant , i.e. (c) is satisfied.

Given , pick with . Choose and finitely supported functions for satisfying the requirements of (3) with respect to the pair and . By choosing large enough, we may also assume that the distance between any two of the subsets and for is at least in . Define . Then for all and , and covers .

We claim that for any , there are and such that . Indeed, since is in the support of at most members of the partition of unity (at most one for each ), it follows that there exist and such that . Thus for any ,

 μ(l)(g′⋅h−1)≥μ(l)(g⋅h−1)−ε>1s+1−1s+1=0.

This shows that .

Note that in particular, this claim implies that is injective when restricted to . Since and are arbitrary and only depends on , it follows from Lemma 2.7(3) that is semi-conjugacy-separating.

Now for each , define the collection , which consists of disjoint subsets of . Define a cover of by

 V(0):={n−1⋃k=1G/Gk}⊔∞⋃m=nV(0)m and V(l):=∞⋃m=nV(l)mfor l=1,…,s.

The diameters of its members are bounded by

 max{diam(n−1⋃m=1G/Gm),diam(U(0)),…,diam(U(s))}

and each consists of disjoint sets. Finally let us show that its Lebesgue number is at least . Given any non-empty subset with radius at most , if we fix any , then . By our choice of , we know that falls entirely in one of the subsets or for . In the first case, . In the case where for , choose some such that . By the claim we proved above, there are and such that , whence . Therefore is a uniformly bounded cover of with Lebesgue number at least and each made up of disjoint members, which shows that . ∎

###### Corollary 2.14.

Let be a countable, discrete, residually finite group and a regular approximation. Let be equipped with some proper, right-invariant metric, which gives it the structure of a coarse metric space. Then .

###### Proof.

This follows immediately from 2.13(2). ∎

###### Corollary 2.15.

Let be a countable, discrete, residually finite group and a regular approximation. Let be a subgroup. Then

1. defines a regular approximation of ;

2. we have ;

3. if has finite index, then and are coarsely equivalent, and in particular .

###### Proof.

To prove (1), we observe that for any , we have a natural embedding , and for any , we have . Consequently is also semi-conjugacy-separating by definition.

Statement (2) follows directly from 2.13(2) by restricting the obtained cover to .

As for (3), we note that given any proper, right-invariant metric on , the finite-index condition implies that there exists such that is an -net in : for any , there exists such that . It follows that as quotient metric spaces, is an -net for every . Fixing a concrete metric space realization of , we may realize as the metric subspace , which is then also an -net. Since an -net is always coarsely equivalent to the ambient space, the assertion is then a consequence of the coarse invariance of asymptotic dimension. ∎

###### Definition 2.16.

Let be a countable, discrete and residually finite group. The set of all decreasing sequences of finite-index subgroups of carries a natural preorder: we say dominates , and write , if for all , there exists with . We say that two sequences are order equivalent, and write , if and .

We denote by the set of all regular approximations (cf. 2.6) of . It follows from Lemma 2.7(2) that is upward closed: if and , then .

We call a regular approximation dominating, if it is dominating with respect to the above order, i.e.  for all . This is the case if and only if, for every finite-index subgroup , there exists such that .

###### Example 2.17.

Let us consider from the point of view of the above definition. Every element in has the form for an increasing sequence of natural numbers with . One has if and only if for all , there exists such that .

Recall that a supernatural number is a formal product , where the product runs over all primes and . Another supernatural number divides if for all . We can then establish the following correspondence.

 (Λ(G)/∼ , ≾)≅(supernatural numbers ~{},~{} % dividing relation),

by sending to the supernatural number defined by

 pm|p⟺pm|kn for some n,

where is any prime natural number. Hence, a sequence is dominating if and only if every prime power devides some member of the sequence . An immediate example of this is .

###### Example 2.18.

We may generalize the last statement of the previous example to any : the sequence is a dominating regular approximation of .

###### Proof.

It suffices to show that any finite-index subgroup of