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# Multivariate Calculus MA 26100

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CONSTRUCTIONS PRESERVING HILBERT SPACE UNIFORM EMBEDDABILITY OF DISCRETE GROUPS MARIUS DADARLAT AND ERIK GUENTNER ABSTRACTi Uniform embeddability in a Hilbert space introduced by Gromov is a geo metric property of metric spaces As applied to countable discrete groups it has important consequences to the Novikov conjecturei Exactness introduced and studied extensively by Kirchberg Wassermann is a functional analytic property of locally compact groupsi Re cently it has become apparent that as properties of countable discrete groups uniform em beddability and exactness are closely related We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of per manence properties with the class of exact groups In particular we prove that it is closed under direct and free products with and without amalgam inductive limits and certain extensions 1 INTRODUCTION Gromov introduced the notion of uniform embeddability of metric spaces and suggested that nitely generated discrete groups that are uniformly embeddable in a Hilbert space when viewed as metric spaces might satisfy the Novikov Conjecture 12 10 Yu proved that this is indeed the case 20 18 Kirchberg Wassermann de ned the notion of exactness of a locally compact group in terms of the behavior of its reduced crossed product functor Subsequently they developed the main properties of exact groups In particular they showed that in the case of a countable discrete group exactness can be reformulated entirely in terms ofthe reduced C algebra 14 that is that exactness is a property of the harmonic analysis of the left regular representation of such a group The starting point of this work is the startling fact that for countable discrete groups uniform embeddability in a Hilbert space a geometric property and exactness an ana lytic property are closely related The rst indications of the relationship between uniform Date July 18 2002 The rst author was supported in part by an MSRI Research Professorship and NSF Grant DMS9970223i The second author was supported in part by an MSRI Postdoctoral Fellowship and NSF Grant DMS007l402i I 2 MARIUS DADARLAT AND ERIK GUENTNER embeddability and exactness are found in the work of Guentner Kaminker 13 these prelim inary steps were quickly expanded by Ozawa 16 Anantharaman Delaroche 2 and others We are concerned with uniformly embeddable groups Our main results are outlined in the following theorem more precise statements follow in later sections which summarizes the basic permanence properties of the class of uniformly embeddable groups Observe that the properties described are all shared by the class of countable discrete exact groups 15 Indeed in each case it is possible to give a uni ed account of the results for uniform embeddability and exactness in some cases our methods provide alternate proofs of the known results concerning exactness Theorem The class of countable discrete groups that are uniformly embeddable in a Hilbert space is closed urider subgroups arid products direct limits free products with amalgam arid epterisioris by epact groups B The fact that subgroups and products of uniformly embeddable groups are again uniformly embeddable is elementary and quite well known they are included in the statement for completeness The other properties are more dif cult to establish It is possible to construct a uniform embedding of a free product without amalgam directly from uniform embeddings of the factors On the other hand the corresponding result for free products with amalgam is considerably more dif cult in view of the fact that the common subgroup of the amalgam can introduce considerable distortion into the product Our proof is based on a suitable adaptation of an argument given by Tu in his work on Property A 19 although we are not able to verify a number of assertions concerning the metric de ned in Section 9 in Tu7s paper we are able to adapt his arguments to the present context The proof we give works equally well for countable exact groups and is unrelated to Dykema7s original proof that the class of countable exact groups is closed under free products with amalgam 9 8 Again in the case without amalgam a considerably simpler proof of this fact is now available The general problem of uniform embeddability of extensions is intriguing Our proof that the class of uniformly embeddable groups is closed under extensions by exact groups is inspired by the argument of Anantharaman Delaroche Renault showing that the class of countable exact groups is closed under extensions It is unknown whether the class of uniformly embeddable groups is closed under general extensions even the case of a central extension of Z by a uniformly embeddable group remains open At present the behavior with respect to extensions provides the best possibility of distinguishing the classes of uniformly embeddable and exact groups UNIFORM EMBED DA BILITY 3 We draw two immediate corollaries Since they are peripheral to our study we will not establish notation or provide the relevant de nitions rather we provide references Corollary The class of countable discrete groups that are uniformly embeddable in a Hilbert space is closed under the formation of HNN ecctensions Proof An HNN extension is built from free products with amalgam direct limits and a semi direct product by Z which is exact 17 4 See 6 for related results D Corollary The fundamental group of a graph of countable discrete groups is uniformly embeddable in a Hilbert space if and only if each of the groups is uniformly embeddable in a Hilbert space Proof Each constituent group is a subgroup of the fundamental group Conversely the fundamental group of a graph of groups is built from free products with amalgam HNN extensions and direct limits 17 4 See 1 for related remarks D 2 BACKGROUND Let X and Y be metric spaces with metrics dX and dy respectively A function F X a Y is a uniform embedding if there exist non decreasing functions pi R a R such that limtH00 pit 00 and such that pdXxz S dyFzFx S pdXzz for all p x E X 1 The space X is uniformly