Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture
aa r X i v : . [ m a t h . G R ] A ug CONTROLLED ANALYTIC PROPERTIES AND THEQUANTITATIVE BAUM–CONNES CONJECTURE
MARTIN FINN-SELL
Abstract.
We show that the classical Baum–Connes assemblymap is quantitatively an isomorphism for a class of lacunary hy-perbolic groups, and we explain how to see that this class containsmany examples of groups that contain graph sequences of largegirth inside their Cayley graphs and therefore do not have prop-erty (A). This includes the known counterexamples to the Baum–Connes conjecture with coefficients, as well as many other monstergroups that have property (T). Introduction
This paper concerns the Baum–Connes conjecture for a specific sub-class of lacunary hyperbolic groups . This class has appeared all over theliterature at the limits of geometric group theory, and has a thoroughtreatment in the work of Osin–Ol’shanski˘ı–Sapir in [OOS09]. What ledthe author to consider this class is the following vague question: “Whatare discrete groups that contain expanders in their Cayley graph like?”At this point, the only examples we have of groups that containexpanders, or more generally sequences of graphs of large girth, areconstructed using the methods of small cancellation - specifically theycome from different small cancellation labellings of sequences of finitegraphs 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 cancel-lation (although of a different flavour), were shown to contain exam-ples with the Haagerup property - this is due to work of Arzhantseva–Osajda [AO14] and Osajda [Osa14].
Date : August 2019.
Key words and phrases.
Lacunary hyperbolic groups, Gromov monster groups,Baum–Connes conjecture, quantitative K-theory.
Once one has labelled finite graphs where the labellings satisfying asmall cancellation condition, the general machinery of small cancella-tion theory gives rise to an embedding of the graphs into the resultingfinitely generated infinite group. The strength of this embedding de-pends on the controls provided by the corresponding Van Kampen typetheorem that appears for that particular type of labelling. The range isfrom weak embeddings [HLS02], or almost quasi-isometric embeddingsat a large scale [Gro03, AD08], or all the way through to isometricembeddings [Gro03, Oll06, Gru14].Currently, it is possible to use either a geometric small cancella-tion labelling (as in Gromov [Gro03]) or a graphical small cancellationlabelling, and these have been shown to have a very different char-acter, for instance the groups that come from these labellings cannotbe quasi-isometric (this is due to Gruber–Sisto [GS18]). The methoditself is very flexible and compatible with other small cancellation con-structions possible in the literature - see for instance the remarks ofArzhantseva–Delzant [AD08].It is often convenient to work with the graphical small cancellationas it provides either coarse or isometric embeddings - and the detailsof this can be seen nicely in the thesis work of Gruber, which appearsin [Gru14]. However, if one is interested in producing examples withproperty (T) currently one needs to use the method of Gromov [Gro03].The graphical C(7) labellings, which exist due to Osajda [Osa14] canalso be obtained for graphs that satisfy a wall type structure as consid-ered in Arzhantseva–Osajda [AO14], and this means that the resultsof [AO14] can be applied to obtain groups that are not property (A),but do have the Haagerup property. This end of the spectrum is ofless interest to us in this paper purely because it subsumes the directcalculations 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 smallcancellation distinctions. The key thing to remember is that the phi-losophy espoused by small cancellation does not change in the con-structions. The technology exists and there is now a general pattern to
ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 3 how the theory works. The difficulty, after the theory is developed , isshowing 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 theselabelling results fall into this class. By focusing away from the smallcancellation and proving slightly more general results, we hope that itmay be adaptable in the future.The remainder of what comes is interpreting the word “like” in theinitial question above, as it can be done in a variety of settings. We cansay, for instance, that groups with expanders in them do not coarselyembed into a Hilbert space (one of the initial motivations for construct-ing them). However, such groups also give us the only known coun-terexamples for the Baum–Connes conjecture with coefficients - thisis present in the paper of Higson–Lafforgue–Skandalis [HLS02] withvery general statements that rely on a very basic construction theyexploit over and over, in particular in the case of groups that containexpanders.What is unknown, however, is if they satisfy the classical version ofthe Baum–Connes conjecture, which asks only about the trivial coeffi-cients C - that is truly a statement about the representation theory ofthe groups in question.We prove that in this most basic form, the Baum–Connes conjectureholds for lacunary hyperbolic groups when the lacunarity is sufficientlyfast. Theorem A.
Let G be a lacunary hyperbolic group satisfying the Baum–Connes conjecture with asymptotic controls. Then the Baum–Connesconjecture is (quantitatively) true for G . We introduce all the terms throughout the paper at their critical mo-ments, and give plenty of citations for the known properties or notionsthat we recall throughout. Please understand I am not trying to minimise the efforts of people who developthis theory, I am just talking in a “once the dust has settled” practical sense.
MARTIN FINN-SELL
The method we introduce here to prove this statement is inspiredby two main sources: the work of Willett–Yu [WY12] on the coarseBaum–Connes conjecture for large girth graph sequences and the workof Yu [Yu98] on the coarse Baum–Connes conjecture for groups of finiteasymptotic dimension.The former paper takes a method of Higson [Hig99] that appearedalso in Higson–Lafforgue–Skandalis [HLS02] and removed the grouptheory aspects to a large degree. The main concept they introduced,the notion of an asymptotically faithful covering sequence , is defined inSection 2. This is used there to construct lifting maps at the level of theRoe algebra - giving us the so called “Higson trace” (first constructed in[Hig99]). This notion of lifting is exploited to obtain the representationtheory results of Section 3 that we apply in the setting of lacunaryhyperbolic groups.The latter paper of Yu makes use of a quantitative or persistentversion of K-theory for algebras that have some kind of scale or length.This quantitative theory gives more refined information that is verypossible to transfer using localised lifting maps like those that appearin the Higson trace. This quantitative theory has been conveniently andsystematically studied by Oyono-Oyono–Yu [OOY15] and this gives usa framework and some general results to begin taking advantage of.Our strategy, overall, is to cut the K-theory group we are interestedin into pieces that we can lift through neatly constructed maps thatact like homomorphisms on some specific scale. We then convert thesegroups into something workable, then piece them back together againusing 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 exam-ples that satisfy our condition of having asymptotic controls, and statesample theorems that come from applying this new method along withthe other constructions in the literature to obtain interesting groupsfor which the technique presented here provides an optimal outcome.This can be found in Section 5. A summary of those results is statedbelow:
ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 5
Theorem B.
