Embeddings of discrete groups and the speed of random walks

Embeddin gs of discrete groups and the speed of random walks Assaf Naor ∗ Courant Institute naor@ci ms.nyu. edu Y uval Peres † Microsoft Research and UC Berkele y peres@m icrosof t.com Abstract Let G be a g roup ge nerated by a finite set S and eq uipped with the associated left- in variant word metric d G . For a Banach space X let α ∗ X ( G ) (re spectiv ely α # X ( G )) be the supr emum over all α ≥ 0 such that there exists a Lipschitz mappin g (respec ti vely an e quiv ariant mapp ing) f : G → X and c > 0 such that f or all x , y ∈ G we h av e k f ( x ) − f ( y ) k ≥ c · d G ( x , y ) α . I n particu lar , the Hilbert comp r ession expo- nent (respectively the eq uivariant Hilb ert compressi on exponent ) of G is α ∗ ( G ) ≔ α ∗ L 2 ( G ) ( respectively α # ( G ) ≔ α # L 2 ( G )). W e show that if X has mo dulus of smo othness of power type p , th en α # X ( G ) ≤ 1 p β ∗ ( G ) . Here β ∗ ( G ) is the largest β ≥ 0 fo r which th ere exists a set o f g enerator s S of G and c > 0 such that for all t ∈ N we h ave E  d G ( W t , e )  ≥ ct β , whe re { W t } ∞ t = 0 is th e can onical simple rando m walk on th e Cayley graph of G d etermined by S , starting at the identity elemen t. This result is sharp wh en X = L p , gener al- izes a th eorem of Guen tner and Kam inker [20], an d a nswers a qu estion posed by T essera [3 7]. W e also show that if α ∗ ( G ) ≥ 1 2 then α ∗ ( G ≀ Z ) ≥ 2 α ∗ ( G ) 2 α ∗ ( G ) + 1 . This improves the previous bound du e to Stalder an d V a lette [3 6]. W e d educe that if we write Z (1) ≔ Z and Z ( k + 1) ≔ Z ( k ) ≀ Z then α ∗ ( Z ( k ) ) = 1 2 − 2 1 − k , an d use this result to answer a qu estion po sed by T essera in [37] on the relation between the Hilbert co mpression exponent and the isoperimetric p rofile of the b alls in G . W e also sho w t hat the c yclic lamplighter groups C 2 ≀ C n embed into L 1 with uniform ly bou nded distortion, answering a question posed by Lee, Naor and Peres in [26]. Finally , we use these results to show that edge Markov typ e need not imply Enflo type. 1 Introd uction Let G be a finitely generat ed group 1 . Fix a finite set of gener ators S ⊆ G , which we will always assume to be symmetric (i.e. s ∈ S ⇐ ⇒ s − 1 ∈ S ). Let d G be the left-in v arian t word metric induced by S on G . Giv en a Banach space X let α ∗ X ( G ) denote the supremum ov er all α ≥ 0 such that there exists a Lipschitz mapping f : G → X and c > 0 such that for all x , y ∈ G we hav e k f ( x ) − f ( y ) k ≥ c · d G ( x , y ) α . For p ≥ 1 w e write α ∗ L p ( G ) = α ∗ p ( G ) and when p = 2 we write α ∗ 2 ( G ) = α ∗ ( G ). T he paramete r α ∗ ( G ) is called the Hilbert compr essi on e xpone nt of G . This quasi-iso metric group in var iant was introd uced by G uentne r and Kaminker in [20]. W e refer to the papers [20, 11, 3, 14, 37, 2 , 36, 13] and the references therei n for backg round on this topic and se v eral interest ing applications . Analogo usly to the abov e definitio n, one can consider the equivariant compr ession e xpon ent α # X ( G ), which is defined exactly as α ∗ X ( G ) with the additio nal requirement that the embedding f : G → X is equi v aria nt ∗ Research supported in part by NSF grants CCF-0635078 and DMS-0528387 . † Research supported in part by NSF grant