embeddable if there exists a uniform embedding F of X into a Hilbert space H Uniform embeddability in a real Hilbert space is equivalent to uniform embeddability in a complex Hilbert space henceforth we shall deal only with real Hilbert spaces Obviously if X is countable we may assume that the Hilbert space is separable A metric space X is locally nite if for every x E X and R gt 0 the metric ball with center z and radius R is nite In this case the metric is called proper A locally nite metric space is discrete as a topological space in the metric topology In the case of locally nite metric spaces there are a number of equivalent formulations of uniform embeddability 7 13 to these we add the following simple extension which applies to any metric space and which will be our fundamental criterion for uniform embeddability Proposition 21 Let X be a metric space Then X is uniformly embeddable if and only if for every R gt 0 ande gt 0 there epists a Hilbert space valued map X a H apex such that 1 for all z E X and such that 4 MARIUS DADARLAT AND ERIK GUENTNER 239 supuo 7511mm R e X s 6 ii limgH00 supllt m mzgtl ddz 2 S on E X 0 These conditions may be replaced by iii supl17lt m mzgtl ddz S R Lz E X S 8 iv limgnooinf l m 7 mzll dxz 2 S Lz E X 2 respectively Remark We refer to and iii collectively as the convergence condition similarly we refer to ii and iv collectively as the support condition Proof The interchangeability of gt iii and ii gt iv follows from the simple observation that for unit vectors g 77 E H we have Hg 7 77H2 2 7 2 lt i7gt Assume that X is uniformly embeddable and let F X 7 H be a uniform embedding in a real Hilbert space H Let EXpH R H H H H H H 69m and de ne Exp H 7 EXpH by 1 1 EX 1 7 97 EH pC C C C EC CC Note that ltExp EXpCgt em for all C C E H For t gt 0 de ne 1 eitllFmH2 It is easily veri ed that lt m mzgt e tllFw Fll2 Consequently for all x x E X we have 1 and if1tiwgtw 2 3 mg 3 eitp7dwgtw 2 2 Letting t 81 pR2 1 it is easy to verify the conditions iii and ii above Conversely assume that X satis es the conditions in the second part of the statement There exists a sequence of maps 77 X 7 Hn and a sequence of numbers SO 0 lt 51 lt SQ lt increasing to in nity such that for every n 2 1 and every x x E X 1 117749011 1 ii 7 nnz ll S 171 provided dxz S w iii 7 nnz ll 2 1 provided dzx 2 Sn Choose a base point 0 6 X and de ne F X 7 69301Hn by 1 1795 g 77195 771950 69 77296 7 772960 EB UNIFORM EMBEDDABILITY 5 It is not hard to verify that F is well de ned and pdxd S 7 S dzx 17 for all d x 6 X7 where p zio mman and the XSn71sn are the characteristic functions of the sets 5 1Sn lndeed7 let x x E X If n is such that xn 71 S dxz lt we have ma aW it llmm ll2izllmm9 ll2 1 1 2 n711 72 dx 1 Similarly7 if n is such that Sikl S dx7 z lt Sn we have 1 HFW FWW Z Z Z HUM WWW 2 i971 Ll mm D Remark Straightforward modi cations of the above argument produce a uniform embedding F satisfying the sharper estimate 7 S dzx 6 for an arbitrarily chosen 6 gt 0 simply replace the 171 in ii by 62 Two metrics d and d on the set X are coarsely equivalent if for every R gt 0 there exists an S gt 0 such that the dimetric ball with center z and radius R is contained in the d imetric ball with center z and radius S and conversely Equivalently7 two metrics on X are coarsely equivalent if the identity map X a X is a uniform embedding Proposition 22 Let d and d be coarsely equivalent metrics on X Then X is uniformly embeddable with respect to d if and only if it is uniformly embeddable with respect to d Proof If two maps are uniform embeddings7 so is their composition D Remark Below we require only the fact that if Y a H and X a Y are uniform embeddings then the composite X a H is a uniform embedding In other words7 we do not need to know that X and Y are coarsely equivalent to conclude the uniform embeddability of X from that of Y Let P be a countable discrete group A length function on P is a non negative7 real valued function Z satisfying7 for all a and b E P i Nab S 1a 15 6 MARIUS DADARLAT AND ERIK GUENTNER 11 10 1 10 iii la 0 if and only if a 17 A length function Z is proper if for all C gt 0 the subset l 10Cl C P is nite One can construct an integer valued proper length function on P as follows Let S be a symmetric set of generators of P Let l0 S a N be a proper function satisfying ii7iii above Then la infl0a1 l0an a a1an7 al 6 S is a proper length function on P Given a length function Z we de ne a metric d by dab la 1b A metric constructed in this way from a length function is left invariant in the sense that dca7 cb dab7 for all a7 b and c E P Conversely7 every left invariant metric arises in this way from a length function A length function is proper if and only if the corresponding left invariant metric has bounded geometry Recall that a metric space X has bounded geometry if for every R gt 07 there is a uniform bound on the number of elements in the balls of radius R in X We require the following well known proposition compare 197 Lemma 21 Proposition 23 Let P be a countable discrete group and let d and d be metrics on P asso ciated to proper length functions l and l respectively Then d and d are coarsely equivalent Proof Since the metrics are left invariant it suf ces to consider the containment7 as in the de nition7 of balls centered at the identity element By symmetry it suf ces to show that for every R gt 0 there exists an S gt 0 such that for all a E P if la lt R then l a lt S But7 since l is proper this is obvious D As a consequence of the previous two propositions7 uniform embeddability of a countable discrete group P is independent of the particular proper length function used to de ne its metric Consequently7 we systematically omit reference to a speci c length function or metric in the statements of all results and say simply P is uniformly embeddable to mean that P is uniformly embeddable in a Hilbert space for some equivalently all left invariant proper metrics Finally7 we draw two simple consequences An action of a discrete group on a locally nite metric space X is proper if for every bounded subset B C X the set a E P a B B 31 0 is nite Equivalently7 for every x E X and R 2 0 the