Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture

TL;DR

This paper demonstrates that the Baum-Connes assembly map is quantitatively an isomorphism for certain lacunary hyperbolic groups, including many with property (T) and counterexamples with coefficients.

Contribution

It identifies a class of groups where the Baum-Connes conjecture holds quantitatively, encompassing groups with large girth graphs and property (T).

Findings

The Baum-Connes assembly map is a quantitative isomorphism for lacunary hyperbolic groups.

Includes groups with property (T) and counterexamples with coefficients.

Contains many groups with large girth graph sequences in their Cayley graphs.

Abstract

We show that the classical Baum-Connes assembly map is quantitatively an isomorphism for a class of lacunary hyperbolic groups, and we explain how to see that this class contains many examples of groups that contain graph sequences of large girth inside their Cayley graphs and therefore do not have property (A). This includes the known counterexamples to the Baum-Connes conjecture with coefficients, as well as many other monster groups that have property (T).

Equations

$$\mathbb{C}[X{]}_R:=\{T\in C^{\ast }X\mid prop(T)\le R\}.$$

$$\mathbb{C}{G}_R:=\{a\in \mathbb{C}G\mid supp(a)\subset B_R(e)\},$$

$$\Phi :\mathbb{C}[X{]}_R\to (\mathbb{C}[Y{]}^G{)}_R$$

$$\Phi (T{)}_{y,y^{\prime }}=\{\begin{array}{l}{T}_{\pi (y),\pi (y^{\prime })}\text{mbox}ifd(y,y^{\prime })\le R,\\ 0\text{mbox}else.\end{array}$$

$$\Vert a{\Vert }_{\infty }=\underset{m}{\mathrm{limsup}}\Vert {\varphi }_m(a){\Vert }_{{\lambda }_m}.$$

$$c\Vert {\varphi }_m(a)\Vert \le \Vert a_m{\eta }_m\Vert =\Vert a{\tilde{\eta }}_m\Vert \le \Vert a\Vert$$

$$K_0^{r,ϵ}(A)=P_{\infty }^{r,ϵ}(A)\times \mathbb{N}/\sim ,$$

$$K_1^{r,ϵ}(A)=U_{\infty }^{r,ϵ}(A)/\sim .$$

$$f_{\ast }:K_{\ast }^{S,ϵ}(A)\to K_{\ast }^{S,cϵ}(B)\to K_{\ast }^{S,ϵ}(A)$$

$$F^{\ast }F-1,F{F}^{\ast }-1\text{mbox}and[F,\pi (f)]\in \mathcal{K}(\mathfrak{H})$$

$$F^2-1\text{mbox}and[F,\pi (f)]\in \mathcal{K}(\mathfrak{H})$$

$$\tilde{F}=\sum _i\sqrt{{\eta }_i}F\sqrt{{\eta }_i}.$$

$$I(F):=\left(\begin{array}{cc}F{F}^{\ast }+(1-F{F}^{\ast })F{F}^{\ast }& F(1-F^{\ast }F)+(1-F^{\ast }F)F(1-F^{\ast }F)\\ (1-F^{\ast }F)F& (1-F^{\ast }F)\end{array}\right)$$

$$\mu ([F])=[I(\tilde{F})]-\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right).$$

$$\mu :\underset{d}{\lim }{K}_{\ast }^G(P_d(G))\to \underset{d}{\lim }{K}_{\ast }(C^{\ast }(P_d(G){)}^G)=K_{\ast }(C^{\ast }{G}^G).$$

$$f:[0,\infty )\to C^{\ast }G$$

$$\Vert {\Phi }_{m,L}(f)\Vert =\underset{t}{\sup }\Vert {\Phi }_m(f)(t)\Vert \le c\underset{t}{\sup }\Vert f(t)\Vert =c\Vert f\Vert .$$

$$ev({\Phi }_{m}f)=({\Phi }_{m}f)(0)={\Phi }_m(f(0))={\Phi }_m(ev(f))$$

$$\underset{d}{\lim }{K}_{\ast }(C_L^{\ast }(P_d(G),A{)}^G)\to K_{\ast }(C_L^{\ast }(G,A{)}^G)$$

$$(e_{i}y)(t):=e_i(y(t)).$$

$$e_{i,\ast }:K_{\ast }^{r,ϵ}(C_L^{\ast }(G,{\ell }^{\infty }(\mathbb{N},\mathcal{K}){)}^G)\to K_{\ast }^{r,ϵ}(C_L^{\ast }(G{)}^G)$$

$$k_m(d,R,ϵ)=\inf \{R^{\prime }\mid G_m\text{mbox}satisfiesQS(d,R,R^{\prime },ϵ,{ϵ}^{\prime })\text{mbox}forsome{ϵ}^{\prime }\}.$$

$$\prod _i{\tilde{e}}_i:C_L^{\ast }(P_d(G),{\ell }^{\infty }(\mathbb{N},\mathcal{K}){)}^G\to \prod _i{C}_L^{\ast }(P_d(G){)}^G.$$

$$l_m(d,R,ϵ)=\inf \{(R^{\prime },d^{\prime })\mid G_m\text{mbox}satisfiesQI(d,d^{\prime },R,R^{\prime },ϵ)\}.$$

$${\varphi }_m:K_{\ast }^{r,ϵ}(C_L^{\ast }(P_d(G){)}^G)\to K_{\ast }^{r,cϵ}(C_L^{\ast }(P_d(G_m){)}^{G_m})$$

$$\Vert a{\Vert }_{2,s}:=\Vert a(1+l{)}^s{\Vert }_2=\sqrt{\sum _{g\in G}|a_g{|}^2(1+l(g){)}^{2s}}.$$

$$\underset{m}{\mathrm{limsup}}\Vert {\varphi }_m(a){\Vert }_{2,s}=\Vert a{\Vert }_{2,s}$$

$$H^{s}G\hookrightarrow \frac{\prod }{\oplus }{H}^s{G}_m$$

$$\Vert a\Vert =\underset{m}{\mathrm{limsup}}\Vert {\varphi }_m(a)\Vert \le C\underset{m}{\mathrm{limsup}}\Vert {\varphi }_m(a){\Vert }_{2,s}=C\Vert a{\Vert }_{2,s}.$$

$$A_k^R=\{x\in X\mid d(x,x_0)\in [kR,(k+1)R)\}.$$

$$18{\delta }_{m}R+12{\delta }_m\le r_m$$

Full text

1. Introduction

This paper concerns the Baum–Connes conjecture for a specific subclass of lacunary hyperbolic groups. This class has appeared all over the literature at the limits of geometric group theory, and has a thorough treatment in the work of Osin–Ol’shanskiĭ–Sapir in [OOS09]. What led the author to consider this class is the following vague question: “What are discrete groups that contain expanders in their Cayley graph like?”

At this point, the only examples we have of groups that contain expanders, or more generally sequences of graphs of large girth, are constructed using the methods of small cancellation - specifically they come from different small cancellation labellings of sequences of finite graphs that have large girth. This was accomplished first by Gromov [Gro03], which was discussed, detailed and expanded by Arzhantseva–Delzant [AD08]. The later examples, which also rely on small cancellation (although of a different flavour), were shown to contain examples with the Haagerup property - this is due to work of Arzhantseva–Osajda [AO14] and Osajda [Osa14].

Once one has labelled finite graphs where the labellings satisfying a small cancellation condition, the general machinery of small cancellation theory gives rise to an embedding of the graphs into the resulting finitely generated infinite group. The strength of this embedding depends on the controls provided by the corresponding Van Kampen type theorem that appears for that particular type of labelling. The range is from weak embeddings [HLS02], or almost quasi-isometric embeddings at a large scale [Gro03, AD08], or all the way through to isometric embeddings [Gro03, Oll06, Gru14].

Currently, it is possible to use either a geometric small cancellation labelling (as in Gromov [Gro03]) or a graphical small cancellation labelling, and these have been shown to have a very different character, for instance the groups that come from these labellings cannot be quasi-isometric (this is due to Gruber–Sisto [GS18]). The method itself is very flexible and compatible with other small cancellation constructions possible in the literature - see for instance the remarks of Arzhantseva–Delzant [AD08].

It is often convenient to work with the graphical small cancellation as it provides either coarse or isometric embeddings - and the details of this can be seen nicely in the thesis work of Gruber, which appears in [Gru14]. However, if one is interested in producing examples with property (T) currently one needs to use the method of Gromov [Gro03]. The graphical C(7) labellings, which exist due to Osajda [Osa14] can also be obtained for graphs that satisfy a wall type structure as considered in Arzhantseva–Osajda [AO14], and this means that the results of [AO14] can be applied to obtain groups that are not property (A), but do have the Haagerup property. This end of the spectrum is of less interest to us in this paper purely because it subsumes the direct calculations we perform to obtain the isomorphism of classical Baum–Connes assembly, due to the work of Higson–Kasparov [HK97].

The current literature is sometimes confusing concerning these small cancellation distinctions. The key thing to remember is that the philosophy espoused by small cancellation does not change in the constructions. The technology exists and there is now a general pattern to how the theory works. The difficulty, after the theory is developed111Please understand I am not trying to minimise the efforts of people who develop this theory, I am just talking in a “once the dust has settled” practical sense., is showing the existence of labellings that satisfy the various conditions. This is achieved in the work of Gromov [Gro03], Arzhantseva–Delzant [AD08] and Osajda [Osa14].

Our focus in this paper is on the class of lacunary hyperbolic groups, as essentially all the groups that have been constructed using these labelling results fall into this class. By focusing away from the small cancellation and proving slightly more general results, we hope that it may be adaptable in the future.

