A finitely-generated amenable group with very poor compression into Lebesgue spaces
aa r X i v : . [ m a t h . M G ] A p r Amenable groups with very poor compressioninto Lebesgue spaces
Tim Austin
Abstract
We give a construction of finitely-generated amenable groups that do notadmit any coarse -Lipschitz embedding with positive compression exponentinto L p for any ≤ p < ∞ , including some that are four-step solvable,answering positively a question of Arzhantseva, Guba and Sapir. Contents
Given a metric space ( X, ρ ) and a Banach space X with norm k·k , the compressionexponent α ∗ X ( X, ρ ) of X into X is the supremum of those α ≥ for which there1xists an injection f : X ֒ → X such that ρ ( x, y ) α . k f ( x ) − f ( y ) k ≤ ρ ( x, y ) ∀ x, y ∈ X (where, as usual, . and & denote inequalities that hold up to an arbitrary positivemultiplicative constant that is independent of the arguments of the functions inquestion).This can be viewed as a quantitative measure of how well, if at all, the space ( X, ρ ) can be coarsely Lipschitzly embedded into X . It was introduced by Guentner andKaminker in [10], with particular emphasis on the case when ( X, ρ ) is a finitely-generated group equipped with a left-invariant word metric associated to a finitesymmetric generating set, and X is a Hilbert space. In this case the value α ∗ X ( G ) is easily seen to be independent of the particular choice of generating set, sincedifferent generating sets lead to quasi-isometric word metrics. For a group G wecan define analogously to α ∗ X ( G ) the equivariant compression exponent α X ( G ) as the supremum of those α ≥ for which there exists an equivariant injection f : G ֒ → X (that is, an injection given by the orbit of a point under some action G y X by affine isometries) such that ρ ( g, h ) α . k f ( g ) − f ( h ) k ≤ ρ ( g, h ) ∀ g, h ∈ G. We write α ∗ p and α p in place of α ∗ L p and α L p . Since Guentner and Kaminker’swork a number of methods have been brought to bear on the estimation of theseexponents for different groups; see, for instance, [2, 3, 7, 8, 18, 19, 20, 17, 4,15, 16]. A more detailed discussion of these developments and more completereferences can be found in the introduction to [16]. It is also worth noting that if G is amenable then necessarily α ( G ) = α ∗ ( G ) , as was shown by Aharoni, Maureyand Mityagin in [1] for Abelian G and then by Gromov (see [8]) in general.Most known results concern the case of amenable groups. It was shown by Guent-ner and Kaminker in [10] that if α ( G ) > then this actually implies that G isamenable, and another proof of this fact has now been given in [15] along with ageneralization to other Lebesgue target spaces (although it is known that there areboth amenable and non-amenable groups that have Euclidean equivariant compres-sion exponent exactly equal to ). In the reverse direction, Arzhantseva, Guba andSapir asked as Question 1.12 in [3] whether there are amenable groups that havecompression exponent (or, more modestly, strictly less than ) for embeddingsinto Hilbert space, and following progress in other directions versions of this ques-tion have since been re-posed as Question 5.23 of Arzhantseva, Drutu and Sapir [2]and Question 1.5 of Tessera [17]. 2his question is interesting partly because previous methods for bounding Hilbertspace compression exponents from above seem unable in principle to break the -barrier. In particular, a very general upper bound in terms of random walk escapespeed, first introduced in [4] and then considerably generalized in [15, 16], cannotpush below this value. One version of that result (although not quite the mostgeneral) asserts that if X is a Banach space having modulus of smoothness of powertype p , then α X ( G ) ≤ pβ ∗ ( G ) , where β ∗ ( G ) is the supremum of those β ≥ forwhich E ( ρ ( e G , X t )) & t β ∀ t ∈ N , where ( X t ) t ≥ is the symmetric random walk on G starting at the identity e G corresponding to some finite generating set. Since it is known that ≤ β ∗ ≤ always, this bound cannot give a value below .Moreover, in several concrete cases (in particular among certain iterated wreathproducts of cyclic groups) the random walk bound turns out to be the correct valueof the compression exponent, even when this value lies in ( , . These observa-tions led Naor and Peres to ask in [16] (Question 10.3) whether it is always thecase that an amenable group G has compression exponent into L p given by ex-actly min { pβ ∗ , } (which would, in particular, answer negatively the question ofArzhantseva, Guba and Sapir).In this work we decide these questions by proving the following. Theorem 1.1.
There is a finitely-generated amenable group that does not admitany embedding into L p with a positive compression exponent for any p ∈ [1 , ∞ ) . Intuitively, this means we have found a finitely-generated amenable group withmuch worse embedding properties into L p than have been witnessed for suchgroups heretofore. Various notions relating to the uniform embeddability of finitely-generated groups into ‘nice’ Banach spaces (predominantly Hilbert space) havebeen introduced in geometric group theory (see, in particular, Gromov’s discussionof a-T-menability and some relatives in Section 7.E of [9]), and some equivalencesfound among them; on the other hand, some of these properties have been shownto have striking consequences elsewhere in gometric group theory and topology(perhaps most notably Yu’s deduction of the coarse Baum-Connes and NovikovConjectures therefrom in [21]). Gromov asked whether amenable groups are al-ways a-T-menable, and this was then proved by Bekka, Cherix and Valette in [5].Thus from the relatively soft viewpoint of coarse geometry amenable groups areknown to possess such desirable embeddings; the present paper indicates that thisis not a guarantee of any very strong quantitative versions of the same conclusions.3efore launching into technical details, we offer a sketch of our approach to Theo-rem 1.1.Underpinning the proof is a simple observation about how a sequence of finitemetric spaces with growing L p -distortion can serve as an obstruction to the good-compression L p -embedding of an infinite metric space. If ( X, ρ ) is an infinitemetric space, and inside it we can find a family of bi-Lipschitzly embedded finitemetric spaces ( Y n , σ n ) , say with embeddings ϕ n : Y n ֒ → X , then these give riseto a bound on α ∗ p ( X, ρ ) in terms of • the distortions c p ( Y n , σ n ) , and • the ratios according to which distances under σ n are approximately expandedby ϕ n .This interplay between the distortions of the Y n and approximate expansion ratiosof the ϕ n is crucial: the resulting obstruction to L p -embedding of the whole of ( X, ρ ) will be more severe according as the expansion ratios of these ϕ n growmore slowly. A quantitative version of this observation that is suitably tailoredfor our proof of Theorem 1.1 appears as Lemma 3.1 below. With this principle inhand, we will find the specific finite metric spaces that will serve as our obstructionsin the form of certain quotients of high-dimensional Hamming cubes. RegardingHamming cubes as vector spaces over Z , a result of Khot and Naor [12] providesus with certain quotients of these by Z -subspaces whose Euclidean distortion islinearly large in their diameter.Most of our work will then go into constructing a group that contains suitable bi-Lipschitz copies of these cube-quotients. The basic strategy of using a sequenceof poor-distortion finite subsets to bound above the compression exponent of agroup is not new; for example, Arzhantzeva, Drut¸u and Sapir use it in [2] fortheir construction of finitely-generated groups with arbitrary compression expo-nents into uniformly convex Banach spaces, except that their finitary obstructionsare sequences of expanders, and they use free products (hence giving non-amenableexamples) to construct finitely-generated groups containing them.Our construction will give a two-fold Abelian extension of any suitable base group G (which could be, for example, the classical lamplighter Z ≀ Z , so that among ourexamples of groups with very poor compression we find certain four-step solvablegroups). The first of these extensions of G is a wreath product, but the second issomething rather more complicated.More precisely, we begin with any finitely-generated amenable G that has expo-4ential growth, form the wreath product over it H := Z ≀ G and equip it with itsnatural lifted generating set and word metric (we will comment on the modulus shortly). Consider H as acting by translation on the Z -vector space Z ⊕ H . Theseare the ingredients needed to form another wreath product Z ≀ H = Z ⊕ H ⋊ H , butwe will instead examine a slightly more complicated relative of this construction.(Indeed, while the random walk method gives an upper bound α ∗ ( Z ≀ H ) ≤ , anda recent analysis by Li [13] building on a construction from de Cornulier, Stalderand Valette [6] has shown that it is at least , its exact value is currently unknown.Arzhantseva, Guba and Sapir originally proposed the related iterated wreath prod-uct Z ≀ ( Z ≀ Z ) as a candidate for having zero Euclidean compression exponent,but Li’s work now also gives a positive lower bound for this exponent.) We willfirst identify a translation invariant Z -subspace V ≤ Z ⊕ H , and will then forminstead the semidirect product ( Z ⊕ H /V ) ⋊ H , equipped with the natural generat-ing set that consists of lifts of the generators of H and the single new generator δ e H + V ≤ Z ⊕ H /V .The rˆole of this Z -subspace V is to insert copies of a sequence of Khot and Naor’spoor-distortion cube-quotients, say Z I n /C n for some increasingly large index-sets I , I , . . . , into the ‘zero-section’ subgroup Z ⊕ H /V ⊂ ( Z ⊕ H /V ) ⋊ H. More precisely, we will prove that for some sequence of these cube-quotients Z I n /C n , one can effectively choose embeddings ϕ ◦ n : Z I n ֒ → Z ⊕ H and a translation-invariant subgroup V ≤ Z ⊕ H so that ϕ ◦ n ( C n ) ⊆ V , and the resulting quotient maps ϕ n : Z I n /C n ֒ → Z ⊕ H /V ⊂ ( Z ⊕ H /V ) ⋊ H are bi-Lipschitz embeddings with control on their distortions and expansion ratiosas required for them to constrain the compression exponents of L p -embeddings ofthe whole group ( Z ⊕ H /V ) ⋊ H .The principal difficulties in this construction arise from the need to insert a wholeinfinite sequence of these cube-quotients Z I n /C n into Z ⊕ H /V using a single choiceof subgroup V , with a uniform bound on their distortions and slow growth oftheir expansion ratios. It is to address these difficulties that we specifically take H = Z ≀ G for some G of exponential growth. The Z -subspace V will be as-sembled as an infinite sum V + V + . . . of Z -subspaces, each of which createsthe corresponding embedding of Z I n /C n , and we will use the particular algebraicstructure of this wreath product H to choose these V n so that they do not ‘interact’with each other too much: more specifically, so that quotienting by V n ′ for n ′ > n does not disrupt the bi-Lipschitz embedding of Z I n /C n that we introduce upon5uotienting by V n . One feature of this algebraic argument is that it leads to thecareful choice of (rather than, say, or ) as the modulus in the wreath product H = Z ≀ G . This is used via the fact that there exists a non-constant function A : Z → Z with the property that one cannot form the constant function ∈ Z as a Z -linear comination of translates of A by elements of the domain Z . For thiswe take A to be the Z -valued indicator function of { , , , } . Since at a differentstep in the construction we will want a modulus that is even, a quick check showsthat is the smallest possibility. By allowing A to act on different coordinates in Z ⊕ G we can create a large family of functions Z ⊕ G → Z (that is, elements of Z ⊕ Z ⊕ G ) all of whose translates are linearly independent, and this will then be usedin the proof that our choices of translation invariant subspaces V , V , . . . ≤ Z Z ≀ G are ‘well-separated’ (see Lemma 4.7 and its use in the proof of Proposition 4.5). Acknowledgements
I am grateful to Assaf Naor and Yuval Peres for introducingme to the topics touched by this paper and for many subsequent discussions, toSean Li and Assaf Naor for each suggesting a number of improvements to thispaper, and to Microsoft Research Redmond for a period of their hospitality duringwhich most of this work was completed. ⊳ The following nomenclature distinguishing certain kinds of metric space will proveuseful.