set a E P a z E BR is nite Observe that a free action of a discrete group on a locally nite metric space is proper UNIFORM EMBED DA BILITY 7 Corollary 24 Let P be a countable discrete group equipped with a left invariant proper metric Let X be a locally nite metric space equipped with a free isometric action of P Then the inclusion P a X as an orbit is a uniform embedding Proof Let 0 6 X Since the action of P on X is by isornetries la dXa 00 de nes a length function on P Since the action is free and X is locally nite l is a proper length function Let d be the left invariant rnetric associated to l According to the previous proposition the original metric on P is coarsely equivalent to d which is precisely what was to be proved D Corollary 25 Let X and P be as in the statement of the previous corollary If X is uniformly embeddable then so is P B Property A is a condition on metric spaces introduced by Yu 20 We will work with the following characterization of Property A Proposition 26 A discrete metric space X with bounded geometry has Property A if and only iffor euery R gt 0 and 8 gt 0 there eccists an S gt 0 and a Hilbert space valued function g X a H such that for all p x E X we have 1 and 2 d7 S R gt 7 S 8 ii dzx 2 S i lt m mzgt 0 Equivalently for every R gt 0 and 8 gt 0 there epists an S gt 0 and X a l2X such that for allz x E X we have 1 as above and m suppg c Bsa a Remark As in the case of uniform ernbeddability we refer to as the convergence condition and to ii and iii collectively as the support condition It is clear from the proposition that Property A is invariant under coarse equivalence We refer the reader to 15 for an introduction to exact groups Our interest is Prop 26 is motivated by the following result inspired by 13 Theorem 27 16 and also A countable discrete group P is eacact if and only ifP has Property A with respect to some every left invariant proper metric D In analogy with Cor 25 we have the following result in which we do not assume that X itself has Property A 8 MARIUS DADARLAT AND ERIK GUENTNER Corollary 28 Let P be a countable discrete group Assume that P acts freely and isomet rically on a locally nite metric space X satisfying the convergence and support conditions of Prop 26 Then P is eccact Proof Include P C X as an orbit The convergence and support conditions of Prop 26 pass from X to the subspace P Since P has bounded geometry Prop 26 applies D 3 LIMITS We begin to establish the closure properties of the class of countable discrete uniformly embeddable groups In this section we treat direct limits7 our main result being the following proposition Proposition 31 Let P be the limit of a directed system of countable discrete groups G1 a G2 a G3 a in which the the maps G a Gn1 are injectiue If each of the groups Gn is uniformly embeddable then so is P Proof The proof is based on a method of extending Hilbert space valued functions from a subgroup to an ambient group Speci cally7 let G a H be a Hilbert space valued function on a subgroup G of a countable discrete group P Choose and x a family of coset representatives z E X C P for PG having done so each element a E P can be uniquely expressed as a product an ga7 where pa 6 X and ga E G The extension Eof is de ned by E r a H l2PG 2 H 1200 31 5 mac 2 59a 36 Let now P be a direct limit as in the statement of the theorem Equip P with a proper length function 1quot and associated metric d1quot Metrize each of the subgroups Gn as subspaces of P the metric and length function on Gn are simply the restriction of d1quot and l1quot Observe that for a7 b E P we have a lbeonaaon onesz zber and that in this case7 dagagb dr 9127917 drm 919617 9b drab 3 We show that P satis es the convergence and support conditions of Prop 21 Let 8 gt 07 and R gt 0 be given Obtain n such that if a E P has lpa S R then a E G this is possible because the length function l1quot on P is assumed to be proper According to the criterion for uniform embeddability obtain a Hilbert space valued function g G a H and such that for all g7 h E Gn we have H gH 1 and UNIFORM EMBEDDABILITY 9 i if dGng7 h S R then Hg 7 hll lt 8 ii V gt 0 3S gt 0 such that if dGngh 2 S then l g hl lt 6 Let Ebe the extension of g to P de ned above Clearly7 1 for all a E P and it remains to verify the conditions of Prop 21 Let a7 b E P For the convergence condition assume that lpa 1b dra7 b S B By our choice of n we have a lb E Gn and7 according to 37 do gmgb dpab S R Therefore7 ME 7 51716 ta 7 9 6017 11697511 lt a For the support condition let 6 gt 07 obtain S as in ii above and assume dpab 2 S According to lt ga7 gb gt7 aGn ltg17 l7gtlt 9a7 9bgtlt6aan76bangt 0 otherwise 7 we may assume a lb E G But then7 according to 3 again7 dGngagb dpab and llt m bgtllt 5 Remark The previous argument is easily adjusted to yield a new proof of the fact that a direct limit7 as in the statement of the proposition7 of exact groups is again exact 15 lndeed7 under the assumption of exactness7 employing Prop 26 instead of Prop 21 we replace ii by ES gt 0 such that dcng7 h 2 S i 9 h 0 Using 4 and the surrounding discussion conclude that this property is shared by 4 EXTENSIONS Let 1 a H a P a G a 1 be an extension of countable discrete groups We study uniform embeddability of P under various hypotheses on H and G Our primary result7 of which our other results are consequences7 is the following theorem Theorem 41 Let P be an eactension ofH by G as above IfH is uniformly embeddable and G is eacact then P is uniformly embeddable As corollaries we mention the following two results about semi direct products Corollary 42 Let G and H be countable discrete groups Let oz G a AutH be an action ofG on H If both H and G are uniformly embeddable and aG C AutH is eacact then the semi direct product P H gt4 G is uniformly embeddable 10 MARIUS DADARLAT AND ERIK GUENTNER Proof The semi direct product P is the set of pairs 57 6 Hgtlt G7 with product 57t7y samtxy The assignment 5x l gt80xm l1 a H gt4 aG gtlt G de nes an injective homomorphism By the theorem the semi direct product Hgt4aG is uniformly embeddable7 as is H gt4 aG gtlt G In particular7 P is a subgroup of the uniformly embeddable group and is therefore