The remainder of what comes is interpreting the word “like” in the initial question above, as it can be done in a variety of settings. We can say, for instance, that groups with expanders in them do not coarsely embed into a Hilbert space (one of the initial motivations for constructing them). However, such groups also give us the only known counterexamples for the Baum–Connes conjecture with coefficients - this is present in the paper of Higson–Lafforgue–Skandalis [HLS02] with very general statements that rely on a very basic construction they exploit over and over, in particular in the case of groups that contain expanders.

What is unknown, however, is if they satisfy the classical version of the Baum–Connes conjecture, which asks only about the trivial coefficients $\mathbb{C}$ - that is truly a statement about the representation theory of the groups in question.

We prove that in this most basic form, the Baum–Connes conjecture holds for lacunary hyperbolic groups when the lacunarity is sufficiently fast.

Theorem A**.**

Let $G$ be a lacunary hyperbolic group satisfying the Baum–Connes conjecture with asymptotic controls. Then the Baum–Connes conjecture is (quantitatively) true for $G$.

We introduce all the terms throughout the paper at their critical moments, and give plenty of citations for the known properties or notions that we recall throughout.

The method we introduce here to prove this statement is inspired by two main sources: the work of Willett–Yu [WY12] on the coarse Baum–Connes conjecture for large girth graph sequences and the work of Yu [Yu98] on the coarse Baum–Connes conjecture for groups of finite asymptotic dimension.

The former paper takes a method of Higson [Hig99] that appeared also in Higson–Lafforgue–Skandalis [HLS02] and removed the group theory aspects to a large degree. The main concept they introduced, the notion of an asymptotically faithful covering sequence, is defined in Section 2. This is used there to construct lifting maps at the level of the Roe algebra - giving us the so called “Higson trace” (first constructed in [Hig99]). This notion of lifting is exploited to obtain the representation theory results of Section 3 that we apply in the setting of lacunary hyperbolic groups.

The latter paper of Yu makes use of a quantitative or persistent version of K-theory for algebras that have some kind of scale or length. This quantitative theory gives more refined information that is very possible to transfer using localised lifting maps like those that appear in the Higson trace. This quantitative theory has been conveniently and systematically studied by Oyono-Oyono–Yu [OOY15] and this gives us a framework and some general results to begin taking advantage of.

Our strategy, overall, is to cut the K-theory group we are interested in into pieces that we can lift through neatly constructed maps that act like homomorphisms on some specific scale. We then convert these groups into something workable, then piece them back together again using the quantitative asymptotic controls condition.

Finally, we make one contribution to the small cancellation theory, that is quite soft - we show how to construct a multitude of examples that satisfy our condition of having asymptotic controls, and state sample theorems that come from applying this new method along with the other constructions in the literature to obtain interesting groups for which the technique presented here provides an optimal outcome. This can be found in Section 5. A summary of those results is stated below:

Theorem B**.**

The following holds:

1. (1)

Every torsion free graphical small cancellation and geometric small cancellation C(7) lacunary hyperbolic group has a covering that has a quantitatively isomorphic Baum–Connes assembly map. 2. (2)

For every hyperbolic group $G$ there is a graded small cancellation quotient $H$ that

1. (a)

is lacunary hyperbolic; 2. (b)

is a Tarski monster (i.e every proper finitely generated subgroup is cyclic of a fixed prime order); 3. (c)

the Baum–Connes assembly map for $G$ is quantitatively an isomorphism.

We note that this concludes the result for many groups that have property (T), as the starting group $G$ or the seed group in a geometric small cancellation construction can be chosen to have it. This result relies heavily on the work of Lafforgue [Laf12] concerning the Baum–Connes conjecture with coefficients for hyperbolic groups, as it relies on the hyperbolic limiting terms having quantitative Baum–Connes assembly isomorphisms - which at this point in time can only be obtained as a consequence of the assembly conjecture for a set of specific coefficients.

Finally, we remark also that at this point we don’t have a good method for including coefficients into this method. The issue with coefficients (aside from the fact that many of these groups have coefficients for which the conjecture is false) is that they may not be “limits” of coefficients of the hyperbolic limiting terms - and making precise what we mean by that is also a difficult proposition we’ve yet to fully understand.

Acknowledgements

The author would like to thank Goulnara Arzhantseva for patiently reading early versions of this text, in particular for accuracy in the small cancellation theory as this is something in which the author is not an expert, and Rufus Willett for a timely remark concerning Theorem 3.2. He also thanks Ján Špakula for his ongoing support and encouragement, and Alain Valette for his many comments about the text.

2. Asymptotically faithful covering spaces

Definition 2.1**.**

Let $\pi:X\rightarrow Y$ be a covering map of metric spaces. The injectivity radius of $\pi$ is the largest $R>0$ such that balls of radius $R$ in $X$ are mapped isometrically to balls of radius $R$ in $Y$.

Definition 2.2**.**

Consider two metric families $\mathcal{X}=\{X_{m}\}_{m}$ and $\mathcal{Y}=\{Y_{m}\}_{m}$. We say $\mathcal{X}$ asymptotically faithfully covers $\mathcal{Y}$ if there exist covering maps $\pi_{m}:X_{m}\rightarrow Y_{m}$ such that the injectivity radius $R_{m}$ for $\pi_{m}$ tends to infinity in $m$.

This notion was introduced by Willett and Yu in [WY12] - to deal with a more general version of Example 2.1 below. For what we will do in this paper, we will consider situation that either $\mathcal{X}$ or $\mathcal{Y}$ is a constant family - i.e $X_{m}$ (resp. $Y_{m}$) are all the same metric space. This is more general than the situation of [WY12], as it captures certain direct limits or elementary convergences in the space of marked groups (see for instance [CG05]).

2.1. Examples

We begin by covering the main families of examples that motivate our work.

Example 2.1**.**

(Box spaces) For a residually finite finitely generated discrete group $G$, let $\mathcal{N}=\{N_{i}\}_{i}$ be a nested family of normal subgroups of finite index and trivial intersection222the existence of such a family is the definition of a group being residually finite. Then the family $\{\pi_{i}:G\rightarrow G/N_{i}\}_{i}$ is an asymptotically faithful covering family. To see this, we must fix a metric on $G$ from a finite generating set $S$, and then use the metric on each $G/N_{i}$ that comes from $\pi_{i}(S)$. Then for any $R$, we can find some $i$ such that $B_{R}$, the ball of radius $R$ in $G$ is not contained in $N_{i}$ - hence it maps injectively to $G/N_{i}$. By choosing $i$ large enough, this can be made isometric.

This covering family allows us to study the box spaces $\square G$, which are infinite metric spaces constructed from the disjoint union of family of finite quotients $G/N_{i}$ with any metric $d$ satisfying

• •

$d|_{G/N_{i}}$ is the metric coming from $\pi_{i}(S)$,

• •

$d(G/N_{i},G/N_{j})\rightarrow\infty$ as $i+j\rightarrow\infty$.

Any two metrics that satisfy i) and ii) are coarsely equivalent, and thus we need not be more specific about the rates of divergence.

The asymptotically faithful property can be used in this case to construct counterexamples to the coarse Baum–Connes conjecture by either taking groups such that the finite quotients form an expander family [Hig99] (such as groups with property (T)). It can be used explicitly to comprehensively study sequences of graphs of large girth, which is the example above when the group $G$ is taken to be a free group [WY12] [FSW14]. These examples motivate the rest of this text.

Example 2.2**.**

(Lacunary hyperbolic groups) This is a class of finitely generated discrete groups that many monster groups from group theory [OOS09] end up living - it is a place where many outputs of small cancellation theory [Gro03, AD08] naturally occur. The definition is as follows:

Definition 2.3**.**

Let $G$ be a direct limit of discrete groups $G_{m}$ via (necessarily surjective) quotient maps, and let $\pi_{m}:G_{m}\rightarrow G$ be the quotient map to the limit. $G$ is lacunary hyperbolic if:

1. (1)

the injectivity radius $r_{m}$ of $\pi_{m}$ tends to infinity as $m$ does; 2. (2)

each $G_{m}$ is $\delta_{m}-$hyperbolic for some $\delta_{m}>0$; 3. (3)

these $\delta_{m}$’s satisfy: $\delta_{m}=o(r_{m})$.

Note that baked into this definition is the condition that $\{\pi_{m}:G_{m}\rightarrow G\}_{m}$ is an asymptotically faithful covering family.

As a general reference to this important class of groups, one can look at Osin–Ol’shanskiĭ–Sapir [OOS09]. In particular it contains the counterexamples to the Baum–Connes conjecture with coefficients [CTWY08, AD08] as well as many Tarski monster groups or other small cancellation monsters from geometric group theory [OOS09].

Example 2.3**.**

(Limit groups of Sela) Limit groups are a particular class of finitely presented groups that arise in the study of first order logical theory of finitely generated groups - they can be defined as those groups that share the same theory as a free group [CG05]. It is known that such groups are precisely those groups $G$ that admit an asymptotically faithful covering family $\{\pi_{m}:G\rightarrow G_{m}\}_{m}$ where each $G_{m}$ is a finitely generated free group (of a fixed rank) - a condition in this case that appears with the terminology “fully residually free” in the literature. However, using the generators of $G$ to generate each $G_{m}$ is necessary for the asymptotically faithful covering example, and this will not be a free generating set.

After reducing this to a free generating set, one can see that the extra generators we would be including must be very long in the word length of the free group - longer than the injectivity radius. Thus, the sequence is not uniformly coarsely equivalent to a sequence of fixed rank free groups with the standard generating set - thus this condition of asymptotic faithfulness is a bit more subtle than it first appears.