The following holds: (1)
Every torsion free graphical small cancellation and geometricsmall cancellation C(7) lacunary hyperbolic group has a cover-ing that has a quantitatively isomorphic Baum–Connes assem-bly map. (2)
For every hyperbolic group G there is a graded small cancellationquotient H that (a) is lacunary hyperbolic; (b) is a Tarski monster (i.e every proper finitely generated sub-group is cyclic of a fixed prime order); (c) the Baum–Connes assembly map for G is quantitatively anisomorphism. We note that this concludes the result for many groups that haveproperty (T), as the starting group G or the seed group in a geometricsmall cancellation construction can be chosen to have it. This resultrelies heavily on the work of Lafforgue [Laf12] concerning the Baum–Connes conjecture with coefficients for hyperbolic groups, as it relieson the hyperbolic limiting terms having quantitative Baum–Connesassembly isomorphisms - which at this point in time can only be ob-tained as a consequence of the assembly conjecture for a set of specificcoefficients.Finally, we remark also that at this point we don’t have a goodmethod for including coefficients into this method. The issue withcoefficients (aside from the fact that many of these groups have co-efficients for which the conjecture is false) is that they may not be“limits” of coefficients of the hyperbolic limiting terms - and makingprecise what we mean by that is also a difficult proposition we’ve yetto fully understand. Acknowledgements
The author would like to thank Goulnara Arzhantseva for patientlyreading early versions of this text, in particular for accuracy in thesmall cancellation theory as this is something in which the author is
MARTIN FINN-SELL not an expert, and Rufus Willett for a timely remark concerning The-orem 3.2. He also thanks J´an ˇSpakula for his ongoing support andencouragement, and Alain Valette for his many comments about thetext. 2.
Asymptotically faithful covering spaces
Definition 2.1.
Let π : X → Y be a covering map of metric spaces.The injectivity radius of π is the largest R > R in X are mapped isometrically to balls of radius R in Y . Definition 2.2.
Consider two metric families X = { X m } m and Y = { Y m } m . We say X asymptotically faithfully covers Y if there existcovering maps π m : X m → Y m such that the injectivity radius R m for π m tends to infinity in m .This notion was introduced by Willett and Yu in [WY12] - to dealwith a more general version of Example 2.1 below. For what we willdo in this paper, we will consider situation that either X or Y is aconstant family - i.e X m (resp. Y m ) are all the same metric space. Thisis more general than the situation of [WY12], as it captures certaindirect 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 examplesthat motivate our work.
Example 2.1. (Box spaces) For a residually finite finitely generateddiscrete group G , let N = { N i } i be a nested family of normal subgroupsof finite index and trivial intersection . Then the family { π i : G → G/N i } i is an asymptotically faithful covering family. To see this, wemust fix a metric on G from a finite generating set S , and then use themetric on each G/N i that comes from π i ( S ). Then for any R , we canfind some i such that B R , the ball of radius R in G is not containedin N i - hence it maps injectively to G/N i . By choosing i large enough,this can be made isometric. the existence of such a family is the definition of a group being residually finite ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 7
This covering family allows us to study the box spaces (cid:3) G , whichare infinite metric spaces constructed from the disjoint union of familyof finite quotients G/N i with any metric d satisfying • d | G/N i is the metric coming from π i ( S ), • d ( G/N i , G/N j ) → ∞ as i + j → ∞ .Any two metrics that satisfy i) and ii) are coarsely equivalent, and thuswe need not be more specific about the rates of divergence.The asymptotically faithful property can be used in this case toconstruct counterexamples to the coarse Baum–Connes conjecture byeither taking groups such that the finite quotients form an expanderfamily [Hig99] (such as groups with property (T)). It can be used ex-plicitly to comprehensively study sequences of graphs of large girth ,which is the example above when the group G is taken to be a freegroup [WY12] [FSW14]. These examples motivate the rest of this text. Example 2.2. (Lacunary hyperbolic groups) This is a class of finitelygenerated discrete groups that many monster groups from group theory[OOS09] end up living - it is a place where many outputs of smallcancellation theory [Gro03, AD08] naturally occur. The definition is asfollows:
Definition 2.3.
Let G be a direct limit of discrete groups G m via(necessarily surjective) quotient maps, and let π m : G m → G be thequotient map to the limit. G is lacunary hyperbolic if:(1) the injectivity radius r m of π m tends to infinity as m does;(2) each G m is δ m − hyperbolic for some δ m > δ m ’s satisfy: δ m = o ( r m ).Note that baked into this definition is the condition that { π m : G m → G } m is an asymptotically faithful covering family.As a general reference to this important class of groups, one can lookat Osin–Ol’shanski˘ı–Sapir [OOS09]. In particular it contains the coun-terexamples to the Baum–Connes conjecture with coefficients [CTWY08,AD08] as well as many Tarski monster groups or other small cancella-tion monsters from geometric group theory [OOS09]. MARTIN FINN-SELL
Example 2.3. (Limit groups of Sela) Limit groups are a particularclass of finitely presented groups that arise in the study of first orderlogical theory of finitely generated groups - they can be defined asthose groups that share the same theory as a free group [CG05]. It isknown that such groups are precisely those groups G that admit anasymptotically faithful covering family { π m : G → G m } m where each G m is a finitely generated free group (of a fixed rank) - a condition inthis case that appears with the terminology “fully residually free” inthe literature. However, using the generators of G to generate each G m is necessary for the asymptotically faithful covering example, and thiswill not be a free generating set.After reducing this to a free generating set, one can see that theextra generators we would be including must be very long in the wordlength of the free group - longer than the injectivity radius. Thus, thesequence is not uniformly coarsely equivalent to a sequence of fixedrank free groups with the standard generating set - thus this conditionof 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 theoryresult concerning asymptotically faithful covering sequences. Firstthough, we recall the basic algebras we will attach to a group or metricspace and how they’re connected.
Definition 3.1.
Let X be a uniformly discrete metric space of boundedgeometry and let T be a bounded operator on ℓ X . The propagation of T is the smallest positive real number R such that the matrix entriesof h T δ x , δ y i vanish when d ( x, y ) > R .The collection of all finite propagation operators forms a ∗ -subalgebraof B ( ℓ ( X )), denoted by C [ X ]. Its completion, is called the uniformRoe algebra and is denoted by C ∗ u ( X ). Finally, to generate the full Roealgebra , denoted C ∗ X , we do exactly as above except we take operatorswith values in the compact operators on some separable Hilbert space.Let X be a metric space. Then the Roe algebra C ∗ X , (resp. uniformRoe algebra C ∗ u X ) can be graded by propagation using the closed linear ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 9 subspaces: C [ X ] R := { T ∈ C ∗ X | prop ( T ) ≤ R } . If G is a discrete group acting on X by isometries, then the subalge-bra of equivariant operators obtained by closing the finite propagationoperators that are equivariant, denoted by C ∗ X G is also graded byintersecting with the subspaces above. Note that this algebra, if theaction is free and proper, is Morita equivalent to the reduced group C ∗ -algebra of G (see for instance [WY12] - by no means an original orearliest reference).We can grade the group ring of C G of G directly using the followinglinear subspaces: C G R := { a ∈ C G | supp ( a ) ⊂ B R ( e ) } , where the support of a , supp ( a ), is the subset of elements of G suchthat the coefficients a g are not 0.Note that gradings interact in the following way as the subspaces C G R and C [ G ] GR are related by mapping a ∈ C G to the element T ∈ C [ G ] G with matrix entries T x,y = a x − y . This relationship is convenientfor the following construction. Observation 3.1.