uni formly embeddable D Corollary 43 Let P Z gt4 G be a semi direct product IfG is uniformly embeddable then P is uniformly embeddable Proof Apply the previous corollary using the fact that GLnZ is exact 15 D Remark Central extensions are more dif cult to analyze than semi direct products Our theorem applies to central extensions in which the quotient is exact Gersten has shown that if a central extension of Z is described by a bounded cocycle then P is quasi isometric to the product Z gtlt G 11 Consequently7 if G is uniformly embeddable so is P Beyond these two results little is known As remarked earlier7 the property of uniform embeddability of a countable discrete group is independent of the proper length function Consequently7 we are free to choose these for our groups G7 H and P in a convenient manner7 which we do as follows Let lp be a proper length function on P De ne length functions on H and G according to lHs lps7 for all s 6 H7 5 lgz minlpa a E P and a d for all z 6 G7 6 where we have introduced the notation a gt gt a for the quotient map F a G It is easily veri ed that the minimum in the de nition of la is attained7 and that la is a proper length function on G that lH is a proper length function is immediate Denote the associated left invariant metrics by dp d0 and dH7 respectively Observe that the inclusion H gt P is an isometry7 and the quotient map F a G is contractive Finally7 choose a set theoretic section 0 of the quotient map F a G with the property that lpox law7 for all z E G 7 andde nei7l gtltG Hby i7ax 7zquotlaz7 f ld7 for all a E P z E G 8 UNIFORM EMBED DA BILITY 11 Lemma 44 Let a b E P z E G We have dFa7b S dGz7ad07bdH77a7z777b7 dH77a796777b796 S doW da967b dra7b 10 Proof Let a b and z E G be as in the statement From the de nition 8 of 77 we obtain 77a m 177b m 0a 1x 1 ailb 05712 The desired inequalities follow easily from this equality together with 5 7 and the sub additivity of length functions D Proof of Thm 41 We prove that P satis es the conditions of Prop 21 Let 8 gt 0 and R gt 0 be given We show that there exists an f P a l2G7 such that HfaH 1 for all a E P and that satis es the convergence and support properties i if dpab S R then 1 7 fafb lt 8 ii V gt 0 HS gt 0 such that if dpab 2 S then lfafb lt 6 Adapting an argument of Anantharaman Delaroche 3 our strategy is to use a g satisfying the conditions of Prop 26 to average an h satisfying those of Prop 21 to produce f Since G is exact there exists according to Prop 26 a g G a l2G and an SC gt 0 such that 1 for all z E G and such that 19 ll lt979y gt1 lt 827 provided downy S R iig suppgx C BSGQ for all z E G We shall without comment view 9 as a function on G gtlt G whenever convenient Since H is uniformly embeddable there exists according to Prop 21 an h H a H such that 1 for all s E H and such that ih l1 7 hs ht lt 82 provided dHst S 250 R iih V gt 0 SSH gt 0 such that if dHst 2 SH then lhsht lt 6 Having chosen 9 and h de ne f P a l2G7 by faz gdxhnaz for all a E P z E G Note that fa E l2G7 It is elementary to verify that HfaH 1 for all a E F We verify the remaining properties 12 MARIUS DADARLAT AND ERIK GUENTNER For the convergence property let ab E P with dpab S R Consider 17 h axh bp an bp 1iltfa7fbgt Z lt77 77 gt9 9 11 l17lt9d795gtl Since the map F a G is contractive we have dgab S B so that by i9 the second term in 11 is bounded by 82 Observe that according to iig the sum in the rst term is over z E BSGa BSGb Recalling that 1 we therefore bound the rst term by supf l1 lth 077 h 57 gtl 95 6 335161 33500 From 10 we see that for z E B5Ga B5Gb we have dHi7axi7bx S 280 B so that by this supremum and consequently the rst term in 11 is bounded by 82 For the support property let 6 gt 0 be given and obtain SH as in iih Let a b E P be such that dpab 2 250 SH Then lltfa7fbgtl ZWWMUWlth77a7967h77b796gt 160 S Zlgwwga llth77a7967h77b796gtl supf llth77a7z7h77biz H 3 96 E BSGW m 33509 7 where again we use the fact that 1 to obtain the second inequality and note that by iig the sums are in fact over z E B5Ga BSG From 9 we see that for such x we have dHi7axi7bx 2 dpab 7 dga 7 dgxb 2 SH so that by iih the supremum is indeed bounded by 6 D l 5 FREE PRODUCTS The main result of this section is the following theorem Theorem 51 Let A and B be countable discrete groups and let G be a common subgroup If both A and B are uniformly embeddable then the amalgamated free product A 0 B is uniformly embeddable Our strategy for proving the theorem is to construct a locally nite metric space X which is uniformly embeddable and on which P acts freely by isometries The construction of X is based on the notion of a tree of metric spaces which we now recall UNIFORM EMBED DA BILITY 13 A tree T consists of two sets a set V of vertices and a set E of edges together with two endpoint maps E a V associating to each edge its endpoints Every two vertices are connected by a unique geodesic edge path that is a path without backtracking A tree of spaces X with base the tree T consists of a family of metric spaces Xm X5 indexed by the vertices 1 E V and edges e E E of T together with maps UM X8 a Xv whenever 1 is an endpoint of e The UM are the structural maps of X We will assume although this is not strictly necessary that the metrics on the vertex and edge spaces are integer valued The total space X of the tree of spaces X is the metric space de ned as follows The underlying set of X is the disjoint union of the vertex spaces Xv the metric on X is the metric envelope d of the partial metric csee the appendix de ned by CR dzy 31 E V such that x y E Xv 79 1 3e 6 E and z 6 X5 such that z 0395v2 y aewz for all my in the domain D z y E Xv 1 E V U03957J20395w2 2 6 X5 e E E Observe that D is ample see the appendix this follows from our assumption that the underlying tree is connected and that cTis de ned for all pairs Ly where z and y are in the same vertex space We call Ly an adjacency if there exists an edge e with endpoints 1 and w and an element 2 6 X5 such that aevz z and aewz y Using this terminology the partial metric can be described as being the given metric on each vertex space and 1 on adjacencies Example 52 Consider the case in which the