3. An elementary representation theory result

In this section we’re going to prove a basic representation theory result concerning asymptotically faithful covering sequences. First though, we recall the basic algebras we will attach to a group or metric space and how they’re connected.

Definition 3.1**.**

Let $X$ be a uniformly discrete metric space of bounded geometry and let $T$ be a bounded operator on $\ell^{2}X$. The propagation of $T$ is the smallest positive real number $R$ such that the matrix entries of $\langle T\delta_{x},\delta_{y}\rangle$ vanish when $d(x,y)>R$.

The collection of all finite propagation operators forms a $\ast$-subalgebra of $\mathbf{B}(\ell^{2}(X))$, denoted by $\mathbb{C}[X]$. Its completion, is called the uniform Roe algebra and is denoted by $C^{*}_{u}(X)$. Finally, to generate the full Roe algebra, denoted $C^{*}X$, we do exactly as above except we take operators with values in the compact operators on some separable Hilbert space.

Let $X$ be a metric space. Then the Roe algebra $C^{*}X$, (resp. uniform Roe algebra $C^{*}_{u}X$) can be graded by propagation using the closed linear subspaces:

[TABLE]

If $G$ is a discrete group acting on $X$ by isometries, then the subalgebra of equivariant operators obtained by closing the finite propagation operators that are equivariant, denoted by $C^{*}X^{G}$ is also graded by intersecting with the subspaces above. Note that this algebra, if the action is free and proper, is Morita equivalent to the reduced group $C^{*}$-algebra of $G$ (see for instance [WY12] - by no means an original or earliest reference).

We can grade the group ring of $\mathbb{C}G$ of $G$ directly using the following linear subspaces:

[TABLE]

where the support of $a$, $supp(a)$, is the subset of elements of $G$ such that the coefficients $a_{g}$ are not [math].

Note that gradings interact in the following way as the subspaces $\mathbb{C}G_{R}$ and $\mathbb{C}[G]^{G}_{R}$ are related by mapping $a\in\mathbb{C}G$ to the element $T\in\mathbb{C}[G]^{G}$ with matrix entries $T_{x,y}=a_{x^{-1}y}$333or to land in the Roe algebra proper, one would, for instance, map to $T\otimes p$, where $p$ is a rank one projection and $T$ is the operator before constructed from $a$.. This relationship is convenient for the following construction.

Observation 3.1**.**

Let $X$ be a metric space and let $\pi$ be a surjective covering map $Y\rightarrow X$ with injectivity radius $r$ and deck transformation group $G$. Then for all $R<r$ there is a linear map

[TABLE]

constructed using the entries:

[TABLE]

Note that when $Y$ and $X$ are chosen to be discrete groups, and $\pi$ is a quotient map, this can also be used to define, for each $a\in\mathbb{C}X_{R}$, a lifting $\phi(a)\in\mathbb{C}Y_{R}$ by coefficients $\phi(a)_{y}=a_{\pi(y)}$.

These maps induce are linear isomorphisms $\mathbb{C}[X]_{R}\cong(\mathbb{C}[Y]^{G})_{R}$ and $\mathbb{C}X_{R}\cong\mathbb{C}Y_{R}$ that respect composition, in the sense that that if $Prop(T)+Prop(S)\leq r$, then $\Phi(TS)=\Phi(T)\Phi(S)$ in $\mathbb{C}[Y]^{G}$ (resp. $\phi(TS)=\phi(T)\phi(S)$ if the elements $S,T\in\mathbb{C}G$).

The relationship between the group ring and the finite propagation operators using the matrix coefficient definition above commutes with the maps $\phi$ and $\Phi$ above.

Definition 3.2**.**

A family of lifting maps is asymptotically continuous (resp. isometric) if for every $R>0$, there is a large enough $m$ such that if $prop(T)\leq R$ then $\|\Phi_{n}(T)\|\leq\|T\|$ (resp. $\|\Phi_{n}(T)\|=\|T\|$) for all $n\geq m$.

Continuity of this sort of map has been studied in the context of the Baum–Connes assembly conjecture, and was used explicitly by Higson [Hig99], Higson–Lafforgue–Skandalis [HLS02] and later Willett–Yu [WY12] to construct counterexamples to the coarse Baum–Connes conjecture. It also appears in the work of Gong–Wang–Yu [GWY08], Oyono-Oyono–Yu [OOY09] and Guentner–Tessera–Yu [GTY11] in this context, and for the most part all of these results have the common theme that they exploit an idea introduced in [CTWY08] called operator norm localisation:

Definition 3.3**.**

Let $X$ be a uniformly discrete metric space with bounded geometry. Then $X$ has operator norm localisation with constant $c\in(0,1)$ and control function $f$ if: for every $R>0$ and $T\in\mathbb{C}[X]_{R}$ there is a unit vector $\eta$ satisfying:

1. (1)

$diam(supp(\eta))\leq f(R)$, 2. (2)

$c\|T\|_{op}\leq\|T\eta\|$.

It’s known now due to work of Sako [Sak14] that this property is equivalent to Yu’s property (A), initially given in [Yu00] and surveyed thoroughly by Willett [Wil09].

One useful fact we will recall from the literature is Proposition 3.3 from Chen–Tessera–Wang–Yu [CTWY08], which states that the operator norm localisation constant (i,e, the $c$ in definition 3.3), once known to exist can be taken to be any value:

Lemma 3.1**.**

If a metric space $X$ has the operator norm localisation property then it has it with any constant $c\in(0,1)$.∎

The cost of performing this trick is that the diameter of the support of the vector increases dramatically. In particular if one has operator norm localisation with constant $c$ and control function $f$, then one can get $c^{\prime}\in(0,1)$ by using control function $g$, where $g(k)=(n-1)k+f(nk)$ - where $n$ is the smallest number such that $c^{n}\geq c^{\prime}$.

We now introduce the main tool we plan to use to study the regular representation of well approximated groups.

Definition 3.4**.**

(The lifting limsup representation) Let $\{\pi_{m}:G_{m}\rightarrow H_{m}\}$ be an asymptotically faithful covering family where either the $G_{m}$ or $H_{m}$ are equal to a single finitely generated group $G$. Then for every element of the group ring $a\in\mathbb{C}G$ we defined lifts $\phi_{m}(a)\in\mathbb{C}G_{m}$ for each large enough $m$. Thus, if we equip each $\mathbb{C}G_{m}$ with a norm, for instance the reduced one, we can define a norm on $\mathbb{C}G$ using the limsup:

[TABLE]

This representation exists by the Gelfand–Naimark–Segal theorem, (for instance found in [Dav96]).

One can ask if the representation $\lambda_{\infty}$ obtained this way is related to the left regular representation. We introduce the following notion to control its behaviour and answer this question.

Definition 3.5**.**

Let $\{\pi_{m}:G_{m}\rightarrow H_{m}\}$ be an asymptotically faithful covering family where either the $G_{m}$ or $H_{m}$ are equal to a single finitely generated group $G$.

If each $G_{m}$ has operator norm localisation with constant $c$ and functions $f_{m}$ such that the number $R_{m}:=\sup\{R\mid R+f_{m}(R)\leq r_{m}\}$ tends to infinity in $m$ then we say the sequence has operator norm localisation with asymptotic controls.

The goal of the remainder of this section is to prove the following result:

Theorem 3.2**.**

Let $\{\pi_{m}:G_{m}\rightarrow H_{m}\}$ be an asymptotically faithful covering family where either the $G_{m}$ or $H_{m}$ are equal to a single finitely generated group $G$. Then the asymptotically faithful covering family $\{\pi_{m}:G_{m}\rightarrow G\}_{m}$ has operator norm localisation with asymptotic controls then $\lambda_{\infty}$ is weakly equivalent to $\lambda$.

Proof.

Let $a\in(\mathbb{C}G)_{R}$ be given. Pick $m$ large enough such that $R+R_{m}\leq r_{m}$. Then operator norm localisation for $G_{m}$ provides a vector $\eta_{m}$ with diameter of its support smaller than $f_{m}(R)\leq r_{m}$. This vector satisfies:

[TABLE]

where $\tilde{\eta}_{m}$ is the unit vector pushed through the quotient map to $G$ and observing that $a\tilde{\eta}_{m}$ lifts to $a_{m}\eta_{m}$ since the support is smaller than $R+R_{m}$. It follows that $\limsup_{m}\|a_{m}\|\leq\frac{1}{c}\|a\|$, which shows continuity. The same argument works for $\Phi$.

Now note that for any $a\in(\mathbb{C}G)_{R}$, there is always a vector of finite support $v$ a constant $c_{a}\in(0,1)$ such that $c_{a}\|a\|\leq\|av\|$. Choosing $m$ large enough, we can as before lift $a$, $v$ and $av$ coefficient wise using $\phi_{m}$ such that $\phi_{m}(av)=\phi_{m}(a)\phi_{m}(v)$. Thus $c_{a}\|a\|\leq\|av\|=\|\phi_{m}(a)\phi_{m}(v)\|\leq\|\phi_{m}(a)\|$. As $c_{a}$ is independent of $m$, we obtain the inequality also for the limsup. ∎

We’ll make use of this result in the context of the Baum–Connes conjecture in the next section.

4. What about the Baum–Connes conjecture?

Our goal in this section is to assemble a proof of the Baum–Connes conjecture for groups that are the base group in an asymptotically faithful covering where each covering term satisfies the Baum–Connes conjecture with coefficients.

We recall some definitions from Oyono-Oyono–Yu [OOY15].