Let X be a metric space and let π be a surjectivecovering map Y → X with injectivity radius r and deck transformationgroup G . Then for all R < r there is a linear mapΦ : C [ X ] R → ( C [ Y ] G ) R constructed using the entries:Φ( T ) y,y ′ = T π ( y ) ,π ( y ′ ) if d ( y, y ′ ) ≤ R, . Note that when Y and X are chosen to be discrete groups, and π isa quotient map, this can also be used to define, for each a ∈ C X R , alifting φ ( a ) ∈ C Y R by coefficients φ ( a ) y = a π ( y ) . or to land in the Roe algebra proper, one would, for instance, map to T ⊗ p , where p is a rank one projection and T is the operator before constructed from a . These maps induce are linear isomorphisms C [ X ] R ∼ = ( C [ Y ] G ) R and C X R ∼ = C Y R that respect composition, in the sense that that if P rop ( T )+ P rop ( S ) ≤ r , then Φ( T S ) = Φ( T )Φ( S ) in C [ Y ] G (resp. φ ( T S ) = φ ( T ) φ ( S ) if the elements S, T ∈ C G ).The relationship between the group ring and the finite propagationoperators using the matrix coefficient definition above commutes withthe maps φ and Φ 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 suchthat if prop ( T ) ≤ R then k Φ n ( T ) k ≤ k T k (resp. k Φ n ( T ) k = k T k ) forall n ≥ m .Continuity of this sort of map has been studied in the context of theBaum–Connes assembly conjecture, and was used explicitly by Hig-son [Hig99], Higson–Lafforgue–Skandalis [HLS02] and later Willett–Yu [WY12] to construct counterexamples to the coarse Baum–Connesconjecture. It also appears in the work of Gong–Wang–Yu [GWY08],Oyono-Oyono–Yu [OOY09] and Guentner–Tessera–Yu [GTY11] in thiscontext, and for the most part all of these results have the commontheme that they exploit an idea introduced in [CTWY08] called oper-ator norm localisation : Definition 3.3.
Let X be a uniformly discrete metric space withbounded geometry. Then X has operator norm localisation with con-stant c ∈ (0 ,
1) and control function f if: for every R > T ∈ C [ X ] R there is a unit vector η satisfying:(1) diam ( supp ( η )) ≤ f ( R ),(2) c k T k op ≤ k T η k .It’s known now due to work of Sako [Sak14] that this property isequivalent to Yu’s property (A), initially given in [Yu00] and surveyedthoroughly by Willett [Wil09].One useful fact we will recall from the literature is Proposition 3.3from Chen–Tessera–Wang–Yu [CTWY08], which states that the opera-tor norm localisation constant (i,e, the c in definition 3.3), once knownto exist can be taken to be any value: ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 11
Lemma 3.1.
If a metric space X has the operator norm localisationproperty then it has it with any constant c ∈ (0 , . (cid:3) The cost of performing this trick is that the diameter of the supportof the vector increases dramatically. In particular if one has operatornorm localisation with constant c and control function f , then one canget c ′ ∈ (0 ,
1) by using control function g , where g ( k ) = ( n − k + f ( nk )- where n is the smallest number such that c n ≥ c ′ .We now introduce the main tool we plan to use to study the regularrepresentation of well approximated groups. Definition 3.4. (The lifting limsup representation) Let { π m : G m → 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 forevery element of the group ring a ∈ C G we defined lifts φ m ( a ) ∈ C G m for each large enough m . Thus, if we equip each C G m with a norm,for instance the reduced one, we can define a norm on C G using thelimsup: k a k ∞ = lim sup m k φ m ( a ) k λ m . This representation exists by the Gelfand–Naimark–Segal theorem, (forinstance found in [Dav96]).One can ask if the representation λ ∞ obtained this way is related tothe left regular representation. We introduce the following notion tocontrol its behaviour and answer this question. Definition 3.5.
Let { π m : G m → H m } be an asymptotically faithfulcovering family where either the G m or H m are equal to a single finitelygenerated group G .If each G m has operator norm localisation with constant c and func-tions f m such that the number R m := sup { R | R + f m ( R ) ≤ r m } tendsto infinity in m then we say the sequence has operator norm localisationwith asymptotic controls .The goal of the remainder of this section is to prove the followingresult: Theorem 3.2.
Let { π m : G m → H m } be an asymptotically faithful cov-ering family where either the G m or H m are equal to a single finitelygenerated group G . Then the asymptotically faithful covering family { π m : G m → G } m has operator norm localisation with asymptotic con-trols then λ ∞ is weakly equivalent to λ .Proof. Let a ∈ ( C G ) R be given. Pick m large enough such that R + R m ≤ r m . Then operator norm localisation for G m provides a vector η m with diameter of its support smaller than f m ( R ) ≤ r m . This vectorsatisfies: c k φ m ( a ) k ≤ k a m η m k = k a ˜ η m k ≤ k a k where ˜ η m is the unit vector pushed through the quotient map to G and observing that a ˜ η m lifts to a m η m since the support is smaller than R + R m . It follows that lim sup m k a m k ≤ c k a k , which shows continuity.The same argument works for Φ.Now note that for any a ∈ ( C G ) R , there is always a vector of finitesupport v a constant c a ∈ (0 ,
1) such that c a k a k ≤ k av k . Choos-ing m large enough, we can as before lift a , v and av coefficient wiseusing φ m such that φ m ( av ) = φ m ( a ) φ m ( v ). Thus c a k a k ≤ k av k = k φ m ( a ) φ m ( v ) k ≤ k φ m ( a ) k . As c a is independent of m , we obtain theinequality also for the limsup. (cid:3) We’ll make use of this result in the context of the Baum–Connesconjecture in the next section.4.
What about the Baum–Connes conjecture?
Our goal in this section is to assemble a proof of the Baum–Connesconjecture for groups that are the base group in an asymptoticallyfaithful covering where each covering term satisfies the Baum–Connesconjecture with coefficients.We recall some definitions from Oyono-Oyono–Yu [OOY15].