vertex spaces are metric graphs or rather the set of vertices of a graph equipped with the graph metric and the edges spaces are singletons The total space X is itself a metric graph Indeed it is the disjoint union of the graphs together with additional edges coming from the underlying tree Precisely the edges of X are rst the edges in the individual Xv and second the edges e of the underlying tree the endpoints of such an edge e are the images of the maps X8 a Xv to the endpoints 1 of e in T Remark The generality of the de nition above is mandated by the fact that when considering amalgamated free products we will encounter vertex spaces that are not metric graphs 14 MARIUS DADARLAT AND ERIK GUENTNER Theorem 53 Let X be a tree of metric spaces iri which the structural maps are isometries If the vertep spaces Xv are uniformly embeddable iri a Hilbert space with commori distortiori bourid theri the total space X is uniformly embeddable iri a Hilbert space Remark Quite explicitly7 the hypothesis is that there exist maps pi as in 1 such that for every vertex i E V there exists FE Xv a H satisfying deXUWJ D S HEW FuWMl S Pdxt9cx 7 for all x 96 E Xv ln other words7 the same distortion bounds may be used for every vertex space Remark If there exist only nitely many distinct isometry types of vertex spaces as will be the case for the amalgamated free product this simply means that every vertex space is uniformly embeddable Remark When we prove the theorem we will use the fact that the existence of a common distortion bound on the uniform embeddings of the vertex spaces implies that the estimates in the fundamental criterion for embeddability are uniform This follows from the proof of Prop 217 speci cally from the estimate Assuming Thm 53 we prepare for the proof Thm 51 by associating a tree of metric spaces Xp to the amalgamated free product T A 0 B The tree T is the Bass Serre tree of T 177 4 Precisely7 the vertex and edge sets of T are given by V PAUPB E rO respectively the endpoint maps are the quotient maps PO a TA and TO a TB In other words7 the endpoints of an edge a C coset are the vertices one A and one B coset that contain it We associate a metric space to each vertex and edge of T as follows Equip T with an integer valued proper length function and associated metric Let i E V be a vertex and assume that i E TA ln particular7 i is an A coset in T denote by Xv this coset itself7 metrized as a subspace of T Proceed similarly for vertices i E TB Let e E E be an edge In particular7 e is a C coset in T denote by X5 this coset itself7 again metrized as a subspace of T The structural maps are de ned as follows Let the vertex i be an endpoint of the edge e Inclusion of cosets subsets of T provides the structural map UM X8 a Xv UNIFORM EMBED DA BILITY 15 Remark The metric space Xv is not isometric to the graph metric space of the Cayley graph of A or B or even of the restriction of the Cayley graph of P to the subset A or B Similar remarks apply the edge space X5 Indeed it is important that we metrize Xv and X5 as subspaces ofP and not via their identi cation with A or B and C using the given metrics on these groups Let Xp be the total space of Xp The group P acts on Xp by left multiplication Precisely if z E Xv and a E P we de ne a z 6 XM using ordinary multiplication in P by virtue of the fact that z E v C P This action preserves adjacencies and since in addition the metric on P is left invariant it preserves the partial metric According to Prop 72 of the appendix P acts by isometries on the total space X1quot Proposition 54 LetA andB be countable discrete groups and let G be a common subgroup Let P AgtkcB be the amalgamatedfree product Let Xp be the tree ofmetric spaces associated to P and let Xp be its total space We have i the structural maps of X1quot are isometries ii every vertep space of Xp is isometric to one ofA or B which are metrized with the subspace metric via the inclusions A B C P iii The action ofP by isometries on Xp is free iv Xp is locally nite Proof All of the assertions but iv follow from the previous discussion The condition iv while apparent it will be derived formally in Prop 56 D Proof of Thm 5 According to the previous proposition the tree of metric spaces Xp as sociated to the amalgamated free product P A 0 B satis es the hypothesis of Thm 53 according to which the total space Xp is uniformly embeddable Again according to the previous proposition P acts freely by isometries on X1quot Hence by Cor 25 P is uniformly embeddable D We have reduced our main theorem concerning amalgamated free products Thm 51 to a theorem concerning trees of metric spaces Thm 53 We now turn attention to the proof of that theorem We suitably adapt the method employed by Tu in his study of Property A for discrete metric spaces 19 For a tree of metric spaces X the inclusions Xv a X of vertex spaces into the total space are as a general rule not isometric the presence of shortcuts in neighboring Xw may cause the distance in X between two vertices p y E Xv to be considerably smaller than the 16 MARIUS DADARLAT AND ERIK GUENTNER distance between them in Xv itself Nevertheless for our X1quot these inclusions are isometries7 a fact that may be traced back to the manner in which the vertex and edge spaces of Xp are metrized as subspaces of P which circumvents any distortion that may have otherwise been introduced by the amalgamating subgroup The following proposition has no analog in Tu7s work 19 nevertheless7 we require it in order to complete our arguments Proposition 55 Let X Xw7 X5 be a tree of metric spaces in which the structural maps X5 a Xv are isometries Let X be the total space of X Then i the inclusions Xv a X are isometries and ii if my is an adjacency then dpy 1 Further for all z E Xv y E Xw dpy dTo7 w inf dx07 x1 dz27 x3 dxp17 pp 7 12 where dT is the distance on the tree T and the in mum is taken over all sequences 0 7 where p 2dTo7 w 1 and i z zo y zp ii zgk1x2k is an adjacency for h 1dTow iii zgkz2k1 E Xvk for h 07dT17 LU and i 110 od mw w are the vertices along the unique geodesic path in T from i to w Remark Formula 12 is Tu7s formula 197 Section 9 with slightly different