Definition 4.1**.**

1. (1)

Let $A[0,1]$ be the $C^{*}$-algebra $C([0,1],A)$. This is where homotopies between elements of $A$ live. 2. (2)

A filtered $C^{*}$-algebra $A$ is a $C^{*}$-algebra equipped with a family of linear subspaces indexed by positive real numbers $(A_{r})_{r>0}$ such that:

• •

$A_{r}\subseteq A_{r^{\prime}}$ if $r\leq r^{\prime}$;

• •

$A_{r}$ is $*$-closed;

• •

$A_{r}A_{r^{\prime}}\subseteq A_{r+r^{\prime}}$;

• •

the subalgebra $\cup_{r}A_{r}$ is dense in $A$.

If $A$ is unital, then we include $1$ in $A_{r}$ for every $r>0$. The elements of $A_{r}$ are said to have propagation at most $r$. 3. (3)

This concept also extends to morphisms in multiple ways. Let $A$, and $A^{\prime}$ be filtered $C^{*}$-algebras by $(A_{r})_{r}$ and $(A^{\prime}_{r})_{r}$ respectively. Then a homomorphism $f:A\rightarrow A^{\prime}$ is filtered if $f(A_{r})\subseteq A^{\prime}_{r}$ for every $r$ and $d-filtered$ for some real number $d$ if $f(A_{r})\subseteq A^{\prime}_{dr}$. Both of these maps interact favourably with what is to come. 4. (4)

Let $\epsilon\in(0,\frac{1}{4})$. An element $p$ of a unital filtered $C^{*}$-algebra $A$ is an $(r,\epsilon)$* quasi-projection* if $p\in A_{r}$, $p=p^{*}$ and $\|p^{2}-p\|\leq\epsilon$. Let $P_{n}^{r,\epsilon}(A)$ denote the collection of all $(r,\epsilon)$ quasi-projections in $M_{n}(A)$ - where $(M_{n}A)_{r}=M_{n}(A_{r})$ for every $n$. Note, that the corner inclusions: $p\mapsto diag(p,0)$ are maps $P_{n}^{r,\epsilon}(A)\rightarrow P_{n+1}^{r,\epsilon}(A)$, and we denote by $P_{\infty}^{r,\epsilon}(A)$ the union of the $P_{n}^{r,\epsilon}(A)$’s. 5. (5)

We can now do the same for unitaries: an element $u$ of a unital filtered $C^{*}$-algebra $A$ is a an $(r,\epsilon)$ unitary if $u\in A_{r}$ and both $\|uu^{*}-1\|$ and $\|u^{*}u-1\|$ are smaller than $\epsilon.$ We denote by $U_{n}^{r,\epsilon}(A)$ the set of all $(r,\epsilon)$ unitaries in $M_{n}(A)$. Similar to above, the maps $u\mapsto diag(u,1)$ give maps $U_{n}^{r,\epsilon}(A)\rightarrow U_{n+1}^{r,\epsilon}(A)$, and we denote the union by $U_{\infty}^{r,\epsilon}(A)$. 6. (6)

We define relations on these (semi)groups using homotopies. A homotopy $h$ of $(r,\epsilon)$ projections $p,q$ (resp. unitaries $u,v$) is an $(r,\epsilon)$ projection (resp. unitary) for $A[0,1]$, with $h(0)=p,h(1)=q$ (resp. $h(0)=u,h(1)=v$). 7. (7)

For a unital filtered $C^{*}$-algebra $A$, we define relations on $P^{r,\epsilon}_{\infty}(A)\times\mathbb{N}$ and $U^{r,\epsilon}_{\infty}(A)$:

1. (a)

if $(p,k),(q,l)\in P^{r,\epsilon}_{\infty}(A)\times\mathbb{N}$, then $(p,k)\sim(q,l)$ if there exists $h\in P^{r,\epsilon}_{\infty}(A[0,1])$ such that $h(0)=diag(p,I_{k})$ and $h(1)=diag(q,I_{l})$; 2. (b)

if $u,v\in U^{r,\epsilon}_{\infty}(A)$, then $u\sim v$ if there exists $h\in U^{r,\epsilon}_{\infty}(A[0,1])$ such that $h(0)=u$, $h(1)=v$. 8. (8)

We define the quantitative K-theory groups

[TABLE] 9. (9)

An $R$-quasi $\ast$-homomorphism between filtered $C^{*}$-algebras $A$ and $B$ is a map $f:A_{R}\rightarrow B_{R}$ that is linear continuous map that satisfies $f(a)f(b)=f(ab)$ for any two elements $a\in A_{S},b\in A_{S^{\prime}}$ such that $S+S^{\prime}\leq R$. A $C^{*}$-algebra homomorphism between $A$ and $B$ is a $R$-quasi $\ast$-homomorphism.

We’ve constructed some examples of $R$-quasi $\ast$-homomorphisms - the maps $\phi_{m}$ that we used to lift operators in the group ring $\mathbb{C}G$ to operators in $\mathbb{C}G_{m}$, for an asymptotically faithful covering sequence $G_{m}$. When the sequence has asymptotically controlled operator norm localisation, the maps $\phi_{m}$ are uniformly continuous (if $m$ is large enough).

We record some elementary facts and conclude a basic lemma in quantitative K-theory that we will use analogues of again and again in the sequel.

Lemma 4.1**.**

Let $A$ and $B$ be graded $C^{\ast}$-algebras and let $f:A_{R}\rightarrow B_{R}$ be a bijective $R$-quasi $\ast$-homomorphism, where $A_{R}$ and $B_{R}$ are finite dimensional Banach spaces and such that the norm $\|f\|$ is bounded above by $c\in(1,\infty)$. Then

1. (1)

$f$* induces the following maps of quantitative K-theory groups for $S=R/2$ and $\epsilon\in(0,\frac{1}{4c})$:*

[TABLE] 2. (2)

$f^{-1}_{*}\circ f_{*}=Id_{A_{S},*}$, $f_{*}\circ f_{*}^{-1}=Id_{B_{S},*}$

Proof.

As $A_{R}$ and $B_{R}$ are finite dimensional and $f$ is a continuous linear bijection of Banach spaces, we also know that $f^{-1}$ is continuous with norm bounded below by $\frac{1}{c}$. The second part follows from the fact that $f$ is a bijection, and the induced maps described - the norm estimates follow from the fact that continuous maps are bounded for Banach spaces. ∎

4.1. The lifting map and assembly

The goal of this section is to construct the assembly maps using the coarse geometric picture, introduce localisation algebras and then state the reformulated (coarse) Baum–Connes conjecture using this localisation algebra picture.

4.1.1. A formula for assembly

We begin by defining the K-homology groups in question, then constructing the assembly map.

Definition 4.2**.**

(K-homology) Let $X$ be a metric space. A Hilbert space $\mathfrak{H}$ is called a $X$-module if there is a representation $\pi:C_{0}(X)\rightarrow B(\mathfrak{H})$. It’s a standard $X$-module if $\pi(C_{0}(X))\cap K(\mathfrak{H})=\{0\}$, and non-degenerate if $\pi$ is non-degenerate. An example of such a module would be $\ell^{2}(X)$, when $X$ is uniformly discrete with bounded geometry.

• •

A $K_{0}$-cycle for $X$ is a pair $(\mathfrak{H},F)$, where $\mathfrak{H}$ is a non-degenerate, standard $X$ module and $F$ is a bounded operator on $\mathfrak{H}$ satisfying:

[TABLE]

for all $f\in C_{0}(X)$

• •

A $K_{1}$-cycle for $X$ is a pair $(\mathfrak{H},F)$ where $\mathfrak{H}$ is a non-degenerate, standard $X$ module and $F$ is self adjoint bounded operator on $\mathfrak{H}$ satisfying:

[TABLE]

for all $f\in C_{0}(X)$.

If $X$ is a $G$-metric space, then we can define equivariant K-homology (denoted $K^{G}_{*}(X)$) cycles by asking additionally that there exists a representation $\sigma:G\rightarrow U(\mathfrak{H})$ such that $\sigma$ intertwines $\pi$ and $\sigma(g)$ commutes with $F$ modulo the compact operators.

In dealing with a class for $K^{G}_{*}(X)$ we will suppress the Hilbert space, and consider just the class of the operator $[F]$.

To define the assembly map $\mu$, we will first consider a uniformly bounded cover $\mathcal{U}=\{U_{i}\}$ and let $\eta_{i}$ be a partition of unity subordinate to $\mathcal{U}$. Then define:

[TABLE]

This sum is

• •

convergent in the strong operator topology;

• •

equivalent to $F$ as a $K-homology$ cycle;

• •

it is a multiplier of $\mathbb{C}[X]^{G}$, the equivariant Roe algebra of $X$.

We can now consider the following matrix:

[TABLE]

We observe that this matrix maps to $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$ under the evaluation map to $\mathbb{C}$ from the standard unitisation of $\mathbb{C}[X]$.

Definition 4.3**.**

(The index map) The index map $\mu:K_{*}^{G}(X)\rightarrow K_{*}(C^{*}X^{G})$ is defined by the formula:

[TABLE]

The Baum–Connes assembly map is then obtained by taking sufficiently good limits through a particular type of space. Let $X=P_{d}(G)$, be the Rips complex over $G$ where $d>1$. Then the Baum–Connes assembly map is defined by the limit of the appropriate indices:

Definition 4.4**.**

(The Baum–Connes assembly map) The assembly map $\mu$ is defined by:

[TABLE]

4.1.2. Localisation algebras

This sections definitions and cited theorems are from Appendix A in Guentner–Tessera–Yu [GTY11] as well as Yu [Yu97]. We recall some definitions and begin from there.