Definition 2.1.
A metric space ( X, ρ ) is • infinite if this is so of X as a set; • locally finite if every open ball B X ( x, r ) in X is a finite set; • d min -discrete for some d min > if ρ ( x, y ) ≥ d min whenever x, y ∈ X aredistinct. Given two metric spaces ( X, ρ ) and ( Z, θ ) and a map f : X → Z , the distortion of f is a measure of the extent to which f fails to be a homothety: distortion( f ) := sup u,v ∈ X, u = v θ ( f ( u ) , f ( v )) ρ ( u, v ) · sup u,v ∈ X, u = v ρ ( u, v ) θ ( f ( u ) , f ( v )) , + ∞ unless f is at least a bi-Lipschitz embedding.Relatedly, the map f is M -bi-Lipschitz for some M ≥ if it is bi-Lipschitz withdistortion at most M .The distortion of X into Z is now obtained by infimizing over f : it is convention-ally denoted by c ( Z,θ ) ( X, ρ ) := inf f : X → Z distortion( f ) . The case in which ( Z, θ ) is a Lebesgue space L p with its norm metric for some ≤ p < ∞ is particularly well-studied, and in this case we abbreviate c L p to c p .In addition to the distortion of a map f , we will sometimes need to keep track ofits expansion ratio : in the above notation this is simply the ratio inf u,v ∈ X, u = v θ ( f ( u ) , f ( v )) ρ ( u, v ) . If the expansion ratio is r , then the above definitions combined tell us that r · ρ ( u, v ) ≤ θ ( f ( u ) , f ( v )) ≤ distortion( f ) · r · ρ ( u, v ) ∀ u, v ∈ X, provided all the quantities appearing here are finite.An important class of metrics that will appear repeatedly later is that of the Ham-ming metrics: Definition 2.2 (Hamming metric) . If ( X, ρ ) is a metric space and n ≥ then the Hamming metric on X n associated to ρ is defined by d Ham (cid:0) ( x i ) i ≤ n , ( y i ) i ≤ n (cid:1) := n X i =1 ρ ( x i , y i ) . We will also make use of the construction of quotient metrics. The following defi-nition is discussed, for example, in Section 3 of Khot and Naor [12].
Definition 2.3 (Quotient metrics) . Suppose that ( X, ρ ) is a metric space and that P is a partition of X with the property thatFor any P, Q ∈ P and any x ∈ P there is some y ∈ Q such that ρ ( x, y ) = dist ρ ( P, Q ) (that is, ‘the minimal distance between P and Q can be realized from any starting point on one side’). hen the quotient metric ρ / P on P is defined by ρ / P ( P, Q ) := dist ρ ( P, Q ) . The routine verification that this is indeed a metric on P under the above assump-tion can be found in Section 3 of [12].In particular, this definition always applies to give a metric on the homogeneousspace G/H when ( G, ρ ) is a group carrying a left-invariant metric ρ and H ≤ G .In this case we denote the quotient metric by ρ /H . Before leaving this subsection, let us recall some useful analyst’s notation. Givenpositive quantities A and C and any other parameter B , we write A . B C or A =O B ( C ) if quantity A is bounded above by quantity C up to a positive multiplicativeconstant depending only on B (and simply A . C or A = O( C ) if the constantis truly universal); similarly A = Ω B ( C ) if A is bounded from below by such apositive multiple of C ; and A = Θ( C ) if A is bounded from above and below bythe multiples of C by two universal positive bounds (that is, if A = O( C ) and A = Ω( C ) ). Thus, for instance, an embedding has ‘expansion ratio Θ(1) ’ if itsexpansion can be shown to lie in [ r, R ] for some R ≥ r > , independently of anyother parameters that went into its construction. The construction of this paper makes a double use of a semidirect product. Givena group L and another H that acts on it by automorphisms, H y L , we may formthe semidirect product H ⋉ L or L ⋊ H as the set L × H equipped with the groupoperation ( ℓ , h ) · ( ℓ , h ) := ( ℓ h · ℓ , h h ) . Importantly, the group H ⋉ L is finitely-generated if • H is finitely-generated and • L admits a finite subset S ⊆ L such that L = h{ α h ( s ) : h ∈ H, s ∈ S }i .This can be so even if L itself is not finitely-generated. An important example ofthis is the wreath product K ≀ H := K ⊕ H ⋊ H , formed from the action H y K ⊕ H (the group of H -indexed families of members of K with cofinitely many entriesequal to e K ) by coordinate right-translation so that (cid:0) ( k ,h ) h ∈ H , h (cid:1) · (cid:0) ( k ,h ) h ∈ H , h (cid:1) = (cid:0) ( k ,hh − k ,h ) h ∈ H , h h (cid:1) . (1)8his is always finitely-generated if this is so of H and K , even though if H isinfinite then L = K ⊕ H is not finitely-generated.The semidirect product L ⋊ H always contains a canonical copy of L in the formof the subgroup { ( ℓ, e H ) : ℓ ∈ L } : following Naor and Peres [16] we refer to thisas the zero section of L ⋊ H and sometimes denote it by ( L ⋊ H ) .In this paper we will use a construction similar to that of the wreath product, exceptthat the place of K ⊕ H will sometimes be taken by one of its quotients. In thefollowing the group K will always be one of the cyclic groups Z or Z . Wewill later form first an extension of a suitable base group by a power of Z , andthen a further extension by a power of Z , and our choice of notation is geared todistinguish between these two extensions as clearly as possible. In particular: • Given a base group G , we will usually denote elements of Z ⊕ G by lowercasebold letters such as w = ( w g ) g ∈ G , and refer to them as vectors . In thissetting we write e g for g ∈ G for the vector with g th entry equal to ∈ Z and all other entries equal to ; we refer to these e g collectively as the standard generators of Z ⊕ G . • On the other hand, given another base group H (which will later be equal to Z ≀ G ), we will denote elements of Z ⊕ H by uppercase calligraphic letters,and will refer to them as functions W : H → Z ; correspondingly anexpression such as W + V refers to a sum of functions defined pointwise.In this setting we denote by δ h the function that takes the value ∈ Z at h and elsewhere. We will sometimes need to work instead with the set { h ∈ H : W ( h ) = 1 } , which we refer to as the support of W and denoteby spt W .The only exceptions to these rules are that will be used for the zero element ineither case, since this should cause no confusion, and that for any finite index set T we write T for the function T → C that identically takes the value ∈ C foreither C = Z or C = Z .If V ≤ Z ⊕ H is a subspace that is invariant under the coordinate right-translationaction of H , then this action of H quotients to a well-defined action H y Z ⊕ H /V .Given also a chosen finite symmetric generating set S for H , we will always endowthe semidirect product ( Z ⊕ H /V ) ⋊ H with the symmetric generating set { ( δ e H + V, e H ) } ∪ { ( , s ) : s ∈ S } . To this enlarged generating set we always associate the left-invariant word metric asusual. In general, if the generating set S of H is understood, the above generating9et will also be understood for ( Z ⊕ H /V ) ⋊ H , and we will (slightly abusively)denote the resulting metric by ρ ( Z ⊕ H /V ) ⋊ H , and will denote the restriction of thisnew metric to the zero section by ρ (( Z ⊕ H /V ) ⋊ H ) .We will be working largely with the restriction of this word metric to the zero sec-tion of ( Z ⊕ H /V ) ⋊ H , and to this end it will be helpful to have a simpler equivalentmetric on the zero section to work with. Such a description can quite easily be givenin terms of traveling salesman tours as a simple extension of the usual heuristic in-terpretation of the wreath product word metric in terms of lamplighter walks. Thedefinition below has been adapted from Section 2 of Naor and Peres [16], where itsconnection with the problem of estimating the length of traveling salesman toursamong points in a metric space (and, in particular, with the work of Jones in [11])is explained in more detail. Definition 2.4 (Pinned traveling salesman metrics) . Given a metric space ( X, ρ ) and a distinguished point x ◦ ∈ X , the associated traveling salesman metricpinned at x is the metric TS ρ,x ◦ on the collection Z ⊕ X of finitely-supported maps X → Z (equivalently, finite subsets of X ) defined by TS ρ,x ◦ ( A , B ):= if A = B min P ℓi =1 ρ ( x i , x i +1 ) : the cycle ( x = x ◦ , x , . . . , x ℓ , x ℓ +1 = x ◦ ) covers spt( A + B ) + 1 else.This is clearly a metric in view of the inclusion spt( A + C ) ⊆ spt( A + B ) ∪ spt( B + C ) and the ability to concatenate covering cycles.Abbreviating, it will be understood that the notation TS Ham on Z ⊕ Z ⊕ S refers tothe Hamming metric d Ham on the underlying group Z ⊕ S (where Z is given theword metric corresponding to the generators { , − } ) and the distinguished point x ◦ := ∈ Z ⊕ S .If T is a finite index set and F ≤ Z Z T is a Z -subspace invariant under the trans-lation action of Z T , then we denote by TS Ham /F the metric on Z Z T /F that resultsfrom quotienting TS Ham . Let us note at once that following trivial consequence of this definition:
Lemma 2.5. If V , V ′ , W , W ′ : X → Z are finitely supported and x ◦ ∈ X then spt ( V ′ + W ′ ) ⊇ spt ( V + W ) ⇒ TS ρ,x ◦ ( V ′ , W ′ ) ≥ TS ρ,x ◦ ( V , W ) . n particular, TS ρ,x ◦ is addition-invariant on Z ⊕ X . A more delicate calculation that we will need later is the following. This closelyresembles the derivation of equation (13) in Naor and Peres [16], and we offer onlya rather terse account here.