notation Proof Let i w E V and let x E Xv and y E Xw A reduced path from x to y is a sequence of elements z 0717 7z2mz2n11 of X for which there exist vertices o0 1 E V such that i W 1941 for j 17 7n7 and ii z2k7x2k1 E Xvk for h 01 n Observe that by appropriately inserting and deleting z7s we may alter a path7 without increasing its length7 so as to obtain a reduced path use the triangle inequality for the metrics on the individual Xv Consequently7 when computing distances in X it suf ces to consider reduced paths Let x 0 7x27 y be a reduced path from p to y According to the de nitions of a reduced path and of the domain D of the partial metric we obtain sequences of i vertices i 110 7o w in V7 ii edges e17 en in E7 and UNIFORM EMBED DA BILITY 17 iii elements 21 2 of the edge spaces X517 7X5 satisfying i z2k7x2k1 E Xvk for h 07 7n7 ii the endpoints of ek are WA and ok for h 17 7n7 and iii oek kil k zgk1 and oewk zk pgk for h 1n The sequences are uniquely determined by these conditions The given reduced path z z07zp y in X lies over the edge path e1en in T We show that in the de nition of d it suf ces to consider reduced paths lying over the unique geodesic edge path in T from i to w the assertions of the proposition follow easily from this fact Let x p07p2n11 y lie over the non geodesic path e17 en in T We show that by successive elimination of certain xi we obtain a shorter path lying over the geodesic path in T lndeed7 there exists i E 17 7n 7 1 such that el and ei1 have the same endpoints7 that is7 such that ill1 iii1 We have z2122 Mi 2142 6 Xvi e Xvi e Xvi1 XvH 2121 2141 2143 We eliminate xgi1p2i 2111 2112 from the given path and claim that the resulting path i is shorter than 0 7p2n11 and ii lies over an edge path with fewer backtracks than e17 7en Of these ii is obvious the new path lies over the edge path e17 ei1ei2 7 en obtained by eliminating el and ei1 Further7 follows from the fact that the structural maps are isometries In particular we have dlt 2i717 2142 d2i72i1 d2i72i17 from which follows d21227 2143 S dlt 2igt27 2igt1gt dlt 2i717 2142 d2i27 2143 d21227 2121 d2i7 2i1gt d2i272i3 D Proposition 56 Let X be the total space of a be a tree of metric spaces X in which the structural maps are isometries Assume that each uertem space Xv is locally nite and there is a uniform bound for the number of adjacencies of each point in X Then X is locally nite Proof In view of formula 12 a nite radius ball in X can intersect only nitely many vertex spaces D 18 MARIUS DADARLAT AND ERIK GUENTNER 6 PROOF OF THEOREM 53 We may enlarge the tree of metric spaces if necessary such that the underlying tree T will contain an in nite geodesic in starting at some basepoint For every 1 E V let 041 6 V be such that the edge 0040 points towards w For each 1 E V let YE 0395vX5 C Xv and fv UBMH O as YE a XDM where e 15040 It follows immediately from Prop 55 that each fv is isometric Using the same proposition7 and the notation just introduced7 we rewrite the distance formula 12 as follows Let 0 6 Xv and 6 6 sz There exist a unique pair of nonnegative integers hZ such that 04111 ov and dT11 h Z7 where again dT denotes the distance in the tree T By symmetry we may assume that h 2 Z If h 2 1 and Z 2 1 then E72 dWOWO Z dltfaivyi7yi1 i0 d9507 956 k Z inf dltfak 1vyhil7 faz 1v yZ71 v 13 l dltfajv y397y391 739 m H 0 subject to the constraints that yi E Yaw and E Yaw If h 2 1 and Z 0 then 1672 d9507 956 k inf 4950790 Z dfaivyi7yi1 dfak1uyk177 14 i0 subject to similar constraints An h eham is a sequence x 031 xn1 such that for each 0 S h S n 7 2 there exists ik E Yam satisfying dkik lt ndMW 1 and fakvik n1 lf x0 6 Xv the n chain starts in Xv Note that for any n 2 17 1 E V and x0 6 Xv there exists an n chain whose initial element is 0 Lemma 61 Let x0 6 Xv and 6 6 sz 1 31 1 with d0xg lt R and let h and Z be as m 1314 with h 2 Z IfZ 0 there em39sts a chain x0731 wk such that max sup dxizi1 dzkzg lt 2BR ogigkir UNIFORM EMBED DA BILITY 19 ft 2 1 there em39st chums zo1xk 955953 J2 and 22 6 XDME XDvz such that max sup axial11 sup dzz1 dkz d2z dzz lt 2BR 09971 09971 Remark In the proof of the lemma and at a number of subsequent points we require the fact that if z E Xv and y E YE then d967fvy dzy 1 This follows from Thm 55 Indeed since y fvy is an adjacency d fvy S dy 1 For the reverse inequality let 8 gt 0 and obtain x E YE such that dx fvy8 2 1dz x dfvz fvy Since fv is an isometry dzx dfvz fvy 2 dxy and we are done Proof Case Z 0 We have k 2 1 since 1 31 1 and k 2 Z Since d0x6 lt R by 14 there is yo 6 YE such that k dx0y0 lt R The sequence 1xk is constructed inductively Let i0 6 YE be such that d0i0 S d0y0 and d0i0 lt d0Yv 1 and de ne 1 fvi0 Then d0x1 d0i0 1 3 dx0y0 1 lt R Thus d0x1 lt R and d1xg S d06 d0x1 lt 2R Repeating the same argu ment for the pair of points 1sz if k 2 2 we nd 2 6 X0420 with 012 being a chain d1x2 lt 2B and d26 lt 22R Continuing in the same way we obtain a chain x01xk such that d1x lt 2i 1R and dx6 lt TR 1 S t S k Since k S R this completes the proof of a Case Z 2 1 Let y0yk1 with y E Yaw and y6y271 with E Yawz be sequences such that the expression whose in mum is taken in 13 is less than R 7 k 7 Z Let 2 fD71Uly271 and z fak71uyk1 Then we have dz 2 lt R dz0 z lt R and dzg 2 lt R The proof is completed by applying part a to the pairs 0 2 and xi 2 D Given an n chain x x0x1xn1 starting in Xv de ne for 0 S k S n 71 6k dzk Yaw and 1 60 V V 6k where we introduce the notation a V b maxab For future notational convenience de ne 191 0 Note that if x 955953 x kl is another n chain starting in Xv then dk717k S 6k71 2 15 wk 7 ldltkYakv 7 S dk 0 S k S 77 71 20 MARIUS DADARLAT AND ERIK GUENTNER Lemma 62 Let x x0731 7xn1 and x 955953 xnil Xv Then be n ehatns starting in 7 ldxk 7 dx0z6l S 2k6k1 V 0271 2k7 0 S k S n 71 17 Proof Since the statement is obvious in the case k 0 we assume k 2 1 Let ik and i be as in the de nition of n chains Observe that d it E ip CK 952 dwk 95 S dww t 29k V 93 2 dWwit Z dwkwt dWwM 9227952 2 dww t 29k V 93 i 2 Since fakm is an isometry we have dk1x 1 dikik and the lemma follows from these inequalities by induction D Given an n chain x x07x17xn1 and N gt 0 we de ne7 for 0 S k S n 717 ln1 1 1k 7 4 7 N ck aOak117 t7k71177 19 where we introduce the notation a1 maxa0 Note that co 1 n 7117 10 One checks immediately that 10 2 a1 2 2 1W1 egeiiln 00 131 20 6k2eNgt0kZeNgtak0gtck1cn10 21 The coef cients ck were introduced by Tu 19 actually we work with the square root of Tu7s coef cients Since the maps fv are isometries7 it is possible in our case to de ne ak explicitly as in 18 Both ak and ck should be regarded as functions a and C of n chains x x0731 7xn1 We will often write a and 0 instead of a and cf Lemma 63 Let x x0731 7xn1 and x 955953 znil be n ehatns starting in 7 Xv flu many91 lai 7 12 then 2 Z 1 7 6321 S 711 w a laws 3 w and m 2 i let 7 on s Proof This is an exercise For i ii use the inequality lf7 S Qla 7 b D The following continuity property of the coef cients ak is a minor variation of a formula in Tu7s paper 197 formula 92 UNIFORM EMBED DA BILITY 21 Lemma 64 Let x 07x17xn1 and x 955953 x be n chams starting in 7 n71 Xv and assume d0x6 S n Then 7712 7 lt 7 0513371le akl N 22 Proof Denote A ln 771 We are going to show that n l IO aol S N la 7al lt la 76 lA 23 k k 7 k71 kil N 1 A 7 2 From these we immediately obtain lak 7 12 3 LA 3 concluding the proof Of these inequalities7 the rst is straightforward lndeed7 using 16 and the fact that the map t gt gt 1 7 ln1 t1r is Lipschitz we have 1 1 n lao 1le S l o bl S NdWo fJ S N To prove the second inequality denote ft 4nt 671 According to 16 and Lemma 627 we have m 7 6 demo 3 gm v 63H lt24 Denote wt ln1t7 so that we have for all t 2 0 IN S 100 S 1000 A 25 Case 1 V 0 S ft9k1 V 0271 By symmetry we may assume that 0k1 2 0271 Thus 0271 S 0 S f0k1 and 0271 S 0k1 3 0k 3 ft9k1 Using 25 we obtain W924 S W903 7Weed S 71019k71 A7 W924 S W903 7Weed S 71019k71 A Now7 23 follows immediately from these inequalities7 together with the fact that for real numbers a S st S b A we have 117 sNgt 717tNm 117bN 7 lt1 7aNm A Case 1 V0 2 ft9k1 V0271 By symmetry we may assume that 19k 2 1 Then 1 2 f0k1 2 0k1 hence 0k 6k Therefore using 24 1 1 6k 3 let 7 6 6 3 wk 7 6 6 s we V624 6 Eek 9 22 MARIUS DADARLAT AND ERIK GUENTNER hence 0k 3 26 S ft9 From 0 3 0k 3 ft and 25 we obtain W92 S W910 S 1006092 S W92 A From this inequality and the fact that the map t gt gt 1 7 tN1r is Lipschitz we obtain ak 7 a 3 which implies 23 D De nition Given R gt 0 and 8 gt 0 choose and cc n E N such that 1 lnn9 5 8 R 7 2B 26 4n lt3R3 71gt andNENsuch that 6nn1 8 lt7 27 N 3R3 Having done so apply the fundamental criterion for uniform embeddability Prop 2 to choose and cc afamily mhex of unit vectors in a Hilbert space HX BugHy with 1 E H ifz E Xv and such that 8 SUPiH y 5211 3dy7y S 251V 30717 6 Xv E V lt W7 28 sup y y dyy S S yy E va E V 0 29 See the remarks after Thm 53 for comments on why this is possible Finally for every n chain x 01 xn1 in X de ne the unit vector 77quot E HX by 1 n71 77x i Ck z 7 30 a i where of course the ck s are de ned according to 19 and depend on the chain x Lemma 65 Let x x0x1 xn1 and x zbxi z1 be n chains starting in Xi Ifd9607966 n than Mr 7M ProofLetIkck7 0c 7 0andJhck0c 7 0 1f1 h Lthen ak1 1 7 ln1 19 1Jr 31 0 hence 0k1 S eN Similarly 0271 S eN Hence from 17 we have dpkx S 2neN 2n dp0z6 lt n2eN 3 31 UNIFORM EMBED DA BILITY 23 Consider 1 n71 1 n71 ll x xll Ck m i 0 5w g k gkk 1 1 1 7 Bk 7 69 gm 7 7 Z 69 7 Ck 761 lt k0 keJ k 7 1 7 W Z 69 7 gw k keI Observe that the gmks and gags are in orthogonal components of HX We bound the third term on the right using 20 31 and 28 as follows 1 12 1 mil 12 2 2 2 8 S sip H y 52 S W where A yy dyy S n2eN 3 y y E Xv 1 E V We bound the sum of the rst two terms on the right by 1 n71 E l 1 2 3n1 1 3n1 7712 2 8 7 7 lt7 7 lt7 7 lt7 3 71le on 0535 laz ailz 7 N 7 3R3 k0 where the inequalities are from Lemma 63 22 and 27 respectively Combining these observations we obtain the result D Lemma 66 Let x 12 xn and x zozi z1 be n ehams with 0 6 Xv and 1 6 XMW If zox1 is a 2 eham and d0x1 S n then H77quot 7 nxH S s R339 Proof Let i0 fv71z1 and set 2 i0x1 xn1 According to the de nition 30 of 77quot we have 101 n71 77x new dkgmk mg k1 n71 7 Ex 1 1 7 7 UWWWORGm ZQQ k1 E 3 24 MARIUS DADARLAT AND ERIK GUENTNER where dk Ck117 k Val k11n 7 7 310 Jk Cki0717 wk amp0amp1 ampk711 k 11 5k with 30 1730 1 and 5 aki01k minlt1iwgt 1gtgk For 131637171wehaveeitherleak0hencedkiclk0or dkigk W i k 1nik1i k1nik711idk lt l k lt 1 7 2 nik17ampk 7 2xn7k39 Applying the above expressions for 77quot and 77quot with this inequality we estimate n71 1 7 llnx 7 gin E 7 7 n5 W 0 k1 k H Slt ni dkdk2gt2 k A lt lnn9 2lt 8 7 7 4n 3R3 where the nal inequality comes from the choice 26 of 71 Apply Lemma 65 to the chains x and X7 noting that dz07 i0 dz07 x1 71 S 717 and use the previous inequality to conclude H x e XHltH x M 7 XHlt 28 8 e 8 7 7 i 7 7 7 7 3R3 3R3 R339 Lemma 67 Let x x07x17xn1 and x zgzgzg1 be n ehams m X Then lltnx7nxlgtl S supllt y7 ylgtl i 1171 6 XM E Vin711 6106079607 WEN 1 Pmof Let k and Z be as in 13 Considering symmetry we assume that k 2 Z Also7 if k 2 717 then lt77x777xlgt 0 so we assume that k lt 71 With these assumptions ltnx7nxlgt 1 nikil E Z Ckicli 7 6 i0 UNIFORM EMBED DA BILITY 25 Making use of 21 conclude that if 014 V 0671 2 EN then all the products owl06 in 32 are zero and we are done Thus we assume that 014 V 0671 3 EN Let m be the largest number 0 S m S n 7 k 71 with the property that 0km1 V 06m71 3 EN By 17 d2k26 2 d2k26 7 2726N 1 for all 0 S 2 S 772 On the other hand using 15 k71 271 d2k26 2 d2026 7 d221 7 d221 20 j0 1671 271 Z dWo h 5239 2 i Z 2 20 j0 2 619607956 k Z29k71 V 9271 2 2 d2026 7 4726N 1 Combining these two inequalities we obtain d2k26 2 d2026 7 6726N 1 for all 0 S 2 S 772 Finally arguing again on the basis of 21 as above conclude that the terms in 32 for 2 gt m are zero Thus applying 20 and the previous inequality we see that x x 1 m lltn 777 gt1 n E Ewan Sgp llt y7 ylgtl S Sgp llt y7 ogtl7 20 where 9 21721 22 6 X271 6 V7 dy7y 2 610607966 7 6n6N 1 5 Conclusion of proof of Theorem 53 Given R gt 0 and 8 gt 0 let n N and and be con structed as in the De nition For each 20 E X choose and x an n chain x 20 21 2n1 and consider the corresponding vector 77quot We are going to show that the map 20 gt gt nlt 0gtw1gtmgtwwll satis es the convergence and support conditions of Prop 21 The support condition follows immediately from Lemma 67 in conjunction with 29 For the convergence condition we show that for any 20 26 E X with d20 26 lt R and any two n chains x and x starting at 20 and 26 respectively we have H77quot 7 77quot39 S 8 Let k and Z be as in13714 We may assume that k 2 Z Furthermore we treat only the case Z 2 1 the case Z 0 being similar Let x2 x j z and 2 be n chains whose initial elements are the points 2 2 z 2 given by Lemma 61 0 S 2 S k 0 Sj 3 Z Applying Lemmas 65 and 26 MARIUS DADARLAT AND ERIK GUENTNER 66 repeatedly note that since 71 gt 2BR this is possible by Lemma 61 with the convention that all empty sums are zero we have k71 11an 7 WWH 1177x027 W H77z 7 Hz H x x l0 H77 777 H S 71 w 7 Wall Z W 7 WWW j0 k l 3e lt 8 R 3 7 Remark In the proof of Thm 53 we showed that for every R gt 0 and every 8 gt 0 there is D a family 77quot106X of unit vectors in HX such that sup lnquot 7 nxH 3616607966 lt R lt 67 llt77x777xlgtl S supllt y7 y gtl M 6 X717 v 6 V7dy7y 2 616607966 7 6716N 1 Let us argue that in addition to Thm 53 these estimates prove that A 0 B is exact whenever A and B are exact 8 9 19 Indeed if we base the construction 30 of 77quot on a family apex satisfying the support condition ii of Prop 26 uniformly with respect to 1 E V then it follows from the second inequality that lt77x77xgt 0 whenever dx0p6 2 S6neN 1 In particular X satis es the convergence and support conditions of Prop 26 and A 0 B is exact by Cor 28 7 APPENDIX We collect several elementary results on the construction of metrics required for our treat ment of amalgamated free products Let X be a set A partial metric on X is an integer valued function d D a R1 de ned on a domain D C X gtlt X satisfying for all x y E X lt1 mm 7w ii dpy 0 if and only if z y In these statements it is assumed that all relevant pairs belong to the domain D of Gland that the domain D is symmetric and contains the diagonal We now associate a metric to a partial metric The construction is analogous to that of path metrics intuitively the distance between two points is the length of the shortest path between them A path from x to y E X is a sequence x zox1 zn y such that for every 1 S j S n we have plAmi E D The length of a path is 21clxi1zi The domain D is ample if for every x y E D there exists a path from x to y UNIFORM EMBED DA BILITY 27 Proposition 71 Let d be a partial metric on X de ned on an ample domain D De ne for all r y E X dxy inf length of paths from x to y Then d is an integer valued metric on X D The metric d de ned in the proposition is the metric envelope of Example If X is the vertex set of a connected graph and dis constant 1 on the domain of all pairs Ly that represent edges then d is the path metric dry is the smallest number of edges on a path from x to y Remark If my 6 D then dxy if and only if the length of every path from x to y is greater than or equal to dxy If D X gtlt X and dis a metric then d d Proposition 72 Let dbe a partial metric on X and let p X a X be a bijection with the property that i is E D 239f andgnly 2391 WC 9 E D and ii dw My CHM far all MI E D Then p is an isometry for the metric envelope Proof lt suf ces to show that dltprltpy S dxy for all x y E X Let 8 gt 0 be given Let do xn be a path in X from x to y such that 22 dag1 zj lt dry 8 Then Mpg is a path from ltpx to ltpy and we have n Ema4 sea Zc mm lt elm a dltpz m S M i 1 Since 8 gt 0 was arbitrary we are done D REFERENCES 1 J Alonso and M Bridson Semihyperbolic groups Proc London Math Soc 70 1995 567114 2 C Anantharaman Delaroche Amenability and exactness for dynamical systems and their Calgebras to appear in Trans of the AMS 2000 3 C Anantharaman Delaroche and J Renault Amenable groupoids LlEnseignement Mathematique 36 2000 with a foreword by Georges Skandalis and Appendix B by E Germain 4 G Baumslag Topics in combinatorial group theory ETH Lectures in Mathematics Birkhauser Boston 1993 MARIUS DADARLAT AND ERIK GUENTNER I XI Chen MI Dadarlat El Guentner and GI Yu Uniform embeddings into Hilbert space and free products of groups Preprint 2002 I P Cherix MI Cowling PI Jollissaint PI Julg and AI Valette Locally compact groups with the Haagerup property Unpublished manuscript 1998 AI Dranishnikov GI Gong VI Lafforgue and GI Yu Uniform embeddings into Hilbert space and a question of Gromov Preprint 1999I KI Dykema Exactness of reduced amalgamated free product Calgebras Preprint 1999I Free products of exact groups Calgebras Miinster 1999 l Cuntz and SI Echterhoff edsl Springer Berlin 2000 pp 61770 I S Ferry AI Ranicki and JI Rosenberg edsl Novikov conjectures index theorems and rigidity London Mathematical Society Lecture Notes no 226 227 Cambridge University Press 1995 I S Gersten Bounded cohomology and combings of groups unpublished manuscript version 55 1991 I MI Gromov Asymptotic invariants of in nite groups London Mathematical Society Lecture Notes no 182 pp 17295 Cambridge University Press 1993 I E Guentner and JI Kaminker Exactness and the Novikov conjecture Topology 41 2002 no 2 4117418 B Kirchberg and SI Wassermann Exact groups and continuous bundles of Calgebras Mathematische Annalen 315 1999 1697203 7 Permanence properties of Cexact groups Documenta Mathematica 4 1999 5137558 elec tronic I NI Ozawa Amenable actions and exactness for discrete groups C RI Acad Sci Paris S rl 1 Math 330 2000 no 8 6917695 I 1 PI Serre Trees Springer New York 1980 Translation from French of Arbres Amalgames SL2 Ast risque no 46 GI Skandalis 1 LI Tu and GI Yu Coarse BaumConnes conjecture and groupoids Preprint 2000 1 LI Tu Remarks on Yu s Property A for discrete metric spaces and groups Bull Soc Mathl France 129 2001 1157139 GI Yu The Coarse BaumConnes conjecture for spaces which admit a uniform embedding into Hilbert space lnventiones Math 139 2000 2017240 DEPARTMENT OF MATHEMATICS PURDUE UNIVERSITY 1395 MATHEMATICAL SCIENCES BUILDING WEST LAFAYETTE IN 47907 1395 86 E mail address mdd math purdue edu MATHEMATICS DEPARTMENT UNIVERSITY OF HAWAI I MANOA 2565 THE MALL HONOLULU H1 802 E mail address er ik math hawaii edu

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