Definition 4.5**.**

Let $G$ be a discrete group, then the localisation algebra of $G$, denoted $C^{*}_{L}G$ is the collection of uniformly bounded maps:

[TABLE]

such that $prop(f(t))$ tends to [math] as $t$ tends to $\infty$, equipped with the supremum norm: $\|f\|=\sup_{t}\|f(t)\|$.

This definition can be made equivariant by mapping into $C^{*}G^{G}$, and this is denoted by $C^{*}_{L}G^{G}$.

Definition 4.6**.**

We grade the equivariant localisation algebra by considering the linear spaces $(C^{*}_{L}G^{G})_{R}$, by asking that $f(t)$ has propagation less than $R$ for all $t$. This grading allows us to define quantitative K-theory groups.

Note that the lifting map $\Phi_{m}$ induces a map $(C^{*}_{L}(G)^{G})_{R}\rightarrow(C^{*}_{L}(G_{m})^{G_{m}})_{R}$, which we will denote by $\Phi_{m,L}$. We can control the norm of this map using the norm of the original $\Phi_{m}$.

Lemma 4.2**.**

Suppose that $\Phi_{m}:(C^{*}(G)^{G})_{R}\rightarrow(C^{*}(G_{m})^{G_{m}})_{R}$ is a continuous linear bijection with norm $\|\Phi_{m}\|\leq c$, $\|\Phi_{m}^{-1}\|\leq d$, $c,d\in[1,\infty)$ Then $\|\Phi_{m,L}\|\leq c$ and $\|\Phi_{m,L}^{-1}\|\leq d$

Proof.

We compute directly from the definition. Let $f\in(C^{*}_{L}(G)^{G})_{R}$ -

[TABLE]

This shows the desired inequality. The same argument run the other way around completes the proof. ∎

We can use this in conjunction with our norm estimates from asymptotically controlled operator norm localisation and the K-theory statements from Lemma 4.1.

Finally, we can observe that lifting commutes with evaluation.

Lemma 4.3**.**

Lifting commutes with evaluation.

Proof.

Consider the following brief calculation for any $f\in C^{*}_{L}G^{G}_{R}$:

[TABLE]

where the middle equality is the definition of the lifting map on localisation algebras. ∎

We will use this to connect the results concerning quantitative K-theory groups later on in the next section.

4.1.3. The localised assembly map

There is a version of the assembly map that is compatible with localisation algebras that is clearly defined in [Yu97, GHW05]. The main result we need is that the Baum–Connes conjecture for $G$ with coefficients in $A$ can be reformulated in terms of localisation algebras and evaluation maps.

Theorem 4.4**.**

(Theorem A.5 from [GTY11]) The Baum–Connes conjecture for $G$ with coefficients in $A$ is equivalent to the assertion that the evaluation map:

[TABLE]

is an isomorphism.

4.2. A brief sketch of the argument for surjectivity

The plan is as follows.

1. (1)

We reformulate a quantitative statement about the Baum–Connes conjecture into the terminology of localisation algebras and quantitative evaluation maps. 2. (2)

We obtain K-theory isomorphisms from the maps $\phi$ and $\Phi$ in a quantitative way for the groups $G_{m}$ and $G$. 3. (3)

We move K-theory classes around carefully through these isomorphisms, assuming we can control the evaluation isomorphism enough to make that go.

We work through these steps in the following sections.

4.3. Quantitative surjectivity of the evaluation map framed using Baum–Connes with coefficients

In this section we show that the quantitative surjectivity of the evaluation map can be seen as a consequence of the surjectivity of the Baum–Connes assembly map with a specific set of coefficients

Definition 4.7**.**

Let $r^{\prime}>0,\epsilon^{\prime}\in(0,\frac{1}{4})$. The evaluation map $ev:\lim_{d\rightarrow\infty}(C^{*}_{L}G)^{G}\rightarrow\lim_{d\rightarrow\infty}(C^{*}G)^{G}$ satisfies $QS(d,r^{\prime},r,\epsilon^{\prime},\epsilon)$, where $\epsilon>\epsilon^{\prime},$if for every element $x$ in $K_{*}^{r^{\prime},\epsilon^{\prime}}(C^{*}G)^{G})$ there is a $y\in K^{r,\epsilon}_{*}(C^{*}_{L}P_{d}(G)^{G})$ such that $ev^{r,\epsilon}(y)=i_{r^{\prime},r,\epsilon^{\prime},\epsilon}(x)$. The evaluation map is quantitatively surjective if for every $r^{\prime},\epsilon^{\prime}$ we can find $d,r,\epsilon$ such that $QS(d,r^{\prime},r,\epsilon^{\prime},\epsilon)$ holds.

This condition appears stronger than asking for a surjection of the evaluation map itself, and it obviously implies this. We relate it to Baum–Connes assembly map with coefficients in the spirit of Oyono-Oyono–Yu [OOY15], making it tractable to study this condition.

Theorem 4.5**.**

Let $G$ be a countable discrete group. If the Baum–Connes conjecture with coefficients in $\ell^{\infty}(\mathbb{N},\mathcal{K}))$ holds for $G$, then the evaluation map for $G$ is quantitatively surjective.

Proof.

We do $*=0$. Suppose that the evaluation map is not quantitatively surjective. Then there is a $r^{\prime},\epsilon^{\prime}$ such that no matter what $d,r,\epsilon$ we pick, $QS(d,r^{\prime},r,\epsilon^{\prime},\epsilon)$ doesn’t hold for the evaluation map. Fix $(d_{i}),(r_{i})$ with both going to infinity as $i$ does and $\epsilon>0$, and then take elements $x_{i}\in K_{0}^{r^{\prime},\epsilon}(C^{*}(G)^{G})$ such that $i_{r^{\prime},r_{i},\epsilon^{\prime},\epsilon}(x)$ is not in the image of $ev_{d_{i},r_{i},\epsilon}$. We then wrap these elements up into a class $x$ that belongs to the K-theory of $C^{*}(G,\ell^{\infty}(\mathbb{N}))^{G}$ by identifying it as $C^{*}_{r}(G,\ell^{\infty}(G,\ell^{\infty}(\mathbb{N},\mathcal{K})))$.

The surjectivity assumption on the assembly map for $G$ with coefficients in $\ell^{\infty}(\mathbb{N},\mathcal{K})$ and the equivalence provided in Theorem 4.4, provides us with an element $y\in\lim_{d}K_{0}(C^{*}_{L}(P_{d}(G),\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G}$ that maps onto $x$ under $ev$. To complete the proof, we examine $y$ carefully using the evaluation maps from $\ell^{\infty}(\mathbb{N},\mathcal{K})$ to $\mathcal{K}$ to test the element $y$.

Recall that the evaluation maps $e_{i}:\ell^{\infty}(\mathbb{N},\mathcal{K})\rightarrow\mathcal{K}$ are defined, for each $i$, by $e_{i}f=f(i)$. These can be applied directly to elements of the localisation algebra $C^{*}_{L}(G,\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G}$ in the following way: for every $t\in[0,1]$, define:

[TABLE]

These maps are continuous $*$-homomorphisms on $C^{*}_{L}(G,\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G}$ with image in $C^{*}_{L}(G)^{G}$. Thus, they induce quantitative maps on K-theory. Note here that the propagation does not increase - if we start with an element $f$ that has $f(t)$ of propagation at most $r$ for every $t$, then this will be true still the case after evaluating in each coordinate - thus we get a map:

[TABLE]

for each $i$. This will also extend to each Rips complex, in the obvious way. Finally, we note these maps commute with the evaluation map from the localisation algebra to the Roe algebra - since $ev(e_{i}f)=(e_{i}f)(0)=f(i)(0)=e_{i}ev(f)$ for each $i$.

We now return to examining the element $y$. First, by definition of the limit in $d$, we can suppose without loss of generality that $y\in K_{0}C^{*}_{L}(P_{d}(G),\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G}$. Now $y$ is a uniformly continuous map from $[0,\infty)$ to $C^{*}(G,\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G}$, which we know gives a non-zero class in the K-theory. So, by picking a large enough $r$, we can find a quantitative $r,\epsilon$ representative for $[y]$ in $K^{r,\epsilon}_{*}(C^{*}_{L}(P_{d}G,\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G})$.

We’re now done, however. Take the first $i$ such that $d_{i}>d$ and $r_{i}>r>r^{\prime}$ and then the class $i_{r,r_{i},\epsilon,\epsilon}(e_{i}y)=i_{r^{\prime},r,\epsilon,\epsilon}(x)$ by construction. This is impossible, by the properties of $x=(x_{i})_{i}$. This completes the proof. ∎

Definition 4.8**.**

Let $\{G_{m}\rightarrow G\}_{m}$ be an asymptotically faithful covering sequence. In this situation we can define a function $k_{m}:\mathbb{R}_{+}\times\mathbb{R}_{+}\times(0,\frac{1}{4})\rightarrow\mathbb{R}_{+}$ using the formula:

[TABLE]

The family $\{G_{m}\}$ has the surjectivity of the evaluation map with asymptotic controls if for every $\epsilon\in(0,\frac{1}{4})$ there is an increasing sequence $(d_{m})_{m}$ such that the value $R_{m}:=\{R\mid k_{m}(d_{m},R,\epsilon)\leq r_{m}\}$ tends to infinity in $m$.

Finally, we need a lemma that shows the strength of operator norm localisation with asymptotic controls:

Lemma 4.6**.**

Let $\{G_{m}\rightarrow G\}_{m}$ be an asymptotically faithful covering sequence satisfying asymptotically controlled operator norm localisation where either all the $G_{m}$’s or $H_{m}$’s are isomorphic to a fixed group $G$. Then for every $R>0$ the largest natural number $n_{m,R}$ that satisfies $R+f_{m}(n_{m,R}R)\leq r_{m}$ tends to infinity in $m$.