Lemma 2.6. If ( Z ⊕ H /V ) ⋊ H is a semidirect product as above, H has generat-ing set S and corresponding word metric ρ and we lift this to a generating set of ( Z ⊕ H /V ) ⋊ H as above with associated metric ρ ( Z ⊕ H /V ) ⋊ H , then the inclusion Z ⊕ H /V ≡ (( Z ⊕ H /V ) ⋊ H ) ⊂ ( Z ⊕ H /V ) ⋊ H is a -bi-Lipschitz embedding of TS ρ,e H /V into ρ ( Z ⊕ H /V ) ⋊ H with expansion ratio lying in Θ(1) . To find a shortest-length word that evaluates to a given group element ( A + V, e H ) ∈ ( Z ⊕ H /V ) ⋊ H amounts to finding a shortest walk from ( V, e H ) to ( A + V, e H ) that steps only along edges of the Cayley graph defined by the generated set of ( Z ⊕ H /V ) ⋊ H . Intuitively, in order to do this we must take a walk in the Cay-ley graph of H that starts and ends at e H , and along the way pass through finitelymany points of H at which we modify the value taken by the identically-zero func-tion : H → Z to obtain a function that lies in A + V (this much correspondsroughly to an evaluation of a distance in TS ρ,e H ), and finally infimize over thechoice of that element of A + V (which corresponds to working in the quotientmetric TS ρ,e H /V ). The following proof makes this intuition precise. Proof
We will show that for any finitely-supported A , B : H → Z we have TS ρ,e H /V ( A + V, B + V ) ≤ ρ ( Z ⊕ H /V ) ⋊ H (( A + V, e H ) , ( B + V, e H )) ≤ · TS ρ,e H /V ( A + V, B + V ) . To see this, recall that by definition ρ ( Z ⊕ H /V ) ⋊ H (( A + V, e H ) , ( B + V, e H )) is theshortest length of any word in the alphabet given by the generating set { ( δ e H + V, e H ) } ∪ { ( V, s ) : s ∈ S } whose evaluation in ( Z ⊕ H /V ) ⋊ H is equal to ( A + V, e H ) − · ( B + V, e H ) =( B −A + V, e H ) . Set ℓ := ρ ( Z ⊕ H /V ) ⋊ H (( A + V, e H ) , ( B + V, e H )) , let g ℓ g ℓ − · · · g be such a word of minimal length in this alphabet and define G i := g i g i − · · · g =:( C i + V, h i ) for i = 1 , , . . . , ℓ and also G := ( V, e H ) =: ( C + V, h ) .Each G i differs from G i − by left-multiplication by either ( δ e H + V, e H ) or by ( V, s ) for some s ∈ S . In the first case the multiplication rule (1) tells us that11 i = h i − and C i = C i − + δ h i − mod V , and in the second it gives C i = C i − but h i = g i h i − . Letting ≤ i < i < . . . < i k ≤ ℓ be the subsequence of those i at which we are in the first of these situations, we see that overall C ℓ = δ h i − + δ h i − + · · · + δ h ik − = A − B mod V. Therefore, omitting the steps i , i , . . . , i k , the sequence h , h , . . . , h i − , h i +1 ,. . . , h ℓ executes a walk in H that starts from e H , visits every point of spt C ℓ (where C ℓ ∈ A − B + V ) at least once and then returns to e H ; and at those omitted stepsthe multiplication by g i j does not change the position of h i j − but instead altersthe function C i j − to C i j modulo V .On the one hand, it follows immediately from Definition 2.4 that the length of thiswalk is at least TS ρ,e H ( , C ℓ ) ; and on the other, for any such C ℓ , since any twopoints of spt C ℓ are separated by a distance of at least in H under ρ , it followsthat there does exist such a sequence of walk-steps interspersed with additions of δ h ik + V of length at most TS ρ,e H ( , C ℓ ) + |C ℓ | ≤ · TS ρ,e H ( , C ℓ ) . Therefore, first minimizing the above lower bound over the possible choices of C ℓ ∈ A − B + V we obtain TS ρ,e H /V ( A + V, B + V ) = min C∈A−B + V TS ρ,e H ( , C ) ≤ ρ ( Z ⊕ H /V ) ⋊ H (( A + V, e H ) , ( B + V, e H )) , and secondly choosing C = C ℓ that attains this minimum and then selecting asuitable word g ℓ g ℓ − · · · g as described above proves the corresponding upperbound. Before turning to the construction of our group, we examine the relation betweenthe notions of compression for infinite metric spaces and of distortion for finitemetric spaces. In particular, we will see how to use large high-distortion finite sub-sets of an infinite metric space as obstructions to good-compression embeddings ofthe whole space, in the sense made precise by the following.12 emma 3.1.
Suppose that X is a normed vector space, that ( X, ρ ) is an infinite,locally finite, -discrete metric space, and suppose further that we can find a se-quence of finite -discrete metric spaces ( Y n , σ n ) and embeddings ϕ n : Y n ֒ → X such that • the Y n are increasing in diameter: diam( Y n , σ n ) → ∞ ; • the Y n are embedded in X with uniformly-bounded distortion: there aresome fixed L ≥ and some sequence of positive reals ( r n ) n ≥ such that L r n σ n ( u, v ) ≤ ρ ( ϕ n ( u ) , ϕ n ( v )) ≤ Lr n σ n ( u, v ) ∀ u, v ∈ Y n , n ≥ (so the distortion is at most L ); • the Y n do not expand too fast inside X relative to their size: we have ≤ r n . diam( Y n , σ n ) ε for every ε > ; • the Y n have bad distortion into X : for some η > we have c X ( Y n , σ n ) & diam( Y n , σ n ) η for all n ≥ .Then α ∗ X ( X, ρ ) ≤ − η . Proof
We may assume that α ∗ ( X, ρ ) > , since otherwise the result is trivial.Given this, let α < α ∗ ( X, d ) and let f : X → X be a -Lipschitz embedding into X that achieves compression α : ρ ( x, y ) α . k f ( x ) − f ( y ) k ≤ ρ ( x, y ) ∀ x, y ∈ X. Combining this with the bi-Lipschitz condition L r n σ n ( u, v ) ≤ ρ ( ϕ n ( u ) , ϕ n ( v )) ≤ Lr n σ n ( u, v ) by setting x := ϕ n ( u ) and y := ϕ n ( v ) in the compression inequality, we deducethat f ◦ ϕ n : Y n ֒ → X satisfies L α (cid:0) r n σ n ( u, v ) (cid:1) α . k f ◦ ϕ n ( u ) − f ◦ ϕ n ( v ) k ≤ Lr n σ n ( u, v ) ∀ u, v ∈ Y n , n ≥ . It follows that distortion( f ◦ ϕ n ) dfn = max u,v ∈ Y n , u = v σ n ( u, v ) k f ◦ ϕ n ( u ) − f ◦ ϕ n ( v ) k · max u,v ∈ Y n , u = v k f ◦ ϕ n ( u ) − f ◦ ϕ n ( v ) k σ n ( u, v ) . ( max u,v ∈ Y n , u = v L α r − αn σ n ( u, v ) − α ) · Lr n = L α r − αn diam( Y n , σ n ) − α . c X ( Y n , σ n ) & diam( Y n , σ n ) η , it followsthat diam( Y n , σ n ) η . L α r − αn diam( Y n , σ n ) − α . Finally, from our assumption that r n . diam( Y n , σ n ) ε for every ε > and since L is independent of n , it follows that diam( Y n , σ n ) η . diam( Y n , σ n ) (1 − α )(1+ ε ) as n → ∞ for every ε > , and hence that − α ≥ η , or α ≤ − η . Remark
Note that the interplay between the expansion ratio r n incurred by theembedding ϕ n : Y n ֒ → X and the compression assumption on f is crucial for thisproof: if, in a different situation, we knew that we could embed the same metricspaces Y n into X bi-Lipschitzly, but only at the expense of drastically enlargingthem, then the above argument would give no bound on the compression. ⊳ In conjunction with the general principle contained in the above lemma, we willuse the following result of Khot and Naor [12] to supply us with high-distortionfinite metric spaces to serve as obstructions.
Theorem 3.2.