Definition 4.1. (1) Let A [0 ,
1] be the C ∗ -algebra C ([0 , , A ). Thisis where homotopies between elements of A live. ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 13 (2)
A filtered C ∗ -algebra A is a C ∗ -algebra equipped with a familyof linear subspaces indexed by positive real numbers ( A r ) r> such that: • A r ⊆ A r ′ if r ≤ r ′ ; • A r is ∗ -closed; • A r A r ′ ⊆ A r + r ′ ; • the subalgebra ∪ r A r is dense in A .If A is unital, then we include 1 in A r for every r >
0. Theelements of A r are said to have propagation at most r .(3) This concept also extends to morphisms in multiple ways. Let A , and A ′ be filtered C ∗ -algebras by ( A r ) r and ( A ′ r ) r respec-tively. Then a homomorphism f : A → A ′ is filtered if f ( A r ) ⊆ A ′ r for every r and d − f iltered for some real number d if f ( A r ) ⊆ A ′ dr . Both of these maps interact favourably withwhat is to come.(4) Let ǫ ∈ (0 , ). An element p of a unital filtered C ∗ -algebra A is an ( r, ǫ ) quasi-projection if p ∈ A r , p = p ∗ and k p − p k ≤ ǫ .Let P r,ǫn ( A ) denote the collection of all ( r, ǫ ) quasi-projections in M n ( A ) - where ( M n A ) r = M n ( A r ) for every n . Note, that thecorner inclusions: p diag ( p,
0) are maps P r,ǫn ( A ) → P r,ǫn +1 ( A ),and we denote by P r,ǫ ∞ ( A ) the union of the P r,ǫn ( A )’s.(5) We can now do the same for unitaries: an element u of a unitalfiltered C ∗ -algebra A is a an ( r, ǫ ) unitary if u ∈ A r and both k uu ∗ − k and k u ∗ u − k are smaller than ǫ. We denote by U r,ǫn ( A )the set of all ( r, ǫ ) unitaries in M n ( A ). Similar to above, themaps u diag ( u,
1) give maps U r,ǫn ( A ) → U r,ǫn +1 ( A ), and wedenote the union by U r,ǫ ∞ ( A ).(6) We define relations on these (semi)groups using homotopies.A homotopy h of ( r, ǫ ) projections p, q (resp. unitaries u, v )is an ( r, ǫ ) projection (resp. unitary) for A [0 , h (0) = p, h (1) = q (resp. h (0) = u, h (1) = v ).(7) For a unital filtered C ∗ -algebra A , we define relations on P r,ǫ ∞ ( A ) × N and U r,ǫ ∞ ( A ): (a) if ( p, k ) , ( q, l ) ∈ P r,ǫ ∞ ( A ) × N , then ( p, k ) ∼ ( q, l ) if thereexists h ∈ P r,ǫ ∞ ( A [0 , h (0) = diag ( p, I k ) and h (1) = diag ( q, I l );(b) if u, v ∈ U r,ǫ ∞ ( A ), then u ∼ v if there exists h ∈ U r,ǫ ∞ ( A [0 , h (0) = u , h (1) = v .(8) We define the quantitative K-theory groups K r,ǫ ( A ) = P r,ǫ ∞ ( A ) × N / ∼ ,K r,ǫ ( A ) = U r,ǫ ∞ ( A ) / ∼ . (9) An R -quasi ∗ -homomorphism between filtered C ∗ -algebras A and B is a map f : A R → B R that is linear continuous map thatsatisfies f ( a ) f ( b ) = f ( ab ) for any two elements a ∈ A S , b ∈ A S ′ such that S + S ′ ≤ R . A C ∗ -algebra homomorphism between A and B is a R -quasi ∗ -homomorphism.We’ve constructed some examples of R -quasi ∗ -homomorphisms - themaps φ m that we used to lift operators in the group ring C G to op-erators in C G m , for an asymptotically faithful covering sequence G m .When the sequence has asymptotically controlled operator norm local-isation, the maps φ m are uniformly continuous (if m is large enough).We record some elementary facts and conclude a basic lemma inquantitative K-theory that we will use analogues of again and again inthe sequel. Lemma 4.1.
Let A and B be graded C ∗ -algebras and let f : A R → B R be a bijective R -quasi ∗ -homomorphism, where A R and B R are finitedimensional Banach spaces and such that the norm k f k is boundedabove by c ∈ (1 , ∞ ) . Then (1) f induces the following maps of quantitative K-theory groupsfor S = R/ and ǫ ∈ (0 , c ) : f ∗ : K S,ǫ ∗ ( A ) → K S,cǫ ∗ ( B ) → K S,ǫ ∗ ( A )(2) f − ∗ ◦ f ∗ = Id A S , ∗ , f ∗ ◦ f − ∗ = Id B S , ∗ Proof. As A R and B R are finite dimensional and f is a continuous linearbijection of Banach spaces, we also know that f − is continuous withnorm bounded below by c . The second part follows from the fact that ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 15 f is a bijection, and the induced maps described - the norm estimatesfollow from the fact that continuous maps are bounded for Banachspaces. (cid:3) The lifting map and assembly.
The goal of this section isto 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-homologygroups in question, then constructing the assembly map.
Definition 4.2. (K-homology) Let X be a metric space. A Hilbertspace H is called a X -module if there is a representation π : C ( X ) → B ( H ). It’s a standard X -module if π ( C ( X )) ∩ K ( H ) = { } , and non-degenerate if π is non-degenerate. An example of such a module wouldbe ℓ ( X ), when X is uniformly discrete with bounded geometry. • A K -cycle for X is a pair ( H , F ), where H is a non-degenerate,standard X module and F is a bounded operator on H satisfy-ing: F ∗ F − , F F ∗ − F, π ( f )] ∈ K ( H )for all f ∈ C ( X ) • A K -cycle for X is a pair ( H , F ) where H is a non-degenerate,standard X module and F is self adjoint bounded operator on H satisfying: F − F, π ( f )] ∈ K ( H )for all f ∈ C ( 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 rep-resentation σ : G → U ( H ) such that σ intertwines π and σ ( g ) commuteswith 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 µ , we will first consider a uniformlybounded cover U = { U i } and let η i be a partition of unity subordinateto U . Then define: ˜ F = X i √ η i F √ η i . This sum is • convergent in the strong operator topology; • equivalent to F as a K − homology cycle; • it is a multiplier of C [ X ] G , the equivariant Roe algebra of X .We can now consider the following matrix: I ( F ) := F F ∗ + (1 − F F ∗ ) F F ∗ F (1 − F ∗ F ) + (1 − F ∗ F ) F (1 − F ∗ F )(1 − F ∗ F ) F (1 − F ∗ F ) ! We observe that this matrix maps to ! under the evaluationmap to C from the standard unitisation of C [ X ]. Definition 4.3. (The index map) The index map µ : K G ∗ ( X ) → K ∗ ( C ∗ X G ) is defined by the formula: µ ([ F ]) = [ I ( ˜ F )] − ! . The Baum–Connes assembly map is then obtained by taking suffi-ciently 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–Connesassembly map is defined by the limit of the appropriate indices:
Definition 4.4. (The Baum–Connes assembly map) The assemblymap µ is defined by: µ : lim d K G ∗ ( P d ( G )) → lim d K ∗ ( C ∗ ( P d ( G )) G ) = K ∗ ( C ∗ G G ) . Localisation algebras.
This sections definitions and cited theo-rems are from Appendix A in Guentner–Tessera–Yu [GTY11] as wellas Yu [Yu97]. We recall some definitions and begin from there.
ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 17
Definition 4.5.
Let G be a discrete group, then the localisation alge-bra of G , denoted C ∗ L G is the collection of uniformly bounded maps: f : [0 , ∞ ) → C ∗ G such that prop ( f ( t )) tends to 0 as t tends to ∞ , equipped with thesupremum norm: k f k = sup t k f ( t ) k .This definition can be made equivariant by mapping into C ∗ G G , andthis is denoted by C ∗ L G G . Definition 4.6.