Proof.

Suppose not. Then we have an $R>0$ such that $n_{m,R}$ is bounded above by some $o\in\mathbb{N}$ on all $m$ larger than some $m_{R}$. Now fix $S=(o+1)R$. Asymptotically controlled operator norm localisation provides us an $m_{S}$ such that $S+f_{m}(S)\leq R_{m}\leq r_{m}$. However, this shows that $o+1\leq n_{m}\leq o$ for all those $m\geq\max(m_{S},m_{R})$, as $R+f_{m}((o+1)R)=R+f_{m}(S)\leq S+f_{m}(S)\leq r_{m}$ for all such $m$. ∎

A consequence of this is that we can suppose that the constant $c_{m}$, which we supposed was uniform, can actually be chosen to tend to $1$ as $m$ tends to infinity.

Theorem 4.7**.**

Let $\{G_{m}\rightarrow G\}_{m}$ be an asymptotically faithful covering sequence satisfying asymptotically controlled operator norm localisation and suppose that the family $\{G_{m}\}_{m}$ has surjectivity of the evaluation map with asymptotic controls. Then the evaluation map for $G$ is quantitatively surjective.

Proof.

Again, we do $*=0$. Let $x\in K_{0}(C^{*}(G)^{G})$, let $R,\epsilon$ be chosen to witness $x$ in $K^{R,\epsilon}_{0}(C^{*}(G)^{G})$ and let $c_{m}$ be the asymptotic operator norm localisation constants, chosen to tend to $1$ as $m$ tends to infinity. Now, choose $m_{c}$ large enough that $\frac{1}{c_{m}}\epsilon<\frac{1}{4}$ for all $m\geq m_{c}$. Now we have the conditions to apply Lemma 4.1. We show there is an $R^{\prime}$ such that $i_{R,R^{\prime},\epsilon,\frac{\epsilon}{c_{m}}}(x)$ is in the image of $ev_{R^{\prime},\frac{\epsilon}{c_{m}}}$.

To see this, choose an $m$ large enough that $R\leq r_{m}$, $k_{m}(R,\epsilon)\leq r_{m}$ and $m\geq m_{c}$ and consider $\phi_{m}(x)$. As $G_{m}$ satisfies $QS(d_{m},R,R^{\prime},\frac{\epsilon}{c_{m}},\epsilon^{\prime})$ for some $R^{\prime}\leq r_{m}$ and $\epsilon^{\prime}$, we can find a class $y$ in $K_{0}^{R^{\prime},\epsilon^{\prime}}(C^{*}_{L}P_{d_{m}}G_{m}^{G_{m}})$ such that $ev(y)=i_{R,R^{\prime},\frac{\epsilon}{c_{m}}}(\phi_{m}(x))$. Using Lemma 4.2, we know that moving $y$ back into $K_{0}^{R^{\prime},c_{m}\epsilon^{\prime}}(C^{*}_{L}P_{d_{m}}G^{G})$, obtaining an element $\phi_{m}^{-1}(y)\in K_{0}^{R^{\prime},\epsilon}(C^{*}_{L}P_{d}G^{G})$.

If we evaluate $\phi_{m}^{-1}(y)$, then will obtain $i_{R,R^{\prime},\epsilon,c_{m}\epsilon^{\prime}}(x)$ as $\phi_{m}$ commutes with evaluation, and $\phi_{m}^{-1}(y)=\phi^{-1}(y(0))$ by definition. This completes the proof. ∎

4.4. Quantitative injectivity of the evaluation map framed using Baum–Connes with coefficients

In this section we show that the quantitative injectivity of the evaluation map can be seen as a consequence of the injectivity of a Baum–Connes with a specific set of coefficients, as with surjectivity in the previous setting. Again, this is based on an argument presented in Oyono-Oyono–Yu [OOY15].

Definition 4.9**.**

For constants $d,d^{\prime},r,r^{\prime}>0,\epsilon\in(0,\frac{1}{4})$ we define the condition $QI(d,d^{\prime},r,r^{\prime},\epsilon)$ to be: for all $x\in K^{r,\epsilon}_{*}(C^{*}_{L}(P_{d}(G))^{G})$ then $ev_{d,r,\epsilon}(x)=0$ in $K^{r,\epsilon}_{*}(C^{*}(G)^{G})$ implies that $i_{r,r^{\prime},d,d^{\prime},\epsilon}(x)=0$ in $K^{r^{\prime},\epsilon}_{*}(C^{*}_{L}(P_{d^{\prime}}(G))^{G})$. We say a group $G$ satisfies quantitative injectivity of the evaluation map if for all $d,r>0,\epsilon\in(0,\frac{1}{4})$ we can find $d^{\prime}>d,r^{\prime}>r$ such that $QI(d,d^{\prime},r,r^{\prime},\epsilon)$ holds for $G$.

Theorem 4.8**.**

Let $G$ be a finitely generated discrete group. If the Baum–Connes conjecture for $G$ with coefficients in $\ell^{\infty}(\mathbb{N},\mathcal{K})$ holds, then the evaluation map for $G$ is quantitatively injective.

Proof.

As in the corresponding statement from [OOY15] we aim for a contradiction. Suppose that the Baum–Connes conjecture holds for $G$ with coefficients in $\ell^{\infty}(\mathbb{N},\mathcal{K})$, but that the evaluation map is not quantitatively injective.

In this situation, we can find a pair $r,d>0$ such that for some $\epsilon(0,\frac{1}{4})$ such that for every $d^{\prime}>d,r^{\prime}>r$ the statement $QI(d,d^{\prime},r,r^{\prime}\epsilon)$ does not hold, i.e we can find a class $x_{d^{\prime},r^{\prime}}$ that vanishes under $ev_{r,\epsilon,d}$ that never vanishes on the right hand side under $i_{r,r^{\prime},d,d^{\prime},\epsilon}$.

To make use of this, pick an $\epsilon\in(0,\frac{1}{4})$ and obtain the corresponding scales $r$ and $d$. From here, pick sequences $(d_{i})_{i}$, $(r_{i})_{i}$ with each $d_{i}>d$, $r_{i}>r$ that tend to infinity in $i$, and then obtain the collection of classes $x_{i}\in K^{r,\epsilon}_{*}(C^{*}_{L}(P_{d}(G)$ that witness the failure of $QI(d,d_{i},r,r_{i},\epsilon)$ for $G$.

We will wrap these up into a single class in the quantitative K-group $K^{r,\epsilon}_{*}(C^{*}_{L}(P_{d}(G),\ell^{\infty}(\mathbb{N},\ell^{\infty}(G,\mathcal{K}))^{G})$ - to do this we will make use of the the naturality of the localisation algebra with respect to coefficients - recall the isomorphism we used during the proof of Theorem 4.5:

[TABLE]

This was constructed from the projection maps in the coefficient algebras and composition.

The idea is now clear. We take our sequence of classes $(x_{i})_{i}$ (which live in the $(r,\epsilon)$-K-theory of the right hand side) and we pull them back to a single class $z$ in the $(r,\epsilon)$-K-theory on the left hand side under this isomorphism. Since evaluation commutes with this isomorphism (by definition), we know that $ev_{d,r}(z)=0$. Pushing this class into $K_{*}(C^{*}_{L}(G,\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G})$, we know that actually $i_{r,d,\epsilon}(z)=0$ due to our assumptions on the evaluation map being an isomorphism for $G$ with coefficients in $\ell^{\infty}(\mathbb{N},\mathcal{K})$.

We’re done however, as we can now find a pair $d^{\prime},r^{\prime}>0$ such that $i_{r,r^{\prime},d,d^{\prime},\epsilon}(z)$ is [math] in $K^{r^{\prime},\epsilon}_{*}(C^{*}_{L}(P_{d^{\prime}}(G),\ell^{\infty}(\mathbb{N},\mathcal{K}))^{G})$. We obtain a contraction by unpacking via the isomorphism above. After doing this, we know that $i_{r,r^{\prime},d,d^{\prime},\epsilon}(x_{i})=0$, however by considering the first $d_{i}>d^{\prime},r_{i}>r^{\prime}$ we know that $0=i_{r^{\prime},r_{i},d^{\prime},d_{i},\epsilon}(i_{r,r^{\prime},d,d^{\prime},\epsilon}(x_{i}))=i_{r,r_{i},d,d_{i},\epsilon}(x_{i})\not=0$ by construction.

∎

Definition 4.10**.**

Let $\{G_{m}\rightarrow G\}_{m}$ be an asymptotically faithful covering sequence. In this situation we can define a function $l_{m}:\mathbb{R}_{+}\times\mathbb{R}_{+}\times(0,\frac{1}{4})\rightarrow\mathbb{R}_{+}\times\mathbb{R}_{+}$ using the formula:

[TABLE]

The family $\{G_{m}\}$ has the injectivity of the evaluation map with asymptotic controls if for every $\epsilon\in(0,\frac{1}{4})$ the supremum $L_{m}=\sup(d,r)\mid l_{m}(d,r,\epsilon)\leq(r_{m},r_{m})$ tends to infinity in both coordinates as $m$ tends to infinity.

Theorem 4.9**.**

Let $\{G_{m}\rightarrow G\}_{m}$ be an asymptotically faithful covering sequence satisfying asymptotically controlled operator norm localisation and suppose that the family $\{G_{m}\}_{m}$ has injectivity of the evaluation map with asymptotic controls. Then the evaluation map for $G$ is quantitatively injective.

Proof.