There are Z -subspaces V d ≤ Z d for all d ≥ such that c p ( Z d /V d , d Ham /V d ) & p d as d → ∞ for all p ∈ [1 , ∞ ) . Moreover, the subspace V d may be taken of the form C ⊥ d for some other subspace C d ≤ Z d such that = (1 , , . . . , ∈ C d , where C ⊥ d := { ( w i ) i ≤ d ∈ Z d : w v + w v + · · · + w d v d = 0 ∈ Z ∀ ( v i ) i ≤ d ∈ C d } . Proof
The existence of such subspaces is covered directly by Remark 3.1 in [12],and the fact that we may take ∈ C d follows at once from the greedy-algorithmproof given for their Corollary 3.5.With these ingredients to hand, we can now state a more precise result from whichTheorem 1.1 will follow immediately. Theorem 3.3.
For any finitely-generated amenable group G of exponential growth,with finite symmetric generating set S and associated left-invariant word metric ρ ,there are a fixed M ≥ and a Z -subspace V ≤ Z ⊕ ( Z ≀ G )2 such that • V is invariant under the action of Z ≀ G by coordinate right-translation, and the natural choice of generating set for ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) admits asequence of injections ϕ n : Z d n /V d n ֒ → Z ⊕ ( Z ≀ G )2 /V ⊂ ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) that are embeddings of d Ham /V dn into ρ ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) with uniformly-bounded distortions and expansion ratios r n . log d n . Proof of Theorem 1.1 from Theorem 3.3
This now follows by applying Lemma 3.1using the Euclidean distortion bound from Theorem 3.2, according to which we cantake η = 1 when the spaces ( Y n , σ n ) are the cube-quotients ( Z d n /V d n , d Ham /V dn ) .The rest of this paper will be taken up by the proof of Theorem 3.3. All the delicacyin this construction lies in the selection of the subspace V . The formation of a towerof two semidirect products (starting from a base group G of exponential growth,which could also be obtained as another semidirect product) is rather important forthe detailed arguments to come, and leads to examples, at their simplest, that are -step solvable. Note that they are, at least, elementary amenable. The ways inwhich we use each of these semidirect products, and also the reasons for choosingthe particular cyclic groups Z and Z as building blocks, will be examined as theyarise during the subsequent sections. Our first step in the construction of the subspace V promised by Theorem 3.3 isthe following. Lemma 4.1 (Finding not-too-wide sets of many roughly equidistant points) . Let G be a finitely-generated group of exponential growth, S a symmetric generating setand ρ the resulting word metric. Then there are some M ≥ and c > , dependingonly on G and S , such that for any r > there is some r ≥ r for which we canfind a subset { x , x , . . . , x d } with d := 10 r such that r ≤ ρ ( x i , x j ) ≤ M r forall ≤ i < j ≤ d and r ≤ ρ ( x i , e G ) ≤ M r for all ≤ i ≤ d . Proof
Let c > be such that | B ( e, r ) | ≥ c r for infinitely many r ≥ .
15n the reverse direction, a na¨ıve count tells us that | B ( e, r ) | ≤ | S | r ∀ r ≥ . Let M ≥ be a fixed integer such that c M > (11 | S | ) . Then for any r ≥ wecan find some r ≥ r + 1 and ≤ s ≤ M − such that | B ( e, M r ) | ≥ | B ( e, M ( r −
1) + s ) | ≥ ( c M ) r − · c s ≥ (11 | S | ) r − ≥ r · | S | r , and the above inequalities together now imply that | B ( e, M r ) \ B ( e, r ) | = | B ( e, M r ) | − | B ( e, r ) | ≥ (11 r − · | S | r ≥ r · | S | r . Using again that | B ( x, r ) | ≤ | S | r for every ( x, r ) , it follows that B ( e, M r ) \ B ( e, r ) cannot be covered by fewer than r balls of radius r . This implies insteadthat we can find in B ( e, M r ) \ B ( e, r ) a subset containing at least d := 10 r points,any two of which are separated by a distance of at least r . Enumerating this subsetas { x , x , . . . , x d } completes the proof. By considering the subsets n(cid:16) d X i =1 w i e x i , e G (cid:17) : ( w i ) i ≤ d ∈ { , } d o ⊂ ( Z ≀ G ) with { x , x , . . . , x d } as given by Lemmas 4.1, it is not hard to deduce fromLemma 2.6 how to embed copies of the Hamming cubes Z d with uniform dis-tortion inside Z ≀ G so that their diameters grow exponentially faster than theexpansion ratios of their embeddings. However, to obtain cube- quotients requiresa little more work, in this subsection we prepare the ground for this.The difficulty that seems to prevent us from finding these quotients inside Z ≀ G liesin finding a single subgroup of Z ⊕ G that is G -translation invariant and for whichthe resulting quotient will restrict to the desired quotient on each of these Hammingcube copies separately. It is for this reason that we make a second extension of ourgroup, to form our final group of the form ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) for a suitablechoice of ( Z ≀ G ) -translation invariant Z -subspace V . Even in this case we willneed to take some care in our proof that the quotient by V really does lead to thebi-Lipschitz obstructions we seek. 16n this subsection we begin by putting the finite-dimensional quotient spaces ofTheorem 3.2 into a form that is better adapted to embedding them into a left-invariant group metric.First, by an iterated appeal to Lemma 4.1 we can clearly select an increasing se-quence of positive reals ( r n ) n ≥ , a sequence of points ( y n ) n ≥ in G and a sequenceof pairwise-disjoint subsets ( I n ) n ≥ of G such that • I n ⊂ G \ { y , y , . . . } for all n ≥ , • r n ≤ ρ ( x, x ′ ) ≤ M r n for all distinct x, x ′ ∈ I n and r n ≤ ρ ( x, e G ) ≤ M r n for all x ∈ I n , n ≥ , • | I n | = d n := 10 r n for all n ≥ , and • we have r < ρ ( y , e G ) < r < ρ ( y , e G ) < . . . . It will matter at various points later that the sequence of positive reals r , ρ ( y , e G ) , r , ρ ( y , e G ) , . . . grows sufficiently quickly. However, in the remainder of thissubsection we will consider only the data of a single set I n .From this set we form the group Z I n . This is a finite-dimensional factor of Z ⊕ G ∼ =( Z ≀ G ) . We endow it with the Hamming metric d Ham that arises from endow-ing each factor copy of Z separately with the word metric corresponding to thegenerating set {± } .Within this group we will consider the subset B n := { e x : x ∈ I n } comprisingthe standard set of basis vectors. Let C n ≤ Z I n be some Z -subspace as appearsin the Theorem 3.2, so that the cube-quotient Z I n /C ⊥ n endowed with the quotientof the Hamming metric has L p -distortion Ω p ( d n ) for all finite p .The space Z I n embeds into Z Z In through its identification with Z B n and thencewith {W : Z I n → Z : spt W ⊆ B n } ∼ = Z B n ⊕ | Z In \ B n ; let us write ξ : Z I n ֒ → Z Z In for this embedding. In order to avoid confusion wedenote by h a , b i := X x ∈ I n a x b x ∈ Z Z -valued inner product on Z I n , and by hhW , Vii := X w ∈ Z In W ( w ) V ( w ) ∈ Z its analog on the space Z Z In of functions.We will now show how to convert C n into a Z -subspace D n ≤ Z Z In that is • invariant under the translation action of Z I n , and • such that the restriction of TS Ham /D ⊥ n to {V + D ⊥ n : V ∈ Z B n ⊕ | Z In \ B n } is equivalent under ξ to the quotient of d Ham on Z I n /C ⊥ n .These conclusions will be a corollary of the following lemma. Lemma 4.2.
