Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

(1)

Article

Reference

Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

ARZHANTSEVA, Goulnara N., DRUŢU, Cornelia, SAPIR, Mark

Abstract

We construct finitely generated groups with arbitrary prescribed Hilbert space compression alpha from the interval [0,1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.

ARZHANTSEVA, Goulnara N., DRUŢU, Cornelia, SAPIR, Mark. Compression functions of uniform embeddings of groups into Hilbert and Banach spaces. Journal für die reine und angewandte Mathematik , 2009, vol. 2009, no. 633, p. 213-235

arxiv : math/0612378

DOI : 10.1515/CRELLE.2009.066

Available at:

http://archive-ouverte.unige.ch/unige:9746

Disclaimer: layout of this document may differ from the published version.

(2)

(Walter de Gruyter BerlinNew York 2009

Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

ByGoulnara Arzhantsevaat Gene`ve,Cornelia Drut¸uat Oxford, and Mark Sapirat Nashville

Abstract. We construct ﬁnitely generated groups with arbitrary prescribed Hilbert space compression aA½0;1. This answers a question of E. Guentner and G. Niblo.

For a large class of Banach spacesE(including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 2, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (moreover, of finite asymptotic dimension and exact) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.

1. Introduction

1.1. Uniform embeddings. The property of uniform embeddability of groups into Hilbert spaces (and, more generally, into Banach spaces) became popular after Gromov [Gr1] suggested that this property might imply the Novikov conjecture. Indeed, following this suggestion, Yu [Yu] and later Kasparov and Yu [KY] proved that a ﬁnitely generated group uniformly embeddable into a Hilbert space, respectively into a uniformly convex Banach space, satisﬁes the Novikov conjecture.

This raised the question whether every ﬁnitely generated group can be embedded uniformly into a Hilbert space, or more generally, into a uniformly convex Banach space.

Gromov constructed [Gr2] ﬁnitely generatedrandomgroups whose Cayley graphs (quasi)- contain some inﬁnite families of expanders and thus cannot be embedded uniformly into a Hilbert space (or into anyl^p with 1ep<y, e.g. [Roe], Ch.11.3). The recent results of V. La¤orgue [Laf ] yield a family of expanders that is not uniformly embeddable into any

This article was written while all the three authors were visitors at the Max Planck Institute in Bonn. We are grateful to the MPI for its hospitality. The research of the ﬁrst author was supported by the Swiss National Science Foundation, grants PP002-68627, PP002-116899. The research of the third author was supported in part by the NSF grants DMS 0245600 and DMS 0455881 and by a BSF (USA-Israeli) grant.

Brought to you by | Université de Genève - Bibliothèque de Genève Authenticated

(3)

uniformly convex Banach space. Nevertheless, one cannot apply Gromov’s argument to deduce that random groups corresponding to La¤orgue’s family of graphs do not embed uniformly into any uniformly convex Banach space. Indeed, La¤orgue’s expanders are Cayley graphs of ﬁnite quotients of a non-free group, therefore there are loops of bounded size in all of them, and the graphs have bounded girth. On the other hand, Gromov’s argument succeeds only if the girth of a graph in the family of expanders is of the same order as the diameter of the graph.

1.2. Compression and compression gap.

Deﬁnition 1.1 (cf. [GuK]). Let ðX;dXÞ and ðY;dYÞ be two metric spaces and let f:X !Y be a 1-Lipschitz map. The compression of f is the supremum over all af0 such that

dY

fðuÞ;fðvÞ

fdXðu;vÞ^a for allu,vwith large enough dXðu;vÞ.

If E is a class of metric spaces closed under rescaling of the metric, then the E-compression of X is the supremum over all compressions of 1-Lipschitz maps X !Y, Y AE. In particular, ifEis the class of Hilbert spaces, we get theHilbert space compression ofX.

The E-compression measures the least possible distortion of distances when one tries to draw a copy ofX inside a space fromE. It is a quasi-isometry invariant ofX and it takes values in the interval ½0;1. Similar concepts of distortion have been extensively studied (mostly for ﬁnite metric spaces mapped into ﬁnite dimensional Hilbert spaces) by combina- torists for many years (see [Bou], [DL], for example).

Since any ﬁnitely generated groupG can be endowed with a word length metric and all such metrics are quasi-isometric, one can speak about theE-compression of a group G.

Guentner and Kaminker proved in [GuK] that if the Hilbert space compression of a ﬁnitely generated group G is larger than 1=2 then the reduced C-algebra of G is exact (in other words,G isexactorGsatisﬁes Guoliang Yu’spropertyA [Yu]).

One of the goals of this paper is to describe all possible values ofE-compression for ﬁnitely generated groups, whenEis either the class of Hilbert spaces or, more generally, the class of uniformly convex Banach spaces.

A very limited information was known about the possible values of Hilbert space compression of ﬁnitely generated groups. For example, word hyperbolic groups have Hil- bert space compression 1 [BS], and so do groups acting properly and co-compactly on a CAT(0) cube complex [CN]; co-compact lattices in arbitrary Lie groups, and all lattices in semi-simple Lie groups have Hilbert space and, moreover, L^p-compression 1 [Te]; any group that is not uniformly embeddable into a Hilbert space (such groups exist by [Gr2]) has Hilbert space compression 0, etc. (see the surveys in [AGS], [Te]). The ﬁrst groups with Hilbert space compressions strictly between 0 and 1 were found in [AGS]: R. Thompson’s groupF has Hilbert space compression 1=2, the Hilbert space compression of the wreath productZoZis between 1=2 and 3=4 (later it was proved in [ANP] that it is actually 2=3), the Hilbert space compression ofZo ðZoZÞis between 0 and 1=2.

(4)

The notion of E-compression can be generalized to the notion ofE-compression gap of a spaceX, whereEis any class of metric spaces closed under rescaling of the metric (see Definition 2.6). It measures even more accurately than the compression the best possible (least distorted) way of embedding X in a space from E. For example, it is proved in [AGS] that R. Thompson’s group F has Hilbert space compression gap ðpffiffiffix

;pffiffiffix logxÞ.

This means there exists an 1-Lipschitz embedding ofF into a Hilbert space with compression function ffiffiffi

px

, and every 1-Lipschitz embedding ofF into a Hilbert space has compression function at most pffiffiffix

logx. This is much more precise than simply stating that the Hilbert space compression of F is 1=2. Another example: it follows from [Te] that every lattice in a semi-simple Lie group has a Hilbert space compression gap x

ffiffiffiffiffiffiffiffiffiffi logx

p log logx;x

!

and the upper bound of the gap cannot be improved. This is a much more precise statement than the statement that the Hilbert space compression of the lattice is 1.