We grade the equivariant localisation algebra by con-sidering the linear spaces ( C ∗ L G G ) R , by asking that f ( t ) has propaga-tion less than R for all t . This grading allows us to define quantitativeK-theory groups.Note that the lifting map Φ m induces a map ( C ∗ L ( G ) G ) R → ( C ∗ L ( G m ) G m ) R ,which we will denote by Φ m,L . We can control the norm of this mapusing the norm of the original Φ m . Lemma 4.2.
Suppose that Φ m : ( C ∗ ( G ) G ) R → ( C ∗ ( G m ) G m ) R is acontinuous linear bijection with norm k Φ m k ≤ c , k Φ − m k ≤ d , c, d ∈ [1 , ∞ ) Then k Φ m,L k ≤ c and k Φ − m,L k ≤ d Proof.
We compute directly from the definition. Let f ∈ ( C ∗ L ( G ) G ) R - k Φ m,L ( f ) k = sup t k Φ m ( f )( t ) k ≤ c sup t k f ( t ) k = c k f k . This shows the desired inequality. The same argument run the otherway around completes the proof. (cid:3)
We can use this in conjunction with our norm estimates from asymp-totically controlled operator norm localisation and the K-theory state-ments 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 ∈ C ∗ L G GR : ev (Φ m f ) = (Φ m f )(0) = Φ m ( f (0)) = Φ m ( ev ( f ))where the middle equality is the definition of the lifting map on local-isation algebras. (cid:3) 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 assemblymap that is compatible with localisation algebras that is clearly definedin [Yu97, GHW05]. The main result we need is that the Baum–Connesconjecture for G with coefficients in A can be reformulated in terms oflocalisation algebras and evaluation maps. Theorem 4.4. (Theorem A.5 from [GTY11] ) The Baum–Connes con-jecture for G with coefficients in A is equivalent to the assertion thatthe evaluation map: lim d K ∗ ( C ∗ L ( P d ( G ) , A ) G ) → K ∗ ( C ∗ L ( G, A ) G ) is an isomorphism. A brief sketch of the argument for surjectivity.
The planis as follows.(1) We reformulate a quantitative statement about the Baum–Connesconjecture into the terminology of localisation algebras andquantitative evaluation maps.(2) We obtain K-theory isomorphisms from the maps φ and Φ in aquantitative way for the groups G m and G .(3) We move K-theory classes around carefully through these iso-morphisms, assuming we can control the evaluation isomor-phism enough to make that go.We work through these steps in the following sections.4.3. Quantitative surjectivity of the evaluation map framedusing Baum–Connes with coefficients.
In this section we showthat the quantitative surjectivity of the evaluation map can be seen asa consequence of the surjectivity of the Baum–Connes assembly mapwith a specific set of coefficients
Definition 4.7.
Let r ′ > , ǫ ′ ∈ (0 , ). The evaluation map ev :lim d →∞ ( C ∗ L G ) G → lim d →∞ ( C ∗ G ) G satisfies QS ( d, r ′ , r, ǫ ′ , ǫ ), where ǫ >ǫ ′ , if for every element x in K r ′ ,ǫ ′ ∗ ( C ∗ G ) G ) there is a y ∈ K r,ǫ ∗ ( C ∗ L P d ( G ) G ) ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 19 such that ev r,ǫ ( y ) = i r ′ ,r,ǫ ′ ,ǫ ( x ). The evaluation map is quantitativelysurjective if for every r ′ , ǫ ′ we can find d, r, ǫ such that QS ( d, r ′ , r, ǫ ′ , ǫ )holds.This condition appears stronger than asking for a surjection of theevaluation map itself, and it obviously implies this. We relate it toBaum–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 ℓ ∞ ( N , K )) holds for G , then theevaluation map for G is quantitatively surjective.Proof. We do ∗ = 0. Suppose that the evaluation map is not quan-titatively surjective. Then there is a r ′ , ǫ ′ such that no matter what d, r, ǫ we pick, QS ( d, r ′ , r, ǫ ′ , ǫ ) doesn’t hold for the evaluation map.Fix ( d i ) , ( r i ) with both going to infinity as i does and ǫ >
0, andthen take elements x i ∈ K r ′ ,ǫ ( C ∗ ( G ) G ) such that i r ′ ,r i ,ǫ ′ ,ǫ ( x ) is not inthe image of ev d i ,r i ,ǫ . We then wrap these elements up into a class x that belongs to the K-theory of C ∗ ( G, ℓ ∞ ( N )) G by identifying it as C ∗ r ( G, ℓ ∞ ( G, ℓ ∞ ( N , K ))).The surjectivity assumption on the assembly map for G with co-efficients in ℓ ∞ ( N , K ) and the equivalence provided in Theorem 4.4,provides us with an element y ∈ lim d K ( C ∗ L ( P d ( G ) , ℓ ∞ ( N , K )) G thatmaps onto x under ev . To complete the proof, we examine y carefullyusing the evaluation maps from ℓ ∞ ( N , K ) to K to test the element y .Recall that the evaluation maps e i : ℓ ∞ ( N , K ) → K are defined, foreach i , by e i f = f ( i ). These can be applied directly to elements of thelocalisation algebra C ∗ L ( G, ℓ ∞ ( N , K )) G in the following way: for every t ∈ [0 , e i y )( t ) := e i ( y ( t )) . These maps are continuous ∗ -homomorphisms on C ∗ L ( G, ℓ ∞ ( N , 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 startwith 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: e i, ∗ : K r,ǫ ∗ ( C ∗ L ( G, ℓ ∞ ( N , K )) G ) → K r,ǫ ∗ ( C ∗ L ( G ) G )for each i . This will also extend to each Rips complex, in the ob-vious way. Finally, we note these maps commute with the evalu-ation 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 definitionof the limit in d , we can suppose without loss of generality that y ∈ K C ∗ L ( P d ( G ) , ℓ ∞ ( N , K )) G . Now y is a uniformly continuous map from[0 , ∞ ) to C ∗ ( G, ℓ ∞ ( N , K )) G , which we know gives a non-zero class in theK-theory. So, by picking a large enough r , we can find a quantitative r, ǫ representative for [ y ] in K r,ǫ ∗ ( C ∗ L ( P d G, ℓ ∞ ( N , K )) G ).We’re now done, however. Take the first i such that d i > d and r i > r > r ′ and then the class i r,r i ,ǫ,ǫ ( e i y ) = i r ′ ,r,ǫ,ǫ ( x ) by construction.This is impossible, by the properties of x = ( x i ) i . This completes theproof. (cid:3) Definition 4.8.
Let { G m → G } m be an asymptotically faithful cov-ering sequence. In this situation we can define a function k m : R + × R + × (0 , ) → R + using the formula: k m ( d, R, ǫ ) = inf { R ′ | G m satisfies QS ( d, R, R ′ , ǫ, ǫ ′ ) for some ǫ ′ } . The family { G m } has the surjectivity of the evaluation map with as-ymptotic controls if for every ǫ ∈ (0 , ) there is an increasing sequence( d m ) m such that the value R m := { R | k m ( d m , R, ǫ ) ≤ r m } tends toinfinity in m .Finally, we need a lemma that shows the strength of operator normlocalisation with asymptotic controls: Lemma 4.6.