There is a subspace D n ≤ Z Z In that is • invariant under the translation action of Z I n , • such that D ⊥ n ∩ ( Z B n ⊕ | Z In \ B n ) = ξ ( C ⊥ n ) ⊕ | Z In \ B n , • and such that whenever V , V ′ : Z I n → Z are supported on B n , the distance TS Ham /D ⊥ n ( V + D ⊥ n , V ′ + D ⊥ n ) is attained as TS Ham ( V , W ) for some W ∈ V ′ + D ⊥ n that is also supportedon B n (and hence, by the second point above, is identified with the sum of V ′ and a member of ξ ( C ⊥ n ) ). Proof
Given w ∈ Z T for any index set T , let w ∈ Z T be its image under thecoordinatewise application of the quotient homomorphism Z ։ Z .Now, for a = ( a x ) x ∈ Z I n we define the associated linear functional L a : Z I n → Z by composition with this quotient homomorphism and duality: L a ( v ) = h v , a i = X x ∈ I n v x a x . C n from Theorem 3.2, we simply let D n := { η + L a : η ∈ Z , a ∈ C n } .This collection of maps Z I n → Z is clearly a Z -subspace by the Z -linearity of a
7→ L a , and is translation-invariant because Trans u ( L a )( v ) dfn = L a ( v − u ) dfn = h v − u , a i = L a ( v ) − h u , a i . Observe next that if U , U ′ : Z I n → Z are both supported on B n then U − U ′ ∈ D ⊥ n ⇔ hhU , Z In ii = hhU ′ , Z In ii and hhU , L a ii = hhU ′ , L a ii ∀ a ∈ C n ⇔ hhU , L In ii = hhU ′ , L In ii and hhU , L a ii = hhU ′ , L a ii ∀ a ∈ C n (because spt U , spt U ′ ⊆ B n and Z In | B n = L In | B n ) ⇔ hhU , L a ii = hhU ′ , L a ii ∀ a ∈ C n (because we chose C n to contain I n ) ⇔ X x ∈ I n U ( e x ) h e x , a i = X x ∈ I n U ′ ( e x ) h e x , a i ∀ a ∈ C n ⇔ U − U ′ ∈ ξ ( C ⊥ n ) ⊕ | Z In \ B n . This proves the second part of the lemma.Now, by the addition-invariance of TS Ham on Z Z In (that is, invariance under ad-dition of a fixed member of Z Z In , not under translation of the base Z I n ), to provethe third part of the lemma it suffices to show that for any V : Z I n → Z with spt V ⊆ B n the minimum TS Ham /D ⊥ n ( D ⊥ n , V + D ⊥ n ) dfn = min W∈V + D ⊥ n TS Ham ( , W ) is attained (possibly not uniquely) for some W ∈ V + D ⊥ n that also has spt W ⊆ B n . Adjusting V by some B n -supported member of D ⊥ n (thus, effectively, by amember of ξ ( C ⊥ n ) ) if necessary, it further suffices to show that if V already mini-mizes TS Ham ( , · ) among members of its equivalence class modulo D ⊥ n ∩ ( Z B n ⊕ | Z In \ B n ) , then any other W : Z I n → Z with V − W ∈ D ⊥ n has TS Ham ( , W ) ≥ TS Ham ( , V ) . We will deduce this by showing how any W in V + D ⊥ n can be explicitly adjusted toanother member of this equivalence class that is at least as close to under TS Ham B n . Define the map R : Z Z In → Z B n ⊕ | Z In \ B n by R ( W )( e x ) := X w ∈ Z In W ( w ) h e x , w i : that is, R ( W ) is the Z -valued indicator function of the subset of B n containingthose basis vectors that appear with odd coefficients in the basis-decomposition ofan odd number of the members of spt W .Now suppose that ( x = , x , . . . , x ℓ +1 = ) is a cycle in Z I n that covers spt W .Since d Ham is a path metric on Z I n (corresponding to the usual nearest-neighbourgraph on the Hamming cube over the one-dimensional space given by Z with itsnatural word metric), we may interpolate additional points along the shortest pathsjoining each pair ( x i , x i +1 ) and re-label so that the value ℓ X i =1 d Ham ( x i , x i +1 ) in unchanged, but so that the consecutive points x i , x i +1 are now neighbours in thisgraph. Since the path starts and ends at , for any x ∈ I n and w ∈ spt W such that w x = 0 , the cycle must traverse some edge in direction x (that is, move betweentwo points of Z I n that differ by ± e x ) at least once before reaching w , and at leastonce again on its way back to . It follows that the length of the cycle is at leasttwice the number of different basis vectors e i that appear in the basis-representationof some w ∈ W , and this in turn is trivially bounded below by | spt R ( W ) | . Onthe other hand, the same reasoning easily shows that TS Ham ( , U ) = 2 | spt U | + 1 whenever spt
U ⊆ B n (since a cycle that starts at and simply goes up to eachmember of spt U in turn and then straight back down achieves the above lowerbound). Therefore we certainly have TS Ham ( , W ) ≥ | spt R ( W ) | + 1 = TS Ham ( , R ( W )) . It therefore suffices to show that R ( W ) − W ∈ D ⊥ n , since then the minimality of V among B n -supported members of its equivalence class shows that TS Ham ( , V ) ≤ TS Ham ( , R ( W )) , so that concatenating this with the above inequality completes the proof.Thus we must show that hW , η + L a i = hR ( W ) , η + L a i η ∈ Z and a ∈ C n ; in fact we will go slightly further and show that thisholds for all a ∈ Z B n . By the Z -linearity of the map a
7→ L a it suffices to proveseparately that hhW , Z In ii = hhR ( W ) , Z In ii , i.e. | spt W| ≡ | spt R ( W ) | mod 2 , and that hhW , L e x ii = hhR ( W ) , L e x ii ∀ x ∈ I n . To prove the first of these equalities, observe that | spt R ( W ) | = |{ x ∈ I n : e x appears with odd coeff.in an odd number of w ∈ spt W}|≡ X x ∈ I n |{ w ∈ spt W : h e x , w i = 1 }| mod 2 ≡ X x ∈ I n X w ∈ spt W h e x , w i mod 2 ≡ X w ∈ Z In W ( w ) D w , X x ∈ I n e x E mod 2 ≡ hhW , L In ii mod 2 . Since by construction I n ∈ C n and so L In ∈ D n , and also Z In ∈ D n , and weknow that W − V ∈ D ⊥ n , we have hhW , L In ii = hhV , L In ii≡ |V| mod 2 ≡ hhV , Z In ii = hhW , Z In ii≡ | spt W| mod 2 , so concatenating these equations gives the result.To prove the second equality, we simply observe that hhW , L e x ii = X w W ( w ) h w , e x i = D X w X y ∈ I n W ( w ) h e y , w i e y , e x E = X w R ( W )( w ) h e x , w i = hhR ( W ) , L e x ii , as required. 21 orollary 4.3. The map κ ◦ n : Z I n → Z Z In /D ⊥ n : a { e x ∈ B n : h e x , a i =1 } + D ⊥ n has kernel precisely C ⊥ n , and the resulting quotient map κ n : Z I n /C ⊥ n ֒ → Z Z In /D ⊥ n is a bi-Lipschitz embedding of d Ham /C ⊥ n into TS Ham /D ⊥ n with distortion at most and expansion ratio at most . Proof
The identification of the kernel of κ ◦ n is already contained in the secondpart of Lemma 4.2, so we need only prove the bi-Lipschitz and expansion ratiobounds.To this end, suppose that a , b ∈ Z I n . Then the images κ n ( a + C ⊥ n ) and κ n ( b + C ⊥ n ) are represented modulo D ⊥ n by the functions A := { e x ∈ B n : h e x , a i =1 } and B := { e x ∈ B n : h e x , b i =1 } . By the third part of Lemma 4.2 the distance TS Ham /D ⊥ n ( A + D ⊥ n , B + D ⊥ n ) is attained as TS Ham ( A ′ , B ′ ) for some A ′ ∈ A + ( ξ ( C ⊥ n ) ⊕ | Z In \ B n ) and B ′ ∈B + ( ξ ( C ⊥ n ) ⊕ | Z In \ B n ) , and given these it is simply equal to | spt( A ′ + B ′ ) | + 1 .However, we may clearly represent A ′ + D ⊥ n = κ ◦ n ( a ′ ) and B ′ + D ⊥ n = κ ◦ n ( b ′ ) , andhence deduce from the first part of the corollary that a ′ ∈ a + C ⊥ n and b ′ ∈ b + C ⊥ n .It follows that TS Ham ( A ′ , B ′ ) is precisely d Ham ( a ′ , b ′ ) + 1 , and thus that TS Ham /D ⊥ n ( κ n ( a + C ⊥ n ) , κ n ( b + C ⊥ n )) ≥ d Ham /C ⊥ n ( a + C ⊥ n , b + C ⊥ n ) + 1 . The reverse inequality also follows simply because for any a ′ ∈ a + C ⊥ n and b ′ ∈ b + C ⊥ n the images κ ◦ n ( a ′ ) and κ ◦ n ( b ′ ) are candidates for being set equal to A ′ and B ′ above.Thus in fact TS Ham /D ⊥ n ( κ n ( a + C ⊥ n ) , κ n ( b + C ⊥ n )) = 2 d Ham /C ⊥ n ( a + C ⊥ n , b + C ⊥ n ) + 1 , and since all distances involved are at least this gives rise to the asserted boundson bi-Lipschitz constant and expansion ratio.22 .3 Completion of the proof We will now show how the quotients of finite-dimensional product spaces stud-ied in the preceding subsection can be simultaneously bi-Lipschitzly recoveredfrom a single translation-invariant quotient of the metric ρ (( Z ⊕ ( Z ≀ G )2 ) ⋊ ( Z ≀ G )) on (( Z ⊕ ( Z ≀ G )2 ) ⋊ ( Z ≀ G )) = Z ⊕ ( Z ≀ G )2 .The tricky part is that we must find a single subspace V ≤ Z ⊕ ( Z ≀ G )2 such thatquotienting by it mimics the finite-dimensional quotienting by D ⊥ n studied aboveat each of an increasing sequence of ‘scales’, indexed by n , so that these scales donot ‘interact’. This is crucial to our recovery of a copy of ( Z Z In /D ⊥ n , TS Ham /D ⊥ n ) (and hence of ( Z I n /C ⊥ n , d Ham /C ⊥ n ) by Corollary 4.3) for each n . It is here that wewill see most clearly the usefulness of making a second extension to construct ouroverall group.Recall that given the group G and its metric ρ , we can pick positive reals r n , subsets I n and points y n as in the preceding section. Clearly by passing to a subsequenceif necessary, we may always assume that the sequence r < ρ ( y , e G ) < r < ρ ( y , e G ) < . . . . grows as fast as we please. We will henceforth refer to this as the sequence ofscales .We will now introduce the feature of Z (as opposed, say, to Z ) that makes itsuitable for our construction: if is its generator, and if we let A be the Z -valuedindicator function of the subset { , , , } , then A has the property (as may easilybe checked) that it and its distinct translates A ( · − v ) for v ∈ Z (of which thereare only three, since A ( · −
3) = A ) have no linear combination modulo that isequal to the indicator function Z . Let us now fix this indicator function A for therest of this paper.Define the sequence of maps Q n : Z ⊕ ( G \ ( I ∪ I ∪···∪ I n ∪{ y ,y ,...,y n } ))6 × Z Z In × Z → Z ⊕ Z ⊕ G by setting Q n ( u , W , u ) : w δ u ( w | G \ ( I ∪ I ∪···∪ I n ∪{ y ,y ,...,y n } ) ) · W ( w | I n ) · A ( w y n − u ) , and define also ˜ Q n : Z ⊕ ( G \ ( I ∪ I ∪···∪ I n ∪{ y ,y ,...,y n } ))6 × Z Z In × Z → Z ⊕ ( Z ≀ G )2 ˜ Q n ( u , W , u )( w , g ) = (cid:26) Q n ( u , W , u )( w ) if g = e G else.Thus Q n ( u , W , u ) is the Z -valued indicator function of the set of those w ∈ Z ⊕ G such that • the restriction of w to the indices in G \ ( I ∪ I ∪ · · · ∪ I n ∪ { y , y , . . . , y n } ) agrees with u , • the restriction w | I n is a member of spt W , and • the single coordinate w y n ( = w | { y n } ) is in the translated set Trans u (spt A ) .These maps Q n and ˜ Q n will both underpin the construction of V and form thebuilding blocks of our final embeddings into Z ⊕ ( Z ≀ G )2 . The slightly mysteriousinclusion of a copy of A in this definition will be important for keeping track ofcertain possible cancelations when we add translated copies of images under thesemaps Q n , and will be clarified shortly.Some of the immediate properties of Q n seem worth making explicit: Definition 4.4 (Linearity; translation-covariance) . Each map Q n is linear in itssecond argument: Q n ( u , W , u ) + Q n ( u , W ′ , u ) = Q n ( u , W + W ′ , u ) ∀W , W ′ ∈ Z Z In for fixed u and u . We will refer to this as the linearity property of Q n .Each map Q n also behaves well under translations in all three arguments: Trans w Q n ( u , W , u )( · )= Q n (cid:0) u + w | G \ ( I ∪···∪ I n ∪{ y ,...,y n } ) , Trans w | In W , u + w y n (cid:1) ( · ) ∀ w , u , W , u. We will refer to this as the translation-covariance property of Q n . Having made these definitions, we can state the last main result on the route toTheorem 3.3. 24 roposition 4.5. If ( D n ) n ≥ is as in Lemma 4.2 and the sequence of scales growssufficiently fast then there is a Z -subspace V ≤ Z ⊕ ( Z ≀ G )2 that is invariant underthe coordinate right-translation action of Z ≀ G and for which the following holds:if for each n ≥ we define the mapping ψ ◦ n : Z Z In → Z ⊕ ( Z ≀ G )2 /V : W 7→ ˜ Q n ( G \ ( I ∪···∪ I n ∪{ y ,...,y n } ) , W ,
0) + V, then1. ψ ◦ n is Z -linear with kernel D ⊥ n ;2. the resulting quotient map ψ n : Z Z In /D ⊥ n ֒ → Z ⊕ ( Z ≀ G )2 /V inclusion ֒ → ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) is a bi-Lipschitz embedding of TS Ham /D ⊥ n into ρ ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) with dis-tortion at most O( M ) and expansion ratio O( r n ) . We will prove this proposition in several steps in this subsection, but let us first seehow it and Corollary 4.3 together imply Theorem 3.3, and so complete the proofof Theorem 1.1.