In this paper, we show that a large class of functions appears as Hilbert space compression functions of graphs of bounded degree, and as upper bounds of Hilbert space compression gaps of logarithmic size of ﬁnitely generated groups. This class of functions is deﬁned as follows. We use the notationRþfor the interval½0;yÞ.

Deﬁnition 1.2. Let C be the collection of continuous functions r:Rþ!Rþ such that for somea>0:

(1) ris increasing on½a;yÞ, and lim

x!yrðxÞ ¼y. (2) ris subadditive.

(3) The function tðxÞ ¼ x

rðxÞ is increasing, and the function tðxÞ

logxis non-decreasing on½a;yÞ.

Remark 1.3. The collection C contains all functions x^a, for aAð0;1Þ, as well as functions logx, log logx, x

log^bðxþ1Þ, x^a

log^gðxþ1ÞforaAð0;1Þ,b>1,g>0, etc.

1.3. Results of the paper. LetEbe the class of all uniformly convex Banach spaces.

Proposition 1.4(see Proposition 4.2). Letrbe a function in C. There exists a graph Pof bounded degree such thatris the Hilbert space compression function ofP,and also the E-compression function ofP.

In particular, for any aA½0;1 there exists a graph of bounded degree whose Hilbert space compression equals theE-compression,and both are equal toa.

Using the construction of the graph in Proposition 1.4, we realize every function ofC as the upper bound of a Hilbert space compression gap of logarithmic size of a ﬁnitely generated group.

Theorem 1.5(see Theorem 5.5). For every functionrACthere exists a ﬁnitely generated group of asymptotic dimension at most2such that for every >0, r

log^1þðxþ1Þ;r

! is a Hilbert space compression gap and anE-compression gap of the group.

(5)

In particular,for everyaA½0;1there exists a ﬁnitely generated group Gaof asymptotic dimension at most2 and with the Hilbert space compression equal to theE-compression and equal toa.

Since the groups Ga have finite asymptotic dimension, they are all exact and uniformly embeddable1)into Hilbert spaces even when a¼0. Thus we construct the first examples of groups uniformly embeddable into Hilbert spaces, moreover exact and even of finite asymptotic dimension, that have Hilbert space compression 0, and even {uniformly convex Banach space}-compression zero. Note that since the construction in [Gr2] does not immediately extend to uniformly convex Banach spaces, our groups seem to be the only existing examples of groups with {uniformly convex Banach space}-compression 0.

1.4. The plan of the proofs. The plan for proving Proposition 1.4 and Theorem 1.5 is the following. We use a family of V. La¤orgue’s expandersPk,kf1, which are Cayley graphs of ﬁnite factor-groups Mk of a lattice G of SL3ðFÞ for a local ﬁeld F. La¤orgue proved [Laf ] that this family of expanders does not embed uniformly into a uniformly convex Banach space. Now taking any function r in C, we choose appropriate scaling con- stantslk, kf1, such that the family of rescaled metric spaces ðlkPkÞ_kf1 has {uniformly convex Banach space}-compression functionrand Hilbert space compression functionras well. This gives Proposition 1.4.

The group satisfying the conditions of Theorem 1.5 is constructed as a graph of groups. We use the fact that each Mk is generated by finitely many involutions, say m (that can be achieved by choosing a latticeGgenerated by involutions). One of the vertex groups of the graph of groups is the free productF of the groupsMk, other vertex groups aremcopies of the free productH¼Z=2ZZ. Edges connectF with each of themcopies ofH. The edge groups are free products of countably many copies ofZ=2Z. We identify such a subgroup in H with a subgroup of F generated by involutions, one involution from the generating set of each factor Mk. As a result, the groupG is finitely generated, and each finite subgroup Mk in G is distorted by a scaling constant close to lk. Hence the Cayley graph of G contains a quasi-isometric copy of the family of metric spaces ðlkPkÞ_kf1. This allows us to apply Proposition 1.4 and get an upper bound for a compression gap. A lower bound is achieved by a careful analysis of the word metric onG.

In order to show that G has asymptotic dimension at most 2, we use a result by Dranishnikov and Smith [DS] on the asymptotic dimension of countable groups, and results by Bell and Dranishnikov [BD], as well as by Bell, Dranishnikov and Keesling [BDK]

on the asymptotic dimension of groups acting on trees.

1.5. Other Banach spaces. The class of Banach spaces to which our arguments apply cannot be extended much beyond the class of uniformly convex Banach spaces; for instance it cannot be extended to reﬂexive strictly convex Banach spaces. Indeed, our proof is based on the fact that a family of V. La¤orgue’s expanders [Laf ] does not embed uniformly into a uniformly convex Banach space. But by a result of Brown and Guentner [BG], any count-

1) A ﬁnitely generated group G of ﬁnite asymptotic dimension has Guoliang Yu’s property A [HR], Lemma 4.2. That property is equivalent to the exactness of the reducedC-algebra ofG([Oz], [HR]) and guaran- tees uniform embeddability into a Hilbert space ([Yu], Th. 2.2).

(6)

able graph of bounded degree can be uniformly embedded into a Hilbertian sumL l^pⁿðNÞ for some sequence of numbers pnAð1;þyÞ, pn!y. The Banach spaceL

l^pⁿðNÞ is re- ﬂexive and strictly convex, but it is not uniformly convex.

Acknowledgement. The authors are grateful to A. Dranishnikov, A. Lubotzky, A. Rapinchuk, B. Remy, R. Tessera and D. Witte Morris for useful conversations and remarks.

2. Embeddings of metric spaces

Given two metric spaces ðX;dXÞ and ðY;dYÞ and a 1-Lipschitz mapf:X !Y we deﬁne thedistortionoff[HLW] as follows:

dtnðfÞ ¼max

x3y

dXðx;yÞ dY

fðxÞ;fðyÞ: ð1Þ

For a metric spaceðX;dÞand a collection of metric spacesEwe deﬁne theE-distortion of ðX;dÞ, which we denote by dtn_EðX;dÞ, as the inﬁmum over the distortions of all 1-Lipschitz maps fromX to a metric space fromE. Note that givenla positive real number dtn_EðX;dÞ ¼dtn_EðX;ldÞprovided that the classEis closed under rescaling of the metrics byl.

Remark 2.1. IfEcontains a space withnpoints at pairwise distance at least 1 from each other then for every graphX withnvertices and edge-length metric, dtn_EXediamX. The notion of distortion originated in combinatorics is related to the following notion of uniform embedding with origin in functional analysis.

Deﬁnition 2.2. Given two metric spacesðX;dXÞ andðY;dYÞ, and two proper non- decreasing functions rG:Rþ!Rþ, with lim

x!yrGðxÞ ¼y, a map f:X !Y is called a ðr;r_þÞ-embedding(also called auniform embeddingor acoarse embedding) if

r

dXðx1;x2Þ edY

fðx1Þ;fðx2Þ er_þ

dXðx1;x2Þ ð2Þ ;

for allx1,x2inX.