Let $c$ denote the asymptotic operator norm localisation control. Then let $d,r>0,\epsilon\in(0,\frac{1}{4})$ be given, and choose $m$ large enough that $d,r$ are smaller than $r_{m}$ and $(d,r)\leq L_{m}$ so that firstly we can apply Lemma 4.1 in the setting of localisation algebras - that is to say the map $\phi_{m}$ induces a map:

[TABLE]

and secondly we know that for $c\epsilon$, that the evaluation map satisfies $QI(d,d_{m},r,s_{m},c\epsilon)$, where both $d_{m}$ and $s_{m}$ are less than $r_{m}$.

Now we consider an element $x$ in $K^{r,\epsilon}_{*}(C^{*}_{L}(P_{d}(G))^{G})$ and suppose it maps to [math] under $ev_{d,r}$. Then as lifting commutes with evaluation, we know that $\phi_{m}(x)$ evaluates to [math] under $ev_{d,r}$, and so we know that $i_{d,d_{m},r,s_{m},c\epsilon}(\phi_{m}(x))=0$.

The proof is complete however, since we can now apply $\phi_{m}^{-1}$, and we see that $i_{d,d_{m},r,s_{m},\epsilon}(x)=\phi_{m}^{-1}(i_{d,d_{m},r,s_{m},c\epsilon}(\phi_{m}(x)))=0$ by applying Lemma 4.1 and the fact that expanding commutes with lifting. ∎

5. Consequences

5.1. Sobolev norms and rapid decay

Definition 5.1**.**

The Sobolev $(2,s)-$norms for a length $l$ for $a\in\mathbb{C}G$ are defined as follows:

[TABLE]

Let $H^{s}G$ denote the Jolissant algebra on $G$, that is the closure of $\mathbb{C}G$ in the $2,s$-Sobolev norm.

A group $G$ has rapid decay if there is some $s$ such that $H^{s}G\subseteq C^{*}_{r}G$. This variant is a definition given by Lafforgue [Laf00].

The weighted norm is spacial and so plays well with the localised lifting maps $\phi$ and $\Phi$ of Observation 3.1.

Lemma 5.1**.**

If $\pi:G\rightarrow H$ is a surjective group homomorphism with injectivity radius $R$, then the map $\mathbb{C}G\rightarrow\mathbb{C}H$ induced by $\pi$ is isometric in the $\|.\|_{2,s}$ norm for elements with support contained in $B_{R}(e)$.

Note that this can be rephrased as $(H^{s}G)_{R}\cong(H^{s}H)_{R}$.

Proof.

The important thing is that the norm $\|.\|_{2,s}$ is spatially defined. For each $a\in(\mathbb{C}G)_{R}$, the image $\pi(a)=\sum_{h\in H}\sum_{g\in\pi^{-1}(h)}a_{g}[h]$. Since the support is contained in a ball that is injected through $\pi$, there is a unique $g\in G$ in the preimage of each $h\in B_{R}(e)$, and all other coefficients are [math]. A calculation with the norms, given that the length of the elements in the ball doesn’t change (since $\pi$ induces an isometry there), shows that the map is isometric for these elements. ∎

The previous lemma shows that

[TABLE]

is satisfied for every $a\in\mathbb{C}G$.

So we’ve proved:

Lemma 5.2**.**

Let $G$ be a direct limit of an asymptotically faithful covering sequence $\{\pi_{m}:G_{m}\rightarrow G\}_{m}$ where the sequence is an asymptotically faithful covering. Then

[TABLE]

is an isometric embedding.

∎

We can also try to get some control of the norms in the case that the group $G$ or the sequence $G_{m}$ have additional properties - we can then try and connect to the property of Rapid Decay to those properties.

Theorem 5.3**.**

Let $G$ be a direct limit of a sequence $(G_{m})_{m}$ such that

1. (1)

the sequence $\{\pi_{m}:G_{m}\rightarrow G\}_{m}$ is asymptotically faithful 2. (2)

the map $\phi$ is reduced norm continuous, 3. (3)

the $G_{m}$ have (RD) uniformly in $m$, i.e we can choose the $c$ and $s$ from Definition 5.1 independently of $m$.

then $G$ has property (RD).

Proof.

. As each $G_{m}$ has (RD) uniformly with the same $C,s>0$, we have $\|a_{m}\|_{op}\leq C\|a_{m}\|_{2,s}$, and thus the norms on $\mathbb{C}G$ satisfy:

[TABLE]

Where the first equality follows from the observation in the proof of Theorem 3.2, the second is uniform property (RD) and the final one is Lemma 5.1. This completes the proof. ∎

Note that the two conditions on $G$ mean that the map $\phi$ is isometric - in fact property (A) for $G$ gives more. Recall (an equivalent form) of property (A) is operator norm localisation [CTWY08], which is a sufficient condition to lift operator norm estimates through local liftings (something abstracted from Higson’s ideas [Hig99] and implemented also in Willett–Yu [WY12].

5.2. Lacunary hyperbolic groups

Another place where we can produce liftings that are continuous is in the class of lacunary hyperbolic groups. As a general reference to this important class of groups, one can look at Osin–Ol’shanskiĭ–Sapir [OOS09]. The definition can be found in Example 2.2. We work from there.

Using Theorem 3.2 we will prove that lacunary hyperbolic groups satisfy the property that $\lambda$ and $\lambda_{\infty}$ are weakly equivalent, or equivalently the maps $\Phi$ and $\phi$ are reduced norm continuous.

We first recall that hyperbolic groups have finite asymptotic dimension. This was shown initially by Roe [Roe05], who confirmed an idea of Gromov from the monograph [Gro93], but there are now many proofs - we use the one presented in Roe’s paper [Roe05], where for each $R>0$ he constructs a $(2R+2\delta_{m})$-bounded cover into at most $|S|^{6\delta_{m}}$ colours444this is an incredibly bad bound that’s just useful for this calculation.. In this case, the ONL number is at least $\frac{1}{|S|^{6\delta_{m}}}$.

We will need two last facts, the first is Lemma 3.1 and the second fact we need before we can prove the theorem is that a cover by uniformly bounded sets of $r$-multiplicity $k$ can be broken into at most $2k$ $r-$disjoint families. Recall the annular decomposition of a space.

Definition 5.2**.**

Let $X$ be a metric space, $x_{0}\in X$ and let

[TABLE]

Then $X=\cup_{k}A^{R}_{k}$ is called the annular decomposition of $X$.

Remark*.*

The annular decomposition of $X$ is useful as it allows us to divide the space into “odd” and “even” parts (which correspond to the annular parts with odd or even index, respectively). This breaking up makes colourings easier in some cases like below.

Lemma 5.4**.**

Suppose $\mathcal{U}$ is a uniformly bounded cover of $X$ with $2r$-multiplicity at most $k$ defined as a cover $\mathcal{U}_{i}$ of each annular region $A^{2r}_{i}$. Then $\mathcal{U}$ can be broken into at most $2k$ families that are $r$-disjoint.

Proof.

For each annular region, pick a maximal $2r$-separated subset $Y_{i}\subset A_{i}^{2r}$ - the balls of radius $r$ around these points covers $A_{i}^{2r}$, and so each meets at most $k+1$ members of the cover $\mathcal{U}_{i}$. Enumerate the elements of $Y_{i}$ (which is a finite set). For each $y_{j}\in Y_{i}$, let $E_{j}=\{U\in\mathcal{U}_{i}\mid U\cap B_{r}(y_{i})\}$ - note that each $U$ might belong to many $E_{j}$. Now colour the set $E_{0}$ using the colours $c\in\{0,...,k\}$. This induces a partial colouring of the $E_{j}$ for each $j>0$. Now colour the uncoloured elements of $E_{1}$ using colours from $\{0,...,k\}$ that haven’t been used yet. Extend this to a partial colouring of $E_{j}>1$. Proceed this way until all the sets $E_{j}$ have been coloured.

Since each $U\in\mathcal{U}_{i}$ belongs to at least one $E_{j}$, it has a colour - and that colour, once fixed isn’t reassigned - so it’s well defined colouring.

Let $U,V\in c\in\{0,...,k\}$. Then $U$ and $V$ don’t intersect the same ball $B_{r}(y_{j})$ - since they must be in different $E_{j}$. But suppose $d(U,V)<r$, then there is a point $y\in U$ and $x\in V$ such that $d(x,y)\leq r$. Let $B_{r}(y_{k})\cap U$ - but $d(y_{k},x)\leq d(y_{k},y)+d(x,y)<2r$, which is a contradiction to the definition of $Y_{i}$.

To extend this to the whole space, use two sets of colours - one for the odd annular regions and one for the even ones. Since the annular regions are at least $2r$ wide, no sets from one odd (resp. even) of the same colour can ever be $r$-close. ∎

Theorem 5.5**.**

Let $G=\lim_{m}G_{m}$ be a lacunary hyperbolic group. Then the sequence $\{G_{m}\rightarrow G\}_{m}$ has asymptotically controlled operator norm localisation with constant $c=1/2|S|$.

Proof.

It follows, combining the above observations and Lemma 5.4 that $\delta_{m}-$hyperbolic groups generated symmetrically by a set $S$ can have ONL number taken to be $c_{m}\geq\frac{1}{2|S|}$ with vectors that have diameter of support at most $g(R)\leq 6\delta_{m}R+f(6\delta_{m}R)=18\delta_{m}R+2\delta_{m}$.

If we can show that the supremum of $R$ that satisfy the inequality

[TABLE]

goes to infinity as $m$ does, then we’re done. Rearranging, we find that:

[TABLE]

By the assumption on the relationship between $\delta_{m}$ and $r_{m}$ we can conclude that this value tends to infinity as $m$ does.