Let { G m → G } m be an asymptotically faithful coveringsequence satisfying asymptotically controlled operator norm localisationwhere either all the G m ’s or H m ’s are isomorphic to a fixed group G .Then for every R > the largest natural number n m,R that satisfies R + f m ( n m,R R ) ≤ r m tends to infinity in m . ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 21
Proof.
Suppose not. Then we have an
R > n m,R is boundedabove by some o ∈ N on all m larger than some m R . Now fix S =( o +1) R . Asymptotically controlled operator norm localisation providesus an m S such that S + f m ( S ) ≤ R m ≤ r m . However, this shows that o + 1 ≤ n m ≤ o for all those m ≥ max( m S , m R ), as R + f m (( o + 1) R ) = R + f m ( S ) ≤ S + f m ( S ) ≤ r m for all such m . (cid:3) 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 1as m tends to infinity. Theorem 4.7.
Let { G m → G } m be an asymptotically faithful coveringsequence satisfying asymptotically controlled operator norm localisationand suppose that the family { G m } m has surjectivity of the evaluationmap with asymptotic controls. Then the evaluation map for G is quan-titatively surjective.Proof. Again, we do ∗ = 0. Let x ∈ K ( C ∗ ( G ) G ), let R, ǫ be chosento witness x in K R,ǫ ( C ∗ ( G ) G ) and let c m be the asymptotic operatornorm localisation constants, chosen to tend to 1 as m tends to infinity.Now, choose m c large enough that c m ǫ < for all m ≥ m c . Now wehave the conditions to apply Lemma 4.1. We show there is an R ′ suchthat i R,R ′ ,ǫ, ǫcm ( x ) is in the image of ev R ′ , ǫcm .To see this, choose an m large enough that R ≤ r m , k m ( R, ǫ ) ≤ r m and m ≥ m c and consider φ m ( x ). As G m satisfies QS ( d m , R, R ′ , ǫc m , ǫ ′ )for some R ′ ≤ r m and ǫ ′ , we can find a class y in K R ′ ,ǫ ′ ( C ∗ L P d m G G m m )such that ev ( y ) = i R,R ′ , ǫcm ( φ m ( x )). Using Lemma 4.2, we know thatmoving y back into K R ′ ,c m ǫ ′ ( C ∗ L P d m G G ), obtaining an element φ − m ( y ) ∈ K R ′ ,ǫ ( C ∗ L P d G G ).If we evaluate φ − m ( y ), then will obtain i R,R ′ ,ǫ,c m ǫ ′ ( x ) as φ m commuteswith evaluation, and φ − m ( y ) = φ − ( y (0)) by definition. This completesthe proof. (cid:3) Quantitative injectivity of the evaluation map framed us-ing Baum–Connes with coefficients.
In this section we show thatthe quantitative injectivity of the evaluation map can be seen as a con-sequence of the injectivity of a Baum–Connes with a specific set of coefficients, as with surjectivity in the previous setting. Again, this isbased on an argument presented in Oyono-Oyono–Yu [OOY15].
Definition 4.9.
For constants d, d ′ , r, r ′ > , ǫ ∈ (0 , ) we definethe condition QI ( d, d ′ , r, r ′ , ǫ ) to be: for all x ∈ K r,ǫ ∗ ( C ∗ L ( P d ( G )) G )then ev d,r,ǫ ( x ) = 0 in K r,ǫ ∗ ( C ∗ ( G ) G ) implies that i r,r ′ ,d,d ′ ,ǫ ( x ) = 0 in K r ′ ,ǫ ∗ ( C ∗ L ( P d ′ ( G )) G ). We say a group G satisfies quantitative injectiv-ity of the evaluation map if for all d, r > , ǫ ∈ (0 , ) we can find d ′ > d, r ′ > r such that QI ( d, d ′ , r, r ′ , ǫ ) holds for G . Theorem 4.8.
Let G be a finitely generated discrete group. If theBaum–Connes conjecture for G with coefficients in ℓ ∞ ( N , K ) holds,then the evaluation map for G is quantitatively injective.Proof. As in the corresponding statement from [OOY15] we aim fora contradiction. Suppose that the Baum–Connes conjecture holds for G with coefficients in ℓ ∞ ( N , K ), but that the evaluation map is notquantitatively injective.In this situation, we can find a pair r, d > ǫ (0 , )such that for every d ′ > d, r ′ > r the statement QI ( d, d ′ , r, r ′ ǫ ) doesnot hold, i.e we can find a class x d ′ ,r ′ that vanishes under ev r,ǫ,d thatnever vanishes on the right hand side under i r,r ′ ,d,d ′ ,ǫ .To make use of this, pick an ǫ ∈ (0 , ) and obtain the correspondingscales 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 ∈ K r,ǫ ∗ ( C ∗ L ( P d ( G ) that witness the failure of QI ( d, d i , r, r i , ǫ ) for G .We will wrap these up into a single class in the quantitative K-group K r,ǫ ∗ ( C ∗ L ( P d ( G ) , ℓ ∞ ( N , ℓ ∞ ( G, K )) G ) - to do this we will make use of thethe naturality of the localisation algebra with respect to coefficients -recall the isomorphism we used during the proof of Theorem 4.5: Y i ˜ e i : C ∗ L ( P d ( G ) , ℓ ∞ ( N , K )) G → Y i C ∗ L ( P d ( G )) G . This was constructed from the projection maps in the coefficient alge-bras and composition.The idea is now clear. We take our sequence of classes ( x i ) i (whichlive in the ( r, ǫ )-K-theory of the right hand side) and we pull them back ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 23 to a single class z in the ( r, ǫ )-K-theory on the left hand side underthis 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, ℓ ∞ ( N , K )) G ), we know that actually i r,d,ǫ ( z ) = 0 due to ourassumptions on the evaluation map being an isomorphism for G withcoefficients in ℓ ∞ ( N , K ).We’re done however, as we can now find a pair d ′ , r ′ > i r,r ′ ,d,d ′ ,ǫ ( z ) is 0 in K r ′ ,ǫ ∗ ( C ∗ L ( P d ′ ( G ) , ℓ ∞ ( N , K )) G ). We obtain a contrac-tion by unpacking via the isomorphism above. After doing this, weknow that i r,r ′ ,d,d ′ ,ǫ ( x i ) = 0, however by considering the first d i >d ′ , r i > r ′ we know that 0 = i r ′ ,r i ,d ′ ,d i ,ǫ ( i r,r ′ ,d,d ′ ,ǫ ( x i )) = i r,r i ,d,d i ,ǫ ( x i ) = 0by construction. (cid:3) Definition 4.10.
Let { G m → G } m be an asymptotically faithful cov-ering sequence. In this situation we can define a function l m : R + × R + × (0 , ) → R + × R + using the formula: l m ( d, R, ǫ ) = inf { ( R ′ , d ′ ) | G m satisfies QI ( d, d ′ , R, R ′ , ǫ ) } . The family { G m } has the injectivity of the evaluation map with asymp-totic controls if for every ǫ ∈ (0 , ) the supremum L m = sup( d, r ) | l m ( d, r, ǫ ) ≤ ( r m , r m ) tends to infinity in both coordinates as m tendsto infinity. Theorem 4.9.