Proof of Theorem 3.3 from Proposition 4.5
This follows simply by setting ϕ n := ψ n ◦ κ n : Z I n /C ⊥ n ֒ → ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) , since the estimates of Corollary 4.3 and Proposition 4.5 show that ϕ n has distortionat most · (4 M ) = 8 M , which does not depend on n , and expansion ratio O( r n ) ,where the dimension of the cube Z I n is d n ≥ r n and so the diameter of itsquotient Z I n /C ⊥ n , which has L p -distortion proportional to this diameter, is also Ω( d n ) , which grows faster than any power of r n .We will construct the subspace V that we need in a number of steps. First, giventhe subspaces D n ≤ Z Z In obtained in Lemma 4.2, let U n := span Z (cid:8) Q n ( u , W , u ) : u ∈ Z ⊕ ( G \ ( I ∪···∪ I n ∪{ y ,...,y n } ))6 , W ∈ D ⊥ n , u ∈ Z (cid:9) . Let us also define U + n := span Z (cid:8) Q n ( u , W , u ) : u ∈ Z ⊕ ( G \ ( I ∪···∪ I n ∪{ y ,...,y n } ))6 , W ∈ Z Z In , u ∈ Z (cid:9) ,
25o that U n ≤ U + n .It follows at once from the translation-covariance of Q n and the translation-invarianceof D ⊥ n that U n (and similarly U + n ) is a translation-invariant subspace of Z ⊕ Z ⊕ G .Now we let U := P n ≥ U n , so that this is still a translation-invariant Z -subspaceof Z ⊕ Z ⊕ G , and finally we extend this to the subspace V := (cid:8) V ∈ Z ⊕ ( Z ≀ G )2 : the map ( w g ) g ∈ G
7→ V (( w gg − ) g ∈ G , g ) lies in U for every g ∈ G (cid:9) of Z ⊕ ( Z ≀ G )2 , which is manifestly invariant under all right-translations by membersof Z ≀ G . This is the subspace that we will show enjoys the properties listed inProposition 4.5.Our first step is the following simple strengthening of Lemma 2.6, which willreduce our proof of Proposition 4.5 to a study of a simpler map into the space Z ⊕ ( Z ≀ G ) /U . Lemma 4.6.
The composition of mappings Z ⊕ ( Z ≀ G ) /U ֒ → Z ⊕ ( Z ≀ G )2 /V ֒ → ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) U + U ( U ⊕ | ( Z ≀ G ) \ ( Z ≀ G ) ) + V (cid:0) ( U ⊕ | ( Z ≀ G ) \ ( Z ≀ G ) ) + V, ( , e G ) (cid:1) is a well-defined Z -linear injection, and moreover is a -bi-Lipschitz embeddingof the metric TS ρ ( Z ≀ G )0 , ( ,e G ) /U on Z ⊕ ( Z ≀ G ) /U into ρ ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) withexpansion ratio Θ(1) . Proof
We have already seen in Lemma 2.6 that the second part of our composi-tion, Z ⊕ ( Z ≀ G )2 /V ֒ → ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) V + V (cid:0) V + V, ( , e G ) (cid:1) , is a -bi-Lipschitz embedding of TS ρ ( Z ≀ G ) , ( ,e G ) /V into ρ ( Z ⊕ ( Z ≀ G )2 /V ) ⋊ ( Z ≀ G ) withexpansion ratio Θ(1) ; therefore it will suffice to show that the first part, Z ⊕ ( Z ≀ G ) /U ֒ → Z ⊕ ( Z ≀ G )2 /V, is a well-defined Z -linear injection that is an isometric embedding of TS ρ ( Z ≀ G )0 , ( ,e G ) /U into TS ρ ( Z ≀ G ) , ( ,e G ) /V .Linearity is immediate, and the correctness of the definition and injectivity holdbecause from the definition of V we see that if U , U ′ ∈ Z ⊕ ( Z ≀ G )2 are both supported26n ( Z ≀ G ) , then they differ by a member of V if and only if U | ( Z ≀ G ) and U ′ | ( Z ≀ G ) differ by a member of U .Isometricity follows from some simple consideration of the definition of the pinnedtraveling salesman metric. Let us identify Z -valued functions on ( Z ≀ G ) withtheir extensions by to Z ≀ G to lighten notation. Given any W ∈ V , let W be thefunction W · ( Z ≀ G ) obtained by replacing W with the identically-zero functionoutside the zero section ( Z ≀ G ) , and observe from the definition of V that W ∈ V also. Now if U and U ′ are supported on the zero section, it follows that spt( U −U ′ + W ) ⊇ spt( U − U ′ + W ) , and now appealing to the monotonicity propertyof Lemma 2.5 we deduce that for such U and U ′ the quotient-metric distance TS ρ ( Z ≀ G ) , ( ,e G ) /V ( U + V, U ′ + V ) is attained as TS ρ ( Z ≀ G ) , ( ,e G ) ( U , U ′ + W ) for some W also supported on ( Z ≀ G ) ,and hence agrees with TS ρ ( Z ≀ G )0 , ( ,e G ) /U ( U + U, U ′ + U ) , as required.We next prove the two lemmas that will underpin the main estimates involved inthe proof of Proposition 4.5. Lemma 4.7.