We can therefore apply Theorem 3.2 with constants $R_{m}$ and $c=\frac{1}{2|S|}$. ∎

As a consequence of this, we get the following statement:

Theorem 5.6**.**

Let $G$ a lacunary hyperbolic group, such that the family $\{G_{m}\}_{m}$ of limiting terms satisfy the injectivity and surjectivity of the evaluation map with asymptotic controls. Then the Baum–Connes assembly map is quantitatively an isomorphism for $G$.

Proof.

For lacunary hyperbolic groups with the asymptotic controls assumption, the result follows and Theorem 5.5, Theorem 4.7 and Theorem 4.9. ∎

Naturally, this extra condition concerning “Baum–Connes with asymptotic controls” could prove problematic. However, using the following argument we can produce many examples of sequences with this property that also do not have property (A), which allows us to apply this theorem in the wilderness.

5.2.1. Constructing examples with asymptotic controls

We will treat the small cancellation conditions required to make the construction and also the associated small cancellation theory that provides the embeddings as a black box - these results and results of their kind can be found explained in [LS01], or [Ol’91] or [Gru14] - these are all have an analogous flavour and depend on the type of small cancellation condition used in the construction. We recall the three flavours via references below, making note to separate the examples where different.

• •

Graphical small cancellation [Gro03],[Oll06],[Gru14]

• •

Geometric small cancellation [Gro03], [AD08]

• •

Graded small cancellation [Ol’91], [OOS09]

With this idea in mind we outline a black box technique for taking small cancellation lacunary hyperbolic groups and produce coverings of those groups that are again lacunary hyperbolic and satisfy the asymptotic control of the Baum–Connes conjecture in the case of surjectivity. A method for injectivity is not sketched here but is similar to what is below.

Method 5.1**.**

(A general approach for surjectivity)

• •

Let $G=\langle\mathcal{S}\mid\mathcal{R}\rangle$ be a small cancellation presentation that produces a lacunary hyperbolic group - note that in the case that $G$ is classical or graphical small cancellation, this means that the set of lengths $\{|R|\mid R\in\mathcal{R}\}$ is sparse in $\mathbb{R}$ [OOS09], [Gru14] - for graded or other small cancellation conditions, this may be more complicated.

• •

We produce the groups $G_{m}$ we will use by modifying the presentation of $G$. We will build two sequences of real numbers and a sequence of groups as follows.

1. (1)

Let $I=(a,b)$ be an interval. Let $\mathcal{R}_{I}=\{r\in\mathcal{R}\mid|r|\in I\}$. For $a=0$, we just call the set $\mathcal{R}_{b}$. 2. (2)

Fix $r_{0}>0$, $\epsilon_{0}>0$, then let $G_{0}=\langle\mathcal{S}\mid\mathcal{R}_{r_{0}}\rangle$ 3. (3)

$G_{0}$ is hyperbolic, and so it satisfies $QS(r_{0},t_{0},\epsilon_{0},\epsilon^{\prime}_{0})$. Set $\epsilon_{1}=\epsilon^{\prime}_{0}$ and pick $r_{1}>>2t_{0}$. Then take $G_{1}=\langle\mathcal{S}\mid\mathcal{R}_{r_{0}}\sqcup\mathcal{R}_{(2t_{0},r_{1}}\rangle$. This group is again hyperbolic. We can choose $r_{1}$ such that the set of relators contains strictly more elements than the set $\mathcal{R}_{r_{0}}$. 4. (4)

We proceed inductively - let $\mathcal{T}_{m-1}$ be the set of relators of $G_{m-1}$. Let $r_{m-1}$ be the longest relator and let $\epsilon_{m-1}$ be given. Then solve Baum–Connes on the scale of the longest relator as before - find $t_{m-1}$, $\epsilon_{m-1}^{\prime}$ such that $QS(r_{m-1},t_{m-1},\epsilon_{m-1},\epsilon^{\prime}_{m-1})$ holds. Set $\epsilon_{m}=\epsilon^{\prime}_{m-1}$ and pick $r_{m}>>t_{m-1}$, then let $G_{m}$ be defined by the presentation $\langle\mathcal{S}\mid\mathcal{T}_{m-1}\sqcup\mathcal{R}_{(t_{m-1},r_{m})}\rangle$. 5. (5)

Let $H$ be the limit of the sequence of quotients $\dots\rightarrow G_{m-1}\rightarrow G_{m}\rightarrow\dots$. The family $G_{m}$ asymptotically faithfully covers $H$, so $H$ is lacunary hyperbolic. By construction, it satisfies surjectivity of Baum–Connes with asymptotic controls.

The method can be adapted to situations where the labellings that give us relators come from families of finite graphs such as graphical or geometric small cancellation - we outline how to do this - it makes the choices in a more integrated way.

Method 5.2**.**

(in the case of graphs for surjectivity)

• •

Fix $\{X_{m}\}_{m}$ a sequence of finite graphs with bounded degree, $\mathcal{S}$ a finite alphabet and let $\mathcal{L}$ be a small cancellation labelling of the edges of $X=\sqcup X_{m}$, with component labellings $\mathcal{L}_{m}$. Let $\mathcal{R}$ be the set of words that we can read from a cycle in the edges of $X$ and let $\mathcal{R}_{m}$ be the labels of cycles that appear in the $m-th$ graph. Let $G$ be the graphical small cancellation group $\langle\mathcal{S}\mid\mathcal{R}\rangle$, which if the lengths of cycles are sufficiently sparse, is lacunary hyperbolic.

• •

Fix $r_{0}=$ the length of the longest cycle in $G_{0}=\langle\mathcal{S}\mid\mathcal{R}_{0}\rangle$ and take $\epsilon_{0}\in(0,\frac{1}{4})$. Now proceed to find $t_{0},\epsilon^{\prime}_{0}$ such that $QS(r_{0},t_{0},\epsilon_{0},\epsilon^{\prime}_{0})$ holds in $G_{0}$.

• •

The difference between this method and the previous one is that we have an easier way to select the next constants. Pick $m_{1}$ such that the length of the shortest cycle in $X_{m_{1}}$ is much smaller than $t_{0}$, and set $\mathcal{T}_{1}=\mathcal{R}_{0}\sqcup\mathcal{R}_{m_{1}}$. Set $G_{1}=\langle\mathcal{S}\mid\mathcal{T}_{1}\rangle$.

• •

Now proceed inductively as before. Suppose we’ve got $G_{n-1}$ and know $\epsilon^{\prime}_{n-1}=\epsilon_{n}$, then we can construct $\mathcal{T}_{n}$ by finding $t_{n-1},\epsilon^{\prime}_{n}$ that will satisfy $QS(r_{n-1},t_{n-1},\epsilon_{n},\epsilon^{\prime}_{n})$ holds in $G_{n-1}$, then selecting $m_{n}$ large enough that the shortest cycle in $X_{m_{n}}$ is much longer than $t_{n-1}$. We then set $G_{n}$ to be given by the presentation $G_{n}=\langle\mathcal{S}\mid\mathcal{T}_{n-1}\sqcup\mathcal{R}_{m_{n}}\rangle$.

• •

As before, we can take the limit of this inductive construction and denote it $H$. This group is actually what we could obtain by using the subfamily of the finite graphs $\{X_{m_{n}}\}_{n}$ and the associated labelling. It’s a lacunary hyperbolic group that covers $G$ and the sequence satisfies quantitative surjectivity of the Baum–Connes assembly map with asymptotic controls.

Using this, and the fact that both graphical and geometric small cancellations labellings exist (again, in the geometric case this is Gromov [Gro03], or Arzhantseva–Delzant [AD08] and in the graphical case this is Osajda [Osa14]) for certain families of finite graphs of large girth. A similar inductive setup can be used to produce injectivity with asymptotic controls - combining them at each stage would give both situations simultaneously.

Recall that a covering of a group $G$ is a group $H$ that quotients onto $G$. With that in mind we obtain the following theorem:

Theorem 5.7**.**

The following hold:

1. (1)

Every torsion free graphical small cancellation C(7) lacunary hyperbolic group $G$ has a covering $H$ that has a quantitatively isomorphic Baum–Connes assembly map. 2. (2)

Every (torsion free) lacunary hyperbolic group $G$ constructed from a geometric small cancellation C(7) labelling of a sequence of finite graphs of large girth has a covering that has a quantitatively isomorphic Baum–Connes assembly map.

∎

We have two remarks. firstly, following the construction presented in Arzhantseva–Delzant [AD08] carefully, Theorem 5.7.(2) contains many groups with property (T). We also note that in the second case, Gruber–Sisto [GS18] showed the groups occurring in (2) cannot be quasi-isometric to any group obtained from a graphical presentation, i.e in condition (1).

These are not the only situations in which we can apply this idea - we can also apply it in the setting of graded small cancellation as in Osin–Ol’shankiĭ–Sapir [OOS09]. This way, we can construct a Tarski Monster as in Theorem 4.7 of [OOS09]. In particular we can obtain torsion groups that satisfy the Baum–Connes conjecture.

Theorem 5.8**.**

For every hyperbolic group $G$ there is a graded small cancellation quotient $H$ that:

1. (1)

is lacunary hyperbolic; 2. (2)

is a Tarski monster (i.e every proper subgroup is finite cyclic); 3. (3)

the Baum–Connes assembly map for $G$ is quantitatively an isomorphism.

If we take the remarks of Arzhantseva–Delzant after Theorem 7.7 in [AD08] in combination, we can obtain Tarski monsters with property (T), contain expanders and that also satisfy the Baum–Connes conjecture quantitatively.