Let { G m → G } m be an asymptotically faithful coveringsequence satisfying asymptotically controlled operator norm localisationand suppose that the family { G m } m has injectivity of the evaluation mapwith asymptotic controls. Then the evaluation map for G is quantita-tively injective.Proof. Let c denote the asymptotic operator norm localisation control.Then let d, r > , ǫ ∈ (0 , ) be given, and choose m large enough that d, r are smaller than r m and ( d, r ) ≤ L m so that firstly we can applyLemma 4.1 in the setting of localisation algebras - that is to say themap φ m induces a map: φ m : K r,ǫ ∗ ( C ∗ L ( P d ( G )) G ) → K r,cǫ ∗ ( C ∗ L ( P d ( G m )) G m ) and secondly we know that for cǫ , that the evaluation map satisfies QI ( d, d m , r, s m , cǫ ), where both d m and s m are less than r m .Now we consider an element x in K r,ǫ ∗ ( C ∗ L ( P d ( G )) G ) and supposeit maps to 0 under ev d,r . Then as lifting commutes with evaluation,we know that φ m ( x ) evaluates to 0 under ev d,r , and so we know that i d,d m ,r,s m ,cǫ ( φ m ( x )) = 0.The proof is complete however, since we can now apply φ − m , andwe see that i d,d m ,r,s m ,ǫ ( x ) = φ − m ( i d,d m ,r,s m ,cǫ ( φ m ( x ))) = 0 by applyingLemma 4.1 and the fact that expanding commutes with lifting. (cid:3) Consequences
Sobolev norms and rapid decay.Definition 5.1.
The Sobolev (2 , s ) − norms for a length l for a ∈ C G are defined as follows: k a k ,s := k a (1 + l ) s k = sX g ∈ G | a g | (1 + l ( g )) s . Let H s G denote the Jolissant algebra on G , that is the closure of C G in the 2 , s -Sobolev norm.A group G has rapid decay if there is some s such that H s G ⊆ C ∗ r G .This variant is a definition given by Lafforgue [Laf00].The weighted norm is spacial and so plays well with the localisedlifting maps φ and Φ of Observation 3.1. Lemma 5.1. If π : G → H is a surjective group homomorphism withinjectivity radius R , then the map C G → C H induced by π is isometricin the k . k ,s norm for elements with support contained in B R ( e ) . Note that this can be rephrased as ( H s G ) R ∼ = ( H s H ) R . Proof.
The important thing is that the norm k . k ,s is spatially defined.For each a ∈ ( C G ) R , the image π ( a ) = P h ∈ H P g ∈ π − ( h ) a g [ h ]. Sincethe support is contained in a ball that is injected through π , thereis a unique g ∈ G in the preimage of each h ∈ B R ( e ), and all othercoefficients are 0. A calculation with the norms, given that the length ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 25 of the elements in the ball doesn’t change (since π induces an isometrythere), shows that the map is isometric for these elements. (cid:3) The previous lemma shows thatlim sup m k φ m ( a ) k ,s = k a k ,s is satisfied for every a ∈ C G .So we’ve proved: Lemma 5.2.
Let G be a direct limit of an asymptotically faithful cover-ing sequence { π m : G m → G } m where the sequence is an asymptoticallyfaithful covering. Then H s G ֒ → Q ⊕ H s G m is an isometric embedding. (cid:3) We can also try to get some control of the norms in the case that thegroup G or the sequence G m have additional properties - we can thentry 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) the sequence { π m : G m → G } m is asymptotically faithful (2) the map φ is reduced norm continuous, (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, wehave k a m k op ≤ C k a m k ,s , and thus the norms on C G satisfy: k a k = lim sup m k φ m ( a ) k ≤ C lim sup m k φ m ( a ) k ,s = C k a k ,s . Where the first equality follows from the observation in the proof ofTheorem 3.2, the second is uniform property (RD) and the final one isLemma 5.1. This completes the proof. (cid:3)
Note that the two conditions on G mean that the map φ is isometric- in fact property (A) for G gives more. Recall (an equivalent form) of property (A) is operator norm localisation [CTWY08], which is asufficient condition to lift operator norm estimates through local liftings(something abstracted from Higson’s ideas [Hig99] and implementedalso in Willett–Yu [WY12].5.2. Lacunary hyperbolic groups.
Another place where we can pro-duce liftings that are continuous is in the class of lacunary hyperbolicgroups. As a general reference to this important class of groups, onecan look at Osin–Ol’shanski˘ı–Sapir [OOS09]. The definition can befound in Example 2.2. We work from there.Using Theorem 3.2 we will prove that lacunary hyperbolic groupssatisfy the property that λ and λ ∞ are weakly equivalent, or equiva-lently the maps Φ and φ are reduced norm continuous.We first recall that hyperbolic groups have finite asymptotic dimen-sion. This was shown initially by Roe [Roe05], who confirmed an idea ofGromov from the monograph [Gro93], but there are now many proofs -we use the one presented in Roe’s paper [Roe05], where for each R > R + 2 δ m )-bounded cover into at most | S | δ m colours .In this case, the ONL number is at least | S | δm .We will need two last facts, the first is Lemma 3.1 and the secondfact we need before we can prove the theorem is that a cover by uni-formly bounded sets of r -multiplicity k can be broken into at most 2 kr − disjoint families. Recall the annular decomposition of a space. Definition 5.2.
Let X be a metric space, x ∈ X and let A Rk = { x ∈ X | d ( x, x ) ∈ [ kR, ( k + 1) R ) } . Then X = ∪ k A Rk is called the annular decomposition of X . Remark.
The annular decomposition of X is useful as it allows us todivide the space into “odd” and “even” parts (which correspond to theannular parts with odd or even index, respectively). This breaking upmakes colourings easier in some cases like below. this is an incredibly bad bound that’s just useful for this calculation. ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 27
Lemma 5.4.