Suppose that η ∈ Z , m ≥ is an integer and that we are given • a fixed point u ◦ ∈ Z ⊕ ( G \ ( I ∪···∪ I m ∪{ y ,...,y m } ))6 , and • for each n ≤ m , a finite family of functions F n := {Q n ( u ( n )1 , W ( n )1 , u ( n )1 ) , Q n ( u ( n )2 , W ( n )2 , u ( n )2 ) , . . . , Q n ( u ( n ) i n , W ( n ) i n , u ( n ) i n ) } from U + n such that u ( n ) i | ( G \ ( I ∪···∪ I m ∪{ y ,...,y m } )) = u ◦ for all i ≤ i n such that m X n =1 i n X i =1 Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) = η { u ◦ }× Z I ∪···∪ Im ∪{ y ,...,ym } . Then we must have η = 0 and i n X i =1 Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) = 0 for each n ≤ m separately. emark It will be important that we have this lemma available for basis membersof the larger spaces U + n , not just of U n . ⊳ Proof
This will follow by induction on the number among the families F , F ,. . . , F m that are nonempty. Base clause
Suppose that only one F n is nonempty, in which case our assump-tion reads i n X i =1 Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) = η { u ◦ }× Z I ∪···∪ Im ∪{ y ,...,ym } , and we need only show that η = 0 . We prove this by contradiction, so suppose in-stead that η = 1 . The function Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) is supported on the (infinite-dimensional) cylinder { u ( n ) i } × Z ( I ∪···∪ I n ∪{ y ,...,y n } )6 , and so if u ( n ) i = u ( n ) j then Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) and Q n ( u ( n ) j , W ( n ) j , u ( n ) j ) are disjointly supported. Nowsuppose that v is a vector that appears in the list u ( n )1 , . . . , u ( n ) i n (so that by as-sumption, v is an extension of u ◦ ), and let j < j < . . . < j k be those values of i ≤ i n where it appears. It follows from the above equation that k X s =1 Q n ( v , W ( n ) j s , u ( n ) j s ) = 1 on { v } × Z I n ∪{ y n } × { I ∪···∪ I n − ∪{ y ,...,y n − } } . Now pick any w ∈ Z I n ; the above equation requires that W ( n ) j s ( w ) = 1 for atleast one s ≤ k . Letting s < s < . . . < s ℓ be those values of s ≤ k where W ( n ) j s ( w ) = 1 , we deduce from the above equation that ℓ X r =1 Q n ( v , W ( n ) j sr , u ( n ) j sr ) = 1 on { v } × { w } × Z × { I ∪···∪ I n − ∪{ y ,...,y n − } } . However, each of the functions Q n ( v , W ( n ) j sr , u ( n ) j sr ) restricted to the set { v ⊕ w } × Z × { I ∪···∪ I n − ∪{ y ,...,y n − } } ∼ = Z is simply a rotated copy of the function A , and we introduced this precisely so as to have the property that the constantfunction cannot be made as a linear combination of its rotates. This gives thedesired contradiction. Recursion clause
Suppose that ≤ n < n < . . . < n ℓ = m with ℓ ≥ are the values of n for which F n is nonempty. We will show that in this case the28ssumption that ℓ X j =1 i nj X i =1 Q n j ( u ( n j ) i , W ( n j ) i , u ( n j ) i ) = η { u ◦ }× Z I ∪···∪ Im ∪{ y ,...,ym } , implies that also ℓ X j =2 i nj X i =1 Q n j ( u ( n j ) i , W ( n j ) i , u ( n j ) i ) = η { u ◦ }× Z I ∪···∪ Im ∪{ y ,...,ym } , (that is, omitting the first term of the outer sum) so that an induction on ℓ completesthe proof.To see this, observe that each of the functions Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) is sup-ported on the set { u ( n ) i } × Z I n ∪ I n − ∪···∪ I ∪{ y n ,y n − ,...,y } , whereas each of thefunctions Q n j ( u ( n j ) i , W ( n j ) i , u ( n j ) i ) for j ≥ is constant on every set of the form { v } × Z I n ∪···∪ I ∪{ y n ,...,y } with v ∈ Z ⊕ G \ ( I n ∪···∪ I ∪{ y n ,...,y } )6 , as is the func-tion η { u ◦ }× Z I ∪···∪ Im ∪{ y ,...,ym } . Therefore the stated cancelation can take placeonly if for every v that appears in the list u ( n )1 , u ( n )2 , . . . , u ( n ) i n we have X i ≤ i n , u ( n i = v Q n ( v , W ( n ) i , u ( n ) i ) = const . on { v } × Z I n ∪···∪ I ∪{ y n ,...,y } , and applying the argument for the base clause to this sub-sum tells us that theconstant in question must be zero. Summing over v now gives i n X i =1 Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) = 0 , and finally subtracting this sum from the initially-given equation completes therecursion step, so the induction continues. Remark
If we assume that η = 0 a priori then the above proposition assertssimply that the subspaces U + n ≤ Z ⊕ Z ⊕ G , n = 1 , , . . . , are linearly independentover Z . Indeed, given any equation m X n =1 i n X i =1 Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) = 0
29e may first partition the left-hand side into sub-sums according to the values of u ( n ) i | ( G \ ( I ∪···∪ I m ∪{ y ,...,y m } )) , and now these sub-sums must all vanish separatelybecause any two maps Q n ( u ( n ) i , W ( n ) i , u ( n ) i ) and Q n ′ ( u ( n ′ ) j , W ( n ′ ) j , u ( n ′ ) j ) have dis-joint support if these restrictions do not agree. Each of these sub-sums falls withinthe conditions of the proposition. ⊳ We are now ready to deduce the crucial relation between the metric TS ρ ( Z ≀ G )0 , ( ,e G ) on Z ⊕ ( Z ≀ G ) and the metrics TS Ham on some of its finite-dimensional factor spaces.
Lemma 4.8.
Provided the sequence of scales grows sufficiently fast, the metric TS ρ ( Z ≀ G )0 , ( ,e G ) admits the following kind of approximation. Suppose that U n , V n : Z I n → Z are non-zero and distinct, and that U , V : Z ⊕ G → Z are such that spt U ⊆ { } × spt U n × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 , spt V ⊆ { } × spt V n × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 , and spt U ∩ (cid:0) { } × { u } × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 (cid:1) = ∅ ∀ u ∈ spt U n , spt V ∩ (cid:0) { } × { u } × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 (cid:1) = ∅ ∀ u ∈ spt V n , U | { }×{ u }× Z ( I ∪···∪ In − ∪{ y ,...,yn } )6 = V| { }×{ u }× Z ( I ∪···∪ In − ∪{ y ,...,yn } )6 ∀ u ∈ spt U n ∩ spt V n . Then we have r n · TS Ham ( U n , V n ) ≤ TS ρ ( Z ≀ G )0 , ( ,e G ) ( U , V ) ≤ M r n · TS Ham ( U n , V n ) (irrespective of the actual sizes of the sets spt U , spt V ). Proof
We now write simply for the zero of either Z ⊕ G or Z ⊕ G \ ( I ∪···∪ I n ∪{ y ,...,y n } )6 for any n , since among these possibilities the index set will always be clear fromthe context.If the distance r n is sufficiently large compared with r , ρ ( e G , y ) , r , ρ ( e G , y ) ,. . . , r n − and ρ ( e G , y n ) , and if u , u ′ ∈ Z I n are distinct, then the pinned travelingsalesman distance in ( Z ≀ G ) between any points of the cylinder sets { } × { u } × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 { } × { u ′ } × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 will be c · ρ ( Z ≀ G ) ( ⊕ u ⊕ I ∪···∪ I n − ∪{ y ,...,y n } , ⊕ u ′ ⊕ I ∪···∪ I n − ∪{ y ,...,y n } ) ≥ r n , for some ≤ c ≤ , since the length of a cycle in G needed to cover all thecoordinates where u differs from u ′ dwarfs the maximum number of steps thatcould possibly be needed to cover all the differences between two points necessaryat coordinates indexed by I ∪ · · · ∪ I n − ∪ { y , . . . , y n } .Similarly, if r n is sufficiently large then the pinned traveling salesman distancebetween these cylinder sets also dwarfs the maximum pinned traveling salesmanlength of any non-self-intersecting path within either of these cylinder sets.It follows for U and V as given that any cycle in ( Z ≀ G ) starting and ending at ( , e G ) and covering spt( U + V ) needs to cover at least one point of each of thesets spt U ∩ (cid:0) { } × { u } × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 (cid:1) for u ∈ spt U n \ spt V n and spt V ∩ (cid:0) { } × { u } × Z ( I ∪···∪ I n − ∪{ y ,...,y n } )6 (cid:1) for u ∈ spt V n \ spt U n , but no others, and that the distance between any two of these sets is much largerthan the maximum number of steps in a path that could be needed to cover thenecessary points within any one of them.Therefore the length of such a covering traveling salesman cycle for spt( U + V ) isbounded below by the length of a traveling salesman cycle in ( Z ≀ G ) for the set (cid:8) ⊕ u ⊕ | I ∪···∪ I n − ∪{ y ,...,y n } : u ∈ spt U n △ spt V n (cid:9) , and if r n is sufficiently large compared with its predecessors then it may also bebounded from above by twice this number. Finally, since by construction any twoindex points of I n are separated by a ρ -distance that lies in [ r n , M r n ] , and soany covering cycle for this latter set must traverse an additional distance of at least r n and at most M r n for each new point that it must visit, we deduce this latterdistance lies between r n · TS Ham ( U n , V n ) and M r n · TS Ham ( U n , V n ) .Combining the bounds obtained above now completes the proof.31 roof of Proposition 4.5 First, Lemma 4.6 allows us to consider instead themaps λ ◦ n : Z Z In → Z ⊕ ( Z ≀ G ) /U : W 7→ Q n ( , W ,
0) + U, for which it will suffice to show the corresponding properties:1. λ ◦ n is Z -linear with kernel D ⊥ n ;2. the resulting quotient map λ n : Z Z In /D ⊥ n ֒ → Z ⊕ ( Z ≀ G ) /U is a bi-Lipschitz embedding of TS Ham /D ⊥ n into TS ρ ( Z ≀ G )0 , ( ,e G ) /U with dis-tortion at most O( M ) and expansion ratio O( r n ) . It is clear from the definition of U n ≤ U that λ ◦ n annihilates D ⊥ n , so weneed only show that it does not annihilate any larger subspace of Z Z In . However,a special case of Lemma 4.7 tells us that the subspaces U + n ≤ Z ⊕ ( Z ≀ G ) are lin-early independent (as remarked immediately after that lemma). It follows that if λ ◦ n ( V ) = Q n ( , V ,
0) + U is equal to the identity element U in Z ⊕ ( Z ≀ G ) /U , thenthere are elements A ∈ U , A ∈ U , . . . , A m ∈ U m for some m ≥ n such that A + A + · · · + (cid:0) Q n ( , V ,
0) + A n (cid:1) + · · · + A m = , and this is possible only if each A i is individually zero for i = n and also Q n ( , V ,