Ifr_þðxÞ ¼Cx, i.e. iffisC-Lipschitz for some constantC >0, then the embedding is called ar-embedding.

Deﬁnition 2.3. For a family of metric spacesXi,iAI, by aðr;r_þÞ-embedding(resp.

r-embedding) of the family we shall mean theðr;r_þÞ-embedding (resp.r-embedding) of the wedge union ofXi.

LetðX;dXÞbe a quasi-geodesic metric space (e.g., the set of vertices of a graph). Then it is easy to see that anyðr;r_þÞ-embedding ofX is also ar-embedding. The same holds for a family of quasi-geodesic metric spaces.

Convention 2.4. Since in this paper we discuss mainly embeddings of graphs, in what follows we restrict ourselves tor-embeddings, and denote the functionrsimply byr.

(7)

Notation 2.5. For two functions f;g:D!RwhereDORwe write f fgif there exist a;b;c>0 such that fðxÞeagðbxÞ þcfor every xAD. If f fg and gff then we write f g.

Deﬁnition 2.6. Let ðX;dÞ be a metric space, and let E be a collection of metric spaces. Let f;g:Rþ!Rþ be two increasing functions such that f fg and

x!limyfðxÞ ¼ lim

x!ygðxÞ ¼y.

We say thatðf;gÞis anE-compression gap ofðX;dÞif (1) there exists an f-embedding ofX into a space fromE;

(2) for everyr-embedding ofX into a space fromE, we haverfg.

If f ¼gthen we say that f is theE-compression functionofX.

The quotient g=f is called thesize of the gap. The functions f and gare called, the lowerandupper bound of the gaprespectively.

Remark 2.7. Assume that E is closed under rescaling of the metrics.

(i) Ifais the supremum of all numbers such thatx^afr andris theE-compression function ofðX;dÞthenais theE-compression ofðX;dÞ.

(ii) If fis theE-compression function of a metric spaceðX;dÞ, andais the supremum over all numbers b such that x^bff then a is the compression of ðX;dÞ in the sense of Deﬁnition 1.1.

3. Embeddings of expanders

For every ﬁnite m-regular graph, the largest eigenvalue of its incidence matrix ism.

We denote byl2the second largest eigenvalue.

We start by a well known metric property of expanders.

Lemma 3.1 ([Lub], Ch. 1). Let >0 and let Gm; be the family of all m-regular graphs with l2em. Then there exist two constants k, k⁰ such that for any graph P in Gm; with set of vertices V, and any vertex xAV, the set fyAVjdðx;yÞfkdiamPg has cardinality at leastk⁰jVj.

We now recall properties of expanders related to embeddings into Hilbert spaces. We denote byHthe class of all separable Hilbert spaces.

Theorem 3.2 ([LLR], Theorem 3.2(6), [HLW], Theorem 13.8 and its proof). Let

>0and letGm; be the family of all m-regular graphs withl2em.

(i) There exist constants c>0and d >1such that for any graphPinGm; with set of vertices V

clogjVjedtn_HPediamPedlogjVj:

ð3Þ

(8)

(ii) Moreover,there exists r>0such that for every1-Lipschitz embeddingfofPinto a Hilbert space Y there exist two vertices v1and v2 in V withdðv1;v2ÞfkdiamP(wherek is the constant in Lemma3.1)and

kfðv1Þ fðv2Þker:

Remark 3.3. Thus for all expanders inGm; the canonical embedding P,! 1

ffiffiffi2 p l²ðVÞ

has optimal distortion, see Remark 2.1.

V. La¤orgue [Laf ] proved that for some smaller family of expanders one can replace H by a large class of Banach spaces. Namely, for every prime number p and natural number r>0, he deﬁned a class of Banach spaces E^p;^r¼ S

a>0

E^p;r;^a satisfying the following:

(1) For any p,r, the classE^p;rcontains all uniformly convex Banach spaces2)(including all Hilbert spaces, and even all spacesl^q,q>1).

(2) S

r E^2;^r is the set of allB-convex Banach spaces.

La¤orgue’s family of expanders that does not embed uniformly into any Banach space from E^p;^r is constructed as follows. Given numbers p and r as above, let F be a local ﬁeld such that the cardinality of its residual ﬁeld is p^r. Let G be a lattice in SLð3;FÞ.

The groupG is residually ﬁnite with Kazhdan’s property (T). In fact,Gsatisﬁes the Banach versionof property (T) with respect to the family E^p;^r ([Laf ], §3 and Proposition 4.5).

LetðGkÞ_kf1be a decreasing sequence of ﬁnite index normal subgroups ofGsuch that T

kf1

Gk ¼ f1g, and letMk¼G=Gk,kf1, be the sequence of quotient groups. Given a ﬁnite symmetric set of generators ofGof cardinalitymf2, we consider eachMk endowed with the induced set ofmgenerators; we denote by dkthe corresponding word metric onMkand by Pk the corresponding m-regular Cayley graph. Since G has property (T), the family ðPk;dkÞ_kf1is a family of expanders ([Lub], Proposition 3.3.1). Moreover, the Banach version of property (T) for G yields Proposition 3.4 below, which implies that the family ðPk;dkÞ_kf1 cannot be uniformly embedded into a Banach space from E^p;r (see Remark 3.6).

2) A Banach space isuniformly convexif for everyR>0 and everyd>0 there existse¼eðR;dÞ>0 such that ifx, yare two points in the ball around the origin of radiusRat distance at leastdthen their middle point 1

2ðxþyÞis in the ball around the origin of radiusRe.

(9)

Proposition 3.4 ([Laf ], Proposition 5.2). For every a>0 there exists a constant C ¼CðaÞ such that for any kf1 and any space Y from E^p;^r;^a a 1-Lipschitz map f:Pk!Y satisﬁes:

1 ðcardPkÞ²

P

x;yAPk

kfðxÞ fðyÞk²eC 1 cardPk

P

x;yneighbors

kfðxÞ fðyÞk²: ð4Þ

Corollary 3.5. (i)For everya>0there exists a constant D¼DðaÞsuch that for any kf1,

DdiamPkedtn_E^p;r;^aPkediamPk: ð5Þ

(ii) Moreover,there exists R¼RðaÞsuch that for every1-Lipschitz embeddingfofPk

into a space Y AE^p;^r;a there exist two vertices v1 and v2 in Pk withdðv1;v2ÞfkdiamPk, wherekis the constant from Lemma3.1,and

kfðv1Þ fðv2ÞkeR:

ð6Þ

Proof. The second inequality in (5) follows from Remark 2.1 and from the fact that no Banach space can be covered by ﬁnitely many balls of radius 1. We prove the ﬁrst inequality in (5). LetY be an arbitrary space fromE^p;^r;aand letf:Pk!Y be a 1-Lipschitz map. Inequality (4) implies that

1

½dtnðfÞcardPk² P

x;yAPk

dkðx;yÞ²eCm:

ð7Þ

Recall that each graph Pk is m-regular. Lemma 3.1 and (7) imply that Cm½dtnðfÞ²fk²k⁰ðdiamPkÞ².