Suppose U is a uniformly bounded cover of X with r -multiplicity at most k defined as a cover U i of each annular region A ri .Then U can be broken into at most k families that are r -disjoint.Proof. For each annular region, pick a maximal 2 r -separated subset Y i ⊂ A ri - the balls of radius r around these points covers A ri , andso each meets at most k + 1 members of the cover U i . Enumerate theelements of Y i (which is a finite set). For each y j ∈ Y i , let E j = { U ∈U i | U ∩ B r ( y i ) } - note that each U might belong to many E j . Nowcolour the set E using the colours c ∈ { , ..., k } . This induces a partialcolouring of the E j for each j >
0. Now colour the uncoloured elementsof E using colours from { , ..., k } that haven’t been used yet. Extendthis to a partial colouring of E j >
1. Proceed this way until all thesets E j have been coloured.Since each U ∈ U i belongs to at least one E j , it has a colour - andthat colour, once fixed isn’t reassigned - so it’s well defined colouring.Let U, V ∈ c ∈ { , ..., k } . Then U and V don’t intersect the sameball B r ( y j ) - since they must be in different E j . But suppose d ( U, V ) Let G = lim m G m be a lacunary hyperbolic group. Thenthe sequence { G m → G } m has asymptotically controlled operator normlocalisation with constant c = 1 / | S | .Proof. It follows, combining the above observations and Lemma 5.4that δ m − hyperbolic groups generated symmetrically by a set S canhave ONL number taken to be c m ≥ | S | with vectors that have diam-eter of support at most g ( R ) ≤ δ m R + f (6 δ m R ) = 18 δ m R + 2 δ m .If we can show that the supremum of R that satisfy the inequality18 δ m R + 12 δ m ≤ r m goes to infinity as m does, then we’re done. Rearranging, we find that: R ≤ 118 ( r m δ m − δ m and r m we canconclude that this value tends to infinity as m does.We can therefore apply Theorem 3.2 with constants R m and c = | S | . (cid:3) As a consequence of this, we get the following statement: Theorem 5.6. Let G a lacunary hyperbolic group, such that the fam-ily { G m } m of limiting terms satisfy the injectivity and surjectivity ofthe evaluation map with asymptotic controls. Then the Baum–Connesassembly map is quantitatively an isomorphism for G .Proof. For lacunary hyperbolic groups with the asymptotic controls as-sumption, the result follows and Theorem 5.5, Theorem 4.7 and The-orem 4.9. (cid:3) Naturally, this extra condition concerning “Baum–Connes with as-ymptotic controls” could prove problematic. However, using the fol-lowing argument we can produce many examples of sequences with thisproperty that also do not have property (A), which allows us to applythis theorem in the wilderness.5.2.1. Constructing examples with asymptotic controls. We will treatthe small cancellation conditions required to make the constructionand also the associated small cancellation theory that provides theembeddings as a black box - these results and results of their kind canbe found explained in [LS01], or [Ol’91] or [Gru14] - these are all havean analogous flavour and depend on the type of small cancellationcondition used in the construction. We recall the three flavours viareferences 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] ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 29 With this idea in mind we outline a black box technique for takingsmall cancellation lacunary hyperbolic groups and produce coverings ofthose groups that are again lacunary hyperbolic and satisfy the asymp-totic control of the Baum–Connes conjecture in the case of surjectivity.A method for injectivity is not sketched here but is similar to what isbelow. Method 5.1. (A general approach for surjectivity) • Let G = hS | Ri be a small cancellation presentation thatproduces a lacunary hyperbolic group - note that in the casethat G is classical or graphical small cancellation, this meansthat the set of lengths {| R | | R ∈ R} is sparse in R [OOS09],[Gru14] - for graded or other small cancellation conditions, thismay be more complicated. • We produce the groups G m we will use by modifying the pre-sentation of G . We will build two sequences of real numbersand a sequence of groups as follows.(1) Let I = ( a, b ) be an interval. Let R I = { r ∈ R | | r | ∈ I } . For a = 0, we just call the set R b .(2) Fix r > ǫ > 0, then let G = hS | R r i (3) G is hyperbolic, and so it satisfies QS ( r , t , ǫ , ǫ ′ ). Set ǫ = ǫ ′ and pick r >> t . Then take G = hS | R r ⊔ R (2 t ,r i . Thisgroup is again hyperbolic. We can choose r such that the setof relators contains strictly more elements than the set R r .(4) We proceed inductively - let T m − be the set of relators of G m − .Let r m − be the longest relator and let ǫ m − be given. Thensolve Baum–Connes on the scale of the longest relator as before- find t m − , ǫ ′ m − such that QS ( r m − , t m − , ǫ m − , ǫ ′ m − ) holds.Set ǫ m = ǫ ′ m − and pick r m >> t m − , then let G m be defined bythe presentation hS | T m − ⊔ R ( t m − ,r m ) i .(5) Let H be the limit of the sequence of quotients · · · → G m − → G m → . . . . The family G m asymptotically faithfully covers H , so H is lacunary hyperbolic. By construction, it satisfiessurjectivity of Baum–Connes with asymptotic controls. The method can be adapted to situations where the labellings thatgive us relators come from families of finite graphs such as graphical orgeometric small cancellation - we outline how to do this - it makes thechoices 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, S a finite alphabet and let L be a small cancellation labelling ofthe edges of X = ⊔ X m , with component labellings L m . Let R be the set of words that we can read from a cycle in the edges of X and let R m be the labels of cycles that appear in the m − th graph. Let G be the graphical small cancellation group hS | Ri ,which if the lengths of cycles are sufficiently sparse, is lacunaryhyperbolic. • Fix r = the length of the longest cycle in G = hS | R i and take ǫ ∈ (0 , ). Now proceed to find t , ǫ ′ such that QS ( r , t , ǫ , ǫ ′ ) holds in G . • The difference between this method and the previous one isthat we have an easier way to select the next constants. Pick m such that the length of the shortest cycle in X m is muchsmaller than t , and set T = R ⊔ R m . Set G = hS | T i . • Now proceed inductively as before. Suppose we’ve got G n − andknow ǫ ′ n − = ǫ n , then we can construct T n by finding t n − , ǫ ′ n that will satisfy QS ( r n − , t n − , ǫ n , ǫ ′ n ) holds in G n − , then se-lecting m n large enough that the shortest cycle in X m n is muchlonger than t n − . We then set G n to be given by the presenta-tion G n = hS | T n − ⊔ R m n i . • As before, we can take the limit of this inductive constructionand denote it H . This group is actually what we could ob-tain by using the subfamily of the finite graphs { X m n } n andthe associated labelling. It’s a lacunary hyperbolic group thatcovers G and the sequence satisfies quantitative surjectivity ofthe Baum–Connes assembly map with asymptotic controls.Using this, and the fact that both graphical and geometric small can-cellations labellings exist (again, in the geometric case this is Gromov ONTROLLED ANALYTIC PROPERTIES AND ASSEMBLY 31 [Gro03], or Arzhantseva–Delzant [AD08] and in the graphical case thisis Osajda [Osa14]) for certain families of finite graphs of large girth. Asimilar inductive setup can be used to produce injectivity with asymp-totic controls - combining them at each stage would give both situationssimultaneously.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) Every torsion free graphical small cancellation C(7) lacunaryhyperbolic group G has a covering H that has a quantitativelyisomorphic Baum–Connes assembly map. (2) Every (torsion free) lacunary hyperbolic group G constructedfrom a geometric small cancellation C(7) labelling of a sequenceof finite graphs of large girth has a covering that has a quanti-tatively isomorphic Baum–Connes assembly map. (cid:3) We have two remarks. firstly, following the construction presentedin Arzhantseva–Delzant [AD08] carefully, Theorem 5.7.(2) containsmany groups with property (T). We also note that in the second case,Gruber–Sisto [GS18] showed the groups occurring in (2) cannot bequasi-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 inOsin–Ol’shanki˘ı–Sapir [OOS09]. This way, we can construct a TarskiMonster as in Theorem 4.7 of [OOS09]. In particular we can obtaintorsion groups that satisfy the Baum–Connes conjecture. Theorem 5.8. 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