0) = A n .We now express A n as i X i =1 Q n ( u i , W i , u i ) W i ∈ D ⊥ n , u i ∈ Z ∀ i ≤ i , and will show that we can have Q n ( , V , equal to such a sum only if in fact V is itself a member of D ⊥ n . First, we may clearly discard all terms of this sum forwhich u i = , since Lemma 4.7 tells us immediately that these must cancel to ,and relabel so that u i = for every i ≤ i . Therefore, recalling the definition of Q n and omitting the fixed vector , the above equation becomes V ( w ) · A ( w ) = i X i =1 W i ( w ) · A ( w − u i ) ∀ w ∈ Z I n , w ∈ Z . w ∈ spt A , this equation simplifies to V ( w ) = X ≤ i ≤ i , A ( w − u i )=1 W i ( w ) ∀ w ∈ Z I n , and so we have expressed V as a linear combination of members of D ⊥ n , and henceproved that it itself lies in D ⊥ n , as required. We prove the two necessary inequalities separately. First we prove that TS ρ ( Z ≀ G )0 , ( ,e G ) /U ( λ n ( U + D ⊥ n ) , λ n ( V + D ⊥ n )) ≤ M r n · TS Ham /D ⊥ n ( U + D ⊥ n , V + D ⊥ n ) . Indeed, if
W ∈ D ⊥ n minimizes TS Ham ( U , V + W ) , then Q n ( , W , ∈ U n ≤ U and by a simple application of Lemma 4.8 the distance TS ρ ( Z ≀ G )0 , ( ,e G ) ( Q n ( , U , , Q n ( , V ,
0) + Q n ( , W , is at least r n and at most M r n times TS Ham ( U , V + W ) , as required.The necessary reverse inequality TS ρ ( Z ≀ G )0 , ( ,e G ) /U ( λ n ( U + D ⊥ n ) , λ n ( V + D ⊥ n )) & r n · TS Ham /D ⊥ n ( U + D ⊥ n , V + D ⊥ n ) requires a little more work. By addition-invariance we may assume that U = .Now suppose that V ∈ Z Z In and m ≥ n and that F k := {Q k ( u ( k )1 , W ( k )1 , u ( k )1 ) , Q k ( u ( k )2 , W ( k )2 , u ( k )2 ) , . . . , Q k ( u ( k ) i k , W ( k ) i k , u ( k ) i k ) } for k = 1 , , . . . , m are finite collections in each U k . Clearly we may assume in our notation that thefunctions W ( k ) i are all non-zero, that F m = ∅ and finally that m is minimal subjectto all these other restrictions.To complete the proof we will show that for some such finite data the combinedcollection F := S k ≤ m F k actually minimizes the value TS ρ ( Z ≀ G )0 , ( ,e G ) (cid:0) , Q n ( , V ,
0) + Σ F (cid:1) (2)over all such finite tuples of finite collections, where we write Σ F simply for thesum in Z ⊕ ( Z ≀ G ) of all elements of F , and then from this minimizing F we willconstruct some W ∈ D ⊥ n so that TS ρ ( Z ≀ G )0 , ( ,e G ) (cid:0) , Q n ( , V ,
0) + Σ F (cid:1) ≥ r n · TS Ham ( , V + W ) . tep (i) We first show by contradiction that in order to infimize the expres-sion (2) over finite tuples F k , k ≤ m , it suffices to assume that m ≤ n . Indeed,suppose instead that m ≥ n + 1 . First, if any u ( m ) i is not , then we may simplyomit the corresponding member of F m together with all members of any F k with k < m whose supports it dominates, and obtain a new family F ′ for which TS ρ ( Z ≀ G )0 , ( ,e G ) (cid:0) , Q n ( , V , F ′ (cid:1) < TS ρ ( Z ≀ G )0 , ( ,e G ) (cid:0) , Q n ( , V , F (cid:1) . Therefore it suffices to assume that u ( m ) i = for all i ≤ i m . Next, for any given w ∈ Z , we have X i ≤ i m Q m ( , W ( m ) i , u ( m ) i )( ⊕ w ⊕ w ⊕ u ) = X i ≤ i m , A ( w − u ( m ) i )=1 W ( m ) i ( w ) for w ∈ Z I m , u ∈ Z I ∪···∪ I m − ∪{ y ,...,y m − } . Regarded as a function of w ∈ Z I m alone, we see that this is a member of D ⊥ m . Ifit is zero for every w ∈ Z , then summing over w shows that Σ F m = 0 , and so wemay simply discard F m to leave a new family F ′ := S k ≤ m − F k that achieves thesame value for the expression (2) but has a smaller value of m . Hence after makingfinitely many such omissions, we may assume also that there is some w ∈ Z forwhich the above sum specifies a nonzero member of D ⊥ m ; let us call that member Y . Then we also have Y 6 = δ (simply because δ D ⊥ m ), and so to this w therecorresponds some w ∈ Z I m \ { } for which Y ( w ) = 0 .However, it now follows that the function P i ≤ i m Q m ( , W ( m ) i , u ( m ) i ) is non-zeroon the cylinder { } × { w } × { w } × Z I ∪···∪ I m − ∪{ y ,...,y m − } , and so actuallytakes the constant value on that cylinder. By Lemma 4.7 the sum function Q n ( , V ,
0) + Σ( F ∪ · · · ∪ F m − ) cannot be constant and equal to on thiscylinder, and so overall the function Q n ( , V ,
0) + Σ F does not vanish on thiscylinder. On the other hand, if our sequence of scales grows fast enough, then anypoint of this cylinder is by itself much further away from ( , e G ) in ρ ( Z ≀ G ) thanany possible value of TS ρ ( Z ≀ G )0 , ( ,eG ) ( , Q n ( , V , , and so in this case even theempty family gives a smaller value for the expression (2) than does F . Thereforethe infimum of the values (2) is unchanged if we only infimize over families ofcollections F for which m ≤ n . Step (ii)
Having reduced to the case m ≤ n , we can now argue as above toreduce the evaluation of the infimum of (2) further to the case when u ( k ) i | G \ ( I ∪···∪ I n ∪{ y ,...,y n } ) = G \ ( I ∪···∪ I n ∪{ y ,...,y n } ) ∀ k ≤ m, -valued coordinates or is identically on it, and so discardingthose that are identically zero gives another collection F for which the value (2) isnot any larger. This last reduction leaves only finitely many possibilities for the col-lection F , and so now we know that we may pick one that is actually minimizing.One last ‘processing’ step will give us the construction of W from it. Step (iii)
For this, it now follows from another application of Lemma 4.7 thaton any of the cylinders { } × { w } × { } × Z I ∪···∪ I n − ∪{ y ,...,y n − } for w ∈ Z I n the sum function Q n ( , V ,
0) + Σ F can vanish only if each individual sum Σ F k vanishes there for each k ≤ n − , and also Q n ( , V ,
0) + Σ F n vanishes there.Substituting the definition of Q n into this latter condition, it becomes that V ( w ) + X ≤ i ≤ i n , A ( − u ( n ) i )=1 W ( n ) i ( w ) = 0 . Therefore, the indicator function of the set (cid:8) w ∈ Z I n : (cid:0) Q n ( , V ,
0) + Σ F (cid:1) | { }×{ w }×{ }× Z I ∪···∪ In − ∪{ y ,...,yn − } (cid:9) is precisely V + P ≤ i ≤ i n , A ( − u ( n ) i )=1 W ( n ) i , and therefore it is a member of V + D ⊥ n ;let us call it V + W . By Lemma 2.5 and Lemma 4.8 this now implies at once that TS ρ ( Z ≀ G )0 , ( ,e G ) (cid:0) , Q n ( , V ,
0) + Σ F (cid:1) ≥ r n · TS Ham ( , V + W ) . as required. I suspect that the construction above can be extended so far as to give a finitely-generated amenable group G with a word metric ρ for which any -Lipschitz em-bedding f : G ֒ → L p , p ∈ [1 , ∞ ) , must be such that k f ( g ) − f ( h ) k . log ρ ( g, h ) for some collection of pairs g, h ∈ G among which ρ ( g, h ) can be arbitrarily large. Question
Does every finitely-generated amenable group G admit Lipschitz em-beddings f : G ֒ → L p for every p ∈ [1 , ∞ ) such that k f ( g ) − f ( h ) k p & log ρ ( g, h ) for all g, h ∈ G ? 35pplied with a little less violence (that is, with η < in Lemma 3.1), the meth-ods of this paper may also be useful for finding examples of finitely-generatedamenable groups with specified compression exponents in (0 , , although of coursethese methods will need to be complemented with matching lower-bound proofs forthe groups in question (that is, constructions of particular good-distortion embed-dings, on which the ideas of the present paper do not seem to bear directly). Thekind of construction used above may also provide interesting test-cases for Ques-tion 10.6 of Naor and Peres [16], which asks for more general methods for estimat-ing compression exponents of semidirect products; and on the question of whichvalues are realizable as Euclidean compression exponents for finitely-generatedamenable groups, specializing the result of Arzhantseva, Drutu and Sapir in [2]that all values in [0 , can be realized as the compression exponents of finitely-generated groups that are not necessarily amenable.It is interesting to note that in [12] Khot and Naor use inequalities described interms of random walks on Hamming cubes to prove Theorem 3.2. Thus, althoughthe random walk upper bound on compression exponents given by Naor and Perescannot reach the zero-compression-exponent regime of the group ( Z ⊕ H /V ) ⋊ H constructed above, it seems that a modification of their method in which we allowourselves to consider a sequence of random walks on our group supported on theimages of these embedded finite cube-quotients would translate into the correctzero upper bound on the compression exponent. It would be interesting to find anexample of a finitely-generated amenable group for which some other obstructionto embeddings with positive compression exponents is needed, genuinely unrelatedto inequalities concerning random walks.One candidate for such an obstruction would be a sequence of embedded copiesof the cubes ( Z n m , ℓ ∞ ) : given an embedded sequence of these with expansion ra-tios not growing too fast, we could instead use an analog of Lemma 3.1 based onthe notion of non-linear cotype introduced by Mendel and Naor in [14], which ischaracterized in terms of these finite spaces. Given a suitable tradeoff between thesizes and expansion ratios, the arguments leading to Mendel and Naor’s Theorem1.11 of [14] would imply that such a sequence of embedded ℓ ∞ -cubes obstructsgood-compression embeddings into any Banach space with nontrivial type and co-type q < ∞ , provided in addition that the parameter m of the embedded cubescan be chosen to grow faster than n /q . (Note also that whether the assumptionof nontrivial type is necessary is one of the major outstanding problems from theirpaper.) However, at this stage I do not see how to construct a finitely-generatedgroup that contains copies of these ℓ ∞ -cubes with long side-lengths.Finally, we remark that the methods above give a group with poor compression36xponent specifically for embeddings into L p for p < ∞ , because these are theBanach spaces to which Khot and Naor’s analysis of cube-quotients in [12] applies(see Remark 3.1 of their paper). It is natural to ask whether some more ‘purelygeometric’ feature of the choice of Banach target could be responsible for thispoor embeddability. Question
Do the groups ( Z ⊕ H /V ) ⋉ H admit Lipschitz embeddings with posi-tive compression exponents into any Banach space with finite cotype? References [1] I. Aharoni, B. Maurey, and B. S. Mityagin. Uniform embeddings of metricspaces and of Banach spaces into Hilbert spaces.
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