Now take R>1 k

ffiffiffiffiffiffiffiffi Cm k⁰ r

, wherek, k⁰ are the constants from Lemma 3.1 andC is the constant from (4). Assume that there exists a 1-Lipschitz embeddingfof Pk into a space Y AE^p;^r;^a such that for any two vertices v1 and v2 in Pk with dkðv1;v2ÞfkdiamPk the following inequality holds:

dkðv1;v2Þ

kfðv1Þ fðv2Þk< 1

R diamPk: Inequality (4) implies that

Cm> 1 ðcardPkÞ²

P

dkðv1;v2ÞfkdiamPk

R²dkðv1;v2Þ²

½diamPk² fR²k²k⁰: This contradicts the choice of R.

Therefore there exist two vertices v1 and v2 in Pk with dkðv1;v2ÞfkdiamPk and such that:

dkðv1;v2Þ

kfðv1Þ fðv2Þkf 1

RdiamPk: The last inequality implies thatkfðv1Þ fðv2ÞkeR. r

(10)

SinceðPk;dkÞ_kf1is a family of expanders, it is contained in some family Gm; . Then according to Theorem 3.2, we have diamPk logjVkj.

Remark 3.6. Theorem 3.2(ii) implies that no sub-family of the family Gm; can be uniformly embedded into a Hilbert space, in the sense of Deﬁnition 2.3.

Similarly, Corollary 3.5(ii) implies that no sub-family of the family ðPkÞ_kf1 can be uniformly embedded into a space fromE^p;r.

The construction in the following lemma was provided to us by Dave Witte- Morris.

Lemma 3.7. Let F be a nonarchimedean local ﬁeld of characteristic0. There exists a lattice in the groupSLð3;FÞcontaining inﬁnitely many noncentral involutions.

Proof. Choose an algebraic number ﬁeldK, such that F is one of the (nonarchimedean) completions ofK, and F is totally real.

Let pbe the characteristic of the residue field ofF, and letc¼1p³. Thusc<0, butcis a square inF. ConsiderL¼K½ ffiffiffi

pc

, and denote bytthe Galois involution ofLoverK. We likewise denote by t the involution deﬁned on the 33 matrices by applying t to each entry. LetG ¼SUð3;L;tÞ ¼ fgASLð3;LÞjtðg^TÞ g¼Id3g. Then:

(a) Gis aK-form of SLð3Þ.

(b) GF ¼SLð3;FÞ.

(c) Gis compact at each real place.

Properties (a), (b), (c) can be proved using arguments similar to the ones in [Wi], Chapter 10.

LetS be the collection of all the Archimedean places ofK and the place corresponding toF. According to the theorem due to Borel, Harish-Chandra, Behr and Harder ([Ma],

§I.3.2), the S-integer points of G form a lattice G in SLð3;FÞ. By [Ta], any lattice in a p-adic algebraic group is co-compact, in particular it is the case for G. The latticeGobvi- ously contains noncentral diagonal matrices that are involutions.

Letsbe one of these involutions and letZðsÞbe its centralizer in SLð3;FÞ. The subgroup GXZðsÞ has inﬁnite index in G. Otherwise, for some positive integernone would have that gⁿAZðsÞ for any gAG. Since ZðsÞ is an algebraic subgroup in SLð3;FÞ and since Gis Zariski dense in SLð3;FÞ, it would follow that gⁿAZðsÞ for any gASLð3;FÞ.

This is impossible.

For a sequence ðg_nÞ_nAN of representatives of distinct left cosets inG=

GXZðsÞ the involutions in the sequenceðg_nsg¹_n Þ_nAN are pairwise distinct. r

(11)

Lemma 3.8. For every prime number p and natural number r>0, there exists m¼mðp;rÞ>0and a latticeGin SLð3;FÞ such that F is a local field with residue field of cardinality p^r,andGis generated by finitely many involutionss1;. . .;sm.

Proof. LetGbe a lattice of SLð3;FÞcontaining infinitely many non-central involutions. According to Lemma 3.7 such a lattice exists. Consider the (infinite) subgroupG⁰of Ggenerated by involutions. SinceG⁰ is a normal subgroup inG, by Margulis’ Theorem G⁰ has finite index inG, thus it is a lattice itself. r

Notation 3.9. Let Gbe one of the lattices from Lemma 3.8. We keep the notation given before Proposition 3.4, referring, in addition, to the generating set consisting of involutions. More precisely, consider a maximal ideal I in the ring of S-integers of the global ﬁeld deﬁningG(S is the corresponding set of valuations containing all Archimedian ones) and the congruence subgroup Gk of G corresponding to the ideal I^k. Let Mk ¼G=Gk, kf1. Let Uk¼ fs1ðkÞ;. . .;smðkÞg be the image of the generating set of G in Mk (each siðkÞ is an involution). We shall denote the corresponding word metric on Mk again by dk. Since eachsiðkÞis an involution, the Cayley graphPkofMk ism-regular. LetvðkÞde- note the cardinality of the groupMk.

Remark 3.10. It is easy to see thatvðkÞ ¼ jMkjsatisﬁes ckc1<logvðkÞ<ckþc1

for some constantsc,c1.

4. Metric spaces with arbitrary compression functions

Let ðX;dÞ be a metric space and l>0. We denote by lX the metric space ðX;ldÞ.

LetðPnÞ_nf1be the sequence of Cayley graphs of ﬁnite factor groupsMn,nf1, of the latticeGfrom Lemma 3.8, see Notation 3.9.

Letrbe a function inC(see Deﬁnition 1.2). For everynf1, letxnbe a ﬁxed vertex inPn. We are going to choose appropriate scaling constants ln, nf1, so that the wedge unionP of the metric spaceslnPn, obtained by identifying all the verticesxn to the same pointx, has the required compression functionr.

Notation 4.1. Throughout this section we denote logvðnÞ by yn. According to Re- mark 3.10,cnc1< yn<cnþc1for some constantsc,c1.

We are looking for a sequence of rescaling constants lnsatisfyingln¼rðlnynÞ. This is equivalent to the fact thattðlnynÞ ¼ yn, where tðxÞ ¼ x

rðxÞ. SincetðxÞ is increasing by the deﬁnition of C, continuous and lim

x!y tðxÞ ¼ þy, t¹ðxÞ exists for large enough x.

Thus, we can takeln:¼t¹ðynÞ yn

.

(12)

Proposition 4.2. LetPbe the wedge union oflnPn,nf1. Thenris both the Hilbert space compression function and theE^p;^r-compression function ofP(up to the equivalence re- lation).

In particular,ris the{uniformly convex Banach space}-compression function ofP.

The proof of Proposition 4.2 will show that the same conclusions hold if we replace the sequenceðlnÞ_nf1by any sequence of numbersðm_nÞ_nf1satisfyingm_nln.

Proof of Proposition4.2. We are going to prove thatris the Hilbert space compression function ofP. The proof thatris theE^p;^r-compression function is essentially the same (with reference to Theorem 3.2 replaced by a reference to Corollary 3.5).

Let dtbe the canonical distance onP, and letV be the set of vertices ofP. Consider the map f from V tol2ðVÞdeﬁned as follows. For every nf1 and everyvAVnnfxng let

fðvÞbe equal tolndv, wheredvis the Dirac function atv. Also let fðxÞ ¼0.

We prove that the following inequalities are satisﬁed, withdf1 the constant in The- orem 3.2(i).

1ffiffiffi

p2r dtðv;v⁰Þ d

ekfðvÞ fðv⁰Þke ffiffiffi p2

dtðv;v⁰Þ:

ð8Þ

Assume that v;v⁰APn for some n. Then kfðvÞ fðv⁰Þkeln

ffiffiffi2

p (when one of the vertices is xn the ﬁrst term is ln). Since dtðv;v⁰Þfln, we get the second inequality in (8).

To prove the ﬁrst inequality, recall that by Theorem 3.2 the diameter of lnPn is at mostdlnyn. Hence

r dtðv;v⁰Þ d

erðlnynÞ ¼lnekfðvÞ fðv⁰Þk:

Assume now thatvAPmnfxmgandv⁰APnnfxng. Then kfðvÞ fðv⁰Þk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l_m² þl_n² q

elmþlnedtðv;xÞ þdtðx;v⁰Þ ¼dtðv;v⁰Þ:

On the other hand kfðvÞ fðv⁰Þkf 1

ffiffiffi2

p ðlmþlnÞ ¼ 1 ffiffiffi2

p

kfðvÞ fðxÞk þ kfðv⁰Þ fðxÞk :

By the previous case, the last term is at least 1ffiffiffi

p2 r dtðv;xÞ d

þr dtðx;v⁰Þ d

f 1 ffiffiffi2

p r dtðv;v⁰Þ d

:

The latter inequality is due to the sub-additivity and the monotonicity ofr.

(13)

This proves that rhas property (1) of the Hilbert space compression function of P, see Deﬁnition 2.6(1).

To prove property (2) of the Hilbert space compression function, let us consider an increasing functionr:R_þ!R_þ with lim

x!yrðxÞ ¼y, and a 1-Lipschitzr-embedding gof Pinto a Hilbert space. For any pair of verticesv,v⁰inPwe have

r

dtðv;v⁰Þ

ekgðvÞ gðv⁰Þkedtðv;v⁰Þ:

ð9Þ

We shall prove thatrfr.

For everynf1, for any two verticesv, v⁰in the same graphlnPn, the inequality (9) can be re-written as:

r

lndðv;v⁰Þ

ekgðvÞ gðv⁰Þkelndðv;v⁰Þ:

ð10Þ

We denote byhthe restriction ofgtolnPn, rescaled by the factor 1=ln. The sequence of inequalities (10) divided byln yields

1 ln

r

lndðv;v⁰Þ

ekhðvÞ hðv⁰Þkedðv;v⁰Þ:

ð11Þ

Theorem 3.2 implies that for some constants r,d and kthere exist verticesv1andv2

such that

kynedðv1;v2Þedyn and khðv1Þ hðv2Þker:

From (11) and the monotonicity ofr, we get rðklnynÞer

lndðv1;v2Þ

elnkhðv1Þ hðv2Þkerln¼rrðlnynÞ:

Denote byznthe productlnyn, also equal tot¹ðynÞ, by the deﬁnition ofln. Then we get

rðkznÞerrðznÞ:

ð12Þ

We have that ynecnþc1ey_n1þC1whereC1¼cþ2c1.

By property (3) of Deﬁnition 1.2, tðxÞ ¼x=rðxÞ is an increasing map deﬁning a bijection ½a;yÞ ! ½b;yÞ. Moreover, the condition that tðxÞ=logx is non-decreasing easily implies that for some y>1, tðxÞ þ1etðyxÞ for every xfa. It follows that tðxÞ þC1etðCxÞ for every xfa, where C is a power of y depending on C1. This implies that for every yfb, t¹ðyþC1ÞeCt¹ðyÞ. Consequently, for n large enough, zn¼t¹ðynÞet¹ðyn1þC1ÞeCt¹ðyn1Þ ¼z_n1. We thus get

zn

zn1

eC:

(14)

Now take any su‰ciently large x>0. Then for some n, x is between zn1 and zn. Hence (by the monotonicity ofrandr), we get:

rðkxÞerðkznÞerrðznÞerrðCzn1ÞfrðCxÞ:

Therefore,rfras required. r

One can obviously replaceP in Proposition 4.2 by a uniformly proper space (graph of bounded degree) with the same property.

The following corollary immediately follows from Proposition 4.2.

Corollary 4.3. For every numberain½0;1there exists a proper metric space(graph of bounded degree)whose Hilbert space compression and theE^p;r-compression are equal toa.

Note that a¼0 can be obtained by taking, say, rðxÞ ¼logxin Proposition 4.2. To achievea¼1 take the one-vertex graph.

5. Discrete groups with arbitrary Hilbert space compressions

For every prime number pand every natural numberr>0, we denoteE^p;r simply by Ein what follows. Recall thatEcontains all uniformly convex Banach spaces.

Pick a functionrAC. For simplicity we assume thatrð1Þ>0. As in Deﬁnition 1.2, we denote bytthe functionx=rðxÞ.

LetðMk;dkÞ_kf1 be the sequence of finite groups defined as above, for fixed pand r, see Lemma 3.8 and Notation 3.9. Recall that dk is the word metric associated to the generating setUk¼ fs1ðkÞ;. . .;smðkÞgconsisting ofm¼mðp;rÞinvolutions, and that vðkÞ is the cardinality ofMk.

Notation 5.1. We denote byj j_kthe length function onMk associated toUk. Notation 5.2. Let us ﬁx three sequences of numbers:ðlkÞ_kf1,ðmkÞ_kf1, andðmkÞ_kf1. For kf1, we set lk such that r

lklogvðkÞ

¼lk; equivalently lk¼t¹

logvðkÞ logvðkÞ withvðkÞas above. Without loss of generality we assume (by taking a suitable subsequence ofMk) that for everykf1,l_kþ1lkf4.

For every kf2, let mk be the integer part of lk1

2 , and let m1¼0. Then we put m_k¼2mkþ1 for everykf1.

According to our choice, for everykf2

m_kelk<m_kþ2:

ð13Þ

We are now ready to construct our group.

(15)

LetF be the free product of the groupsMk,kf1.

For everyiAf1;. . .;mg, consider a copyHi of the free productZ=2ZZ, where the generator of theZ=2Z-factor ofHi is denoted bysi, while the generator of theZ-factor of Hi is denoted byti.

Notation 5.3. We denote byj j_H_i the length function onHi relative to the generating set fsi;tig. For every kAZ, we denote the element t_i^ksit^k_i in Hi by s^ðkÞ_i . Note that js^ðkÞ_i j_H_i ¼2kþ1, and thatHi is the semidirect product ofhs^ðkÞ_i ;kAZiandhtii.

LetG be the fundamental group of the following graph of groups (see [Ser], Ch. 5.1, for the deﬁnition). The vertex groups areF andH1;. . .;Hm, the only edges of the graph are

ðF;HiÞ, i¼1;. . .;m. The edge groupsFXHi are free products

kf1ðZ=2ZÞ_k of countably many groups of order 2, where thek-th factorðZ=2ZÞ_kis identiﬁed withhsiðkÞi<MkinF and withhs^ðm_i ^k^ÞiinHi.

Notation 5.4. We denote by dG and by j j_G the word metric and respectively the length function onGassociated to the generating setU ¼ fs1ð1Þ;. . .;smð1Þ;t1;. . .;tmg.

Theorem 5.5. With the notation above and the terminology in Deﬁnition2.6,the following hold:

(I) The group G has as a Hilbert space compression gap, and an E-compression gap ðpffiffiffir

;rÞ.

(II) The group G has as a Hilbert space compression gap,and an E-compression gap r

log^1þðxÞ;r

!

,for every >0. Hence the Hilbert space compression and theE-compression of G are equal to the supremum over all the non-negative numbersasuch that x^afr.

(III) The asymptotic dimension of G is at most2.

We start our proof by describing the length functionj j_G.

By the standard properties of amalgamated products ([LS77], Ch. IV, Th. 2.6), the free productF ¼

kf1Mk and the groupsHi,i Af1;2;. . .;mg, are naturally embedded into G. Moreover, the subgroupT ¼hti;1eiemiis free of rankmand a retract ofG, so its length function coincides withj j_G restricted toT.

Deﬁnition 5.6. With every elementsAMk we assign theweightwtðsÞ ¼m_kjsj_k. With every element bAHi we assign the weight wtðbÞ ¼ jbj_H_i. The weight wtðWÞ of any word

W ¼s1b1. . .snbn withs1andbnpossibly trivial andnANis the sum Pⁿ

i¼1

½wtðsiÞ þwtðbiÞ.

Let F⁰ be the free product of F ¼

kf1Mk and all the 2-element groupshs^ðkÞ_i iwith

kAZnfmk;kf1gand iAf1;. . .;mg. Then G is the multiple HNN extension of F⁰ with

free letterst1;. . .;tmshiftings^ðkÞ_i . This is the presentation ofGused in the following lemma.

(16)

A subword t^G_i ¹s^ðkÞ_i t^H_i ¹of a word in generatorss^ðkÞ_i ,ti ofG is called apinch. It can be removed and replaced bys^ðk_i ^G^1Þ; we call this operationremoval of a pinch.

Lemma 5.7. For every kf1and gAMk,jgj_G¼wtðgÞ ¼m_kjgj_k.

Proof. Lets¼ jgj_kand letg¼si1ðkÞsi2ðkÞ. . .sisðkÞbe a shortest representation ofg as a product of generators ofMk. Then we can representgas

t_i^m₁^ksi1ð1Þt^m_i₁ ^kt_i^m₂^ksi2ð1Þt^m_i₂ ^k. . .t_i^m_s^ksisð1Þt^m_i_s ^k: Hencejgj_Geð2mkþ1Þs¼m_kjgj_k.

LetW be any shortest word in the alphabetU representingginG. SincegAF, there exists a sequence of removals of pinches and subwords of the form aa¹ that transfersW into a wordW⁰ in generators of F (i.e. letters of the formsjðlÞ) representingg. Since the weight of everysið1Þandt^G_i ¹is 1, the total weight of the wordW is the length ofW. The removal of a pinch does not increase the total weight, and the removal of a wordaa¹de- creases it. Hence the total weight of the letters inW⁰ is at most jWj. On the other hand, since W⁰ is a word in generators of the free productF representing an element of one of the factorsMk, all letters fromW⁰ must belong toMk, i.e. they have the formsiðkÞwith

iAf1;. . .;mg. Therefore the total weight ofW⁰ism_kjW⁰j. SinceW⁰representsginMk, we

have that the total weight ofW⁰ is at leastm_kjgj_k. Hencejgj_G ¼ jWjfm_kjgj_k as required.

r

The choice ofm_kgiven by (13), Proposition 4.2 and the construction in its proof, and Lemma 5.7 imply the following.

Lemma 5.8. (1) The function r is the E-compression function and the Hilbert space compression function of S¼ S

kf1

Mk endowed with the restriction of the metricdG.

(2) The map f :S !l²ðSÞdeﬁned by fð1Þ ¼0and fðsÞ ¼m_kdsfor every sAMknf1g is ar-embedding ofðS;dGÞ.

Lemma 5.8 immediately implies

Lemma 5.9. The functionris an upper bound for anE-compression gap of G(in particular,also for a Hilbert space compression gap of G).

Deﬁnition 5.10. We are going to use the standard normal forms of elements in the amalgamated productG(see [LS77], Ch. IV.2):

s1b1. . .snbn

ð14Þ

where:

(N0) Possiblys1¼1 orbn¼1.

(N1) si AMki andbi AHli (si,bi are calledsyllables).

(17)

(N2) If bi ¼1, i<n, thensi and siþ1 are not in the same Mk; if si¼1, i>1, then bi1,bi are not in the sameHk.

One can get from one normal form to another by sequences of operations as follows:

(r1) Replacingsibi with

siskðpÞ skðpÞbi

(left insertion).

(r2) Replacingbis_iþ1with

biskðpÞ

skðpÞsiþ1

(right insertion).

(r3) Replacing two-syllable words with syllables from the same factor by the one- syllable word, their product.

One can easily check that this collection of moves is conﬂuent, so one does not need the inverse of rule (r3).

Choose one representative for each left coset ofMk=hsiðkÞiand one representative in each left coset ofHi=hsiðkÞi, the representative ofhsiðkÞiis 1. LetMbe the set of all these representatives. We consider only the normal forms (14) in which all the si and bi (except possibly for the last non-identity syllable) are inM. We shall call these normal formsgood.

Every element ofG has a unique good normal form.

Lemma 5.11. Let V ¼s1b1s2. . .snbnbe the good normal form of an element g in the amalgamated product G. Then the lengthjgj_G is betweenwtðVÞ=3andwtðVÞ,i.e.

Pⁿ

i¼1

m_k_ijsij_k_iþPⁿ

i¼1

jbij_H

li fjgj_Gf1

3 Pⁿ

i¼1

m_k_ijsij_k_i þPⁿ

i¼1

jbij_H

li

ð15Þ :

For every normal form V⁰of g,the length of g is betweenwtðV⁰Þ=9andwtðV⁰Þ.

Proof. The ﬁrst inequality in (15) follows from Lemma 5.7 and it holds for every normal form.

Consider a shortest wordW in the generators ofG representingg. Making moves of type (r3), we rewrite W in a normal form W⁰ without increasing its weight (note that a removal of pinches can be seen as a succession of two (r3)-type moves). We have that jgj_G¼wtðWÞfwtðW⁰Þfjgj_G; the last inequality holds because any word of the form (14) and of weightl is equal inG to a word of length lin the generators in U and their inverses. It follows that wtðW⁰Þ ¼ jgj_G. By doing insertion moves (r1) and (r2), we can rewrite the normal form W⁰ into the good normal form V of g. Note that each syllable is multiplied by at most two involutions during the process. The weight of each of these involutions does not exceed the weight of the syllables si, siþ1 involved in the moves.

Hence the total weight of the word cannot more than triple during the process. Hence wtðVÞe3 wtðW⁰Þ ¼3jgj_G which proves the second inequality in (15).

The second statement is proved in a similar fashion: one needs to analyze the procedure of getting the good normal form from any normal form, and then use the ﬁrst part of the lemma. r

(18)

Deﬁnition 5.12. Lets1b1. . .snbnbe the good normal form ofg. Represent eachbias the shortest wordwðbiÞin the alphabet of generatorsfsli;tligof the corresponding subgroup Hli. Then the words1wðb1Þ. . .snwðbnÞin the alphabet S

kf1

MkWfsi;ti;iAf1;2;. . .;mgg is called theextended normal formofg.

The unicity of the good normal form and of each word wðbiÞimplies that every element inG has a unique extended normal form.

Deﬁnition 5.13. For every pair of elementsgandhinGwith extended normal forms g¼s1wðb1Þ. . .slwðblÞand h¼s₁⁰wðb₁⁰Þ. . .s_n⁰wðb_n⁰Þ let pðg;hÞ ¼ pðh;gÞ be the longest com- mon preﬁx of these words, of lengthiðg;hÞ. Thus

g1pðg;hÞfðg;hÞqðg;hÞ and h1pðg;hÞfðh;gÞqðh;gÞ

where the words fðg;hÞqðg;hÞ, fðh;gÞqðh;gÞ have di¤erent ﬁrst syllables fðg;hÞ and fðh;gÞ respectively. Here the syllables are either in some Mk, kf1, or they are in ft^G₁¹;t^G₂¹;. . .;t^G_m¹g.

Note that pðg;hÞ,qðg;hÞ,qðh;gÞare extended normal forms; fðg;hÞ(resp. fðh;gÞ) is either an element inMk for somekf1 or it is infsi;t^G_i ¹gfor some iAf1;2;. . .;mg.

Deﬁnition 5.14. Lets1wðb1Þ. . .snwðbnÞ be the extended normal form ofg. Then for everyi let g½ibe thei-th letter of the extended normal form, letgg^_i be the preﬁx ending in g½i, and letg_ibe the su‰x starting withg½i of the extended normal form.

Lemma 5.15. For every g;hAG,the distancedGðg;hÞis in the interval½A=9;Awhere A is equal to

(S) m_kdk

fðg;hÞ;fðh;gÞ þwt

qðg;hÞ þwt

qðh;gÞ

if fðg;hÞ;fðh;gÞAMk,or (B) wt

fðg;hÞqðg;hÞ þwt

fðh;gÞqðh;gÞ

otherwise.

Proof. In Case (S), qðg;hÞ¹

fðg;hÞ¹fðh;gÞ

qðh;gÞ becomes a normal form for g¹h if we combine all neighbor letters from the same Hk into one syllable, and fðg;hÞ¹fðh;gÞ into one syllable. In Case (B), qðg;hÞ¹fðg;hÞ¹fðh;gÞqðh;gÞ becomes a normal form forg¹hafter a similar procedure. Then one can use Lemma 5.11. r

For every element g in G whose extended normal form has the last syllable in Y, where Y ¼Mk for some kf1 or Y ¼ fsi;t^G_i ¹g for some iAf1;. . .;mg, we consider a copyf_gof a 1-Lipschitz embedding ofY into a Hilbert spaceHgwith optimal distortion:

either the embedding from Remark 3.3 rescaled by the factorm_kifY ¼Mk(i.e. the embedding sending all non-trivial elements inMk in pairwise orthogonal vectors of length m_k) or the embedding deﬁned by any choice of an orthonormal basis in a copy ofR² otherwise.

Note that sincem₁¼1 this is coherent with the identiﬁcation ofsi AHi withsið1ÞAM1. LetHbe the Hilbertian sum of all Hg. Consider the following map from G intoH:

for everygAG let

cðgÞ:¼P

i

f_g_g_^_iðg½iÞ:

ð16Þ

(19)

We use the mapcto prove the following.

Lemma 5.16. The group G has as a Hilbert space and anE-compression gapðpffiffiffir

;rÞ.

Proof. In view of Lemma 5.9, it su‰ces to show thatcis apffiffiffir

-embedding to ﬁnish the proof.

Let g;hAG. Then, by (16), since any two elements of H of the form f_g_g_^_iðg½iÞ, f^hhjðh½jÞeither coincide or are orthogonal to each other, we have:

kcðgÞ cðhÞk²¼f_pðg;hÞ_f_ðg;_hÞ

fðg;hÞ²þf_pðg;_hÞ_f_ðh;gÞ

fðh;gÞ²

þ P

j>iðg;hÞþ1

kf_g_g_^_jðg½jÞk²þ P

j>iðg;hÞþ1

kf^hhjðh½jÞk²eCdGðg;hÞ²

for some constant C (by Lemma 5.15 and the fact that all embeddings fare 1-Lipschitz).

Thuscis a Lipschitz embedding.

To prove the lower bound, note that forg½iinHk, jg½ij_G ¼ kf_g_g_^_iðg½iÞk ¼1grðjg½ij_GÞ:

Also ifg½iAMk then by Lemma 5.8 (2), and by the deﬁnition off_gwe may write jg½ij_G¼m_kjg½ij_k¼ kfgg_^_iðg½iÞkgrðjg½ij_GÞ:

Thus, in all cases,

kf_g_g_^_iðg½iÞkgrðjg½ij_GÞ:

Now, in case (B), kcðgÞ cðhÞk²gr

dG

fðg;hÞ; fðh;gÞ2

þ P

j>iðg;hÞþ1

rðjg½jj_GÞ²þ P

j>iðg;hÞþ1

rðjh½jj_GÞ²: The fact that rðg½iÞfrð1Þ for non-identityg½i, and the subadditivity of r(plus Lemma 5.15) imply

kcðgÞ cðhÞk²gr

dGðg;hÞ

as desired. In case (S), the proof is similar, only one needs to take into account that f_pðg;_hÞ_f_ðg;hÞ

fðg;hÞ²þf_pðg;hÞ_fðh;_gÞ

fðh;gÞ²

¼f_pðg;hÞ_f_ðg;hÞ

fðg;hÞ

f_pðg;_hÞ_f_ðh;gÞ

fðh;gÞ² fr

dG

fðg;hÞ;fðh;gÞ2

: r

Lemma 5.16 gives part (I) of Theorem 5.5.

(20)

Let us prove part (II). For simplicity, we take¼1. The reader can easily modify the proof to make it work for every >0. Thus, we are going to prove that rðxÞ

log²ðxþ1Þ is a lower bound for a Hilbert space compression gap and anE-compression gap ofG. That is, we are going to prove that there exists a rðxÞ

log²ðxþ1Þ-uniform embedding of G in a Hilbert space.

Consider the same Hilbert spaceHand the same embeddingsf_gforgAG as before.

Let us deﬁne an embedding p ofG into H. Let g be given in an extended normal form.

Then we set

pðgÞ:¼P

j

kjf_g_g_^_jðg½jÞ where the coe‰cientskj are deﬁned as follows:

kj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg_jÞ q log

wtðg_jÞ þ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg½jÞ

p :

We shall need the following two elementary inequalities, the ﬁrst of which is obvious.

Lemma 5.17. For every sequence of positive numbers a1;. . .;an,we have Pⁿ

i¼1

Pⁿ

r¼i

ar

aie

Pⁿ

r¼1

ai

2

:

Lemma 5.18. Let a1;. . .;anbe positive real numbers with anf1. Then a1

ða1þ þanÞlog²ða1þ þanþ1Þ ð17Þ

þ a2

ða2þ þanÞlog²ða2þ þanþ1Þþ þ an

anlog²ðanþ1Þ eC for some constant C.

Proof. For everyk ¼1;. . .;ndenotesk ¼akþ þan. Then (17) can be rewritten as follows:

s1s2

s1log²ðs1þ1Þþ þ sn1sn

sn1log²ðsn1þ1Þþ 1 log²ðsnþ1Þ: ð18Þ

Since the function 1

xlog²ðxþ1Þis decreasing, one can estimate (providedbfaf1) ba

blog²ðbþ1ÞeÐ^b

a

dx

xlog²ðxþ1Þ e2Ð^b

a

dx

ðxþ1Þlog²ðxþ1Þ¼ 2

logðaþ1Þ 2 logðbþ1Þ: Applying this inequality to (18) and usinganf1, we get (17). r

(21)

Let g, hbe two elements in G given in their extended normal forms. Let i¼iðg;hÞ, f ¼ fðg;hÞ, f⁰¼ fðh;gÞ, p¼ pðg;hÞ,q¼qðg;hÞ, q⁰¼qðh;gÞ.

ThenpðgÞ pðhÞ ¼S1þS2þS3where

S1¼ Pⁱ

r¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg_rÞ p log

wtðg_rÞ þ1

ffiffiffiffiffiffiffiffiffiffiffiffiffi wtðhrÞ q log

wtðhrÞ þ1 0

@

1

A 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg½rÞ

p f_g_g_^_rðg½rÞ;

S2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg_iþ1Þ p

log

wtðg_iþ1Þ þ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg½iþ1Þ

p f_pfðfÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðhiþ1Þ q

log

wtðhiþ1Þ þ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðh½iþ1Þ

p f_pf0ðf⁰Þ;

S3¼ P

rfiþ2

ffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg_rÞ p log

wtðg_rÞ þ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg½rÞ

p f_g_g_^_rðg½rÞ P

rfiþ2

ffiffiffiffiffiffiffiffiffiffiffiffiffi wtðhrÞ q log

wtðhrÞ þ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðh½rÞ

p f^hhrðh½rÞ:

Lemma 5.19(Upper bound). kpðgÞ pðhÞkfdGðg;hÞ,that ispis Lipschitz.

Proof. SinceGis ﬁnitely generated, it su‰ces to prove that the normkpðgÞ pðgsÞk is bounded uniformly insAUWU¹ andgAG. In this case the sum S3 does not appear, andS2¼ 1

log 2f_gsðsÞhas norm 1

log 2. By eventually replacinggwithgsands withs¹, we can always assume thath¼gssatisﬁes wtðhrÞ ¼wtðg_rÞ þ1 for allrei. Then

kS1k²¼ Pⁱ

r¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðgrÞ þ1 p

log

wtðgrÞ þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffi wtðgrÞ p log

wtðgrÞ þ1

!2

wtðg½rÞ:

Since the functionx7!

ffiffiffix p

logðxþ1Þis increasing forxfe²1 it follows that

kS1k²fPⁱ

r¼1

½ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg_rÞ þ1

p ffiffiffiffiffiffiffiffiffiffiffiffiffi wtðg_rÞ

p ²

log²

wtðg_rÞ þ1 wtðg½rÞePⁱ

r¼1

wtðg½rÞ wtðg_rÞlog²

wtðg_rÞ þ1: Lemma 5.18 implies now that

kS1k²¼Oð1Þ: r ð19Þ

Lemma 5.20(Lower bound). kpðgÞ pðhÞkg r

dGðg;hÞ log²

dGðg;hÞ þ1if g3h.

Proof. We can write kS3k²g P

rfiþ2

wtðg_rÞ log²

wtðg_rÞ þ1

wtðg½rÞr

wtðg½rÞ2

ð20Þ

þ P

rfiþ2

wtðhrÞ log²

wtðhrÞ þ1

wtðh½rÞr

wtðh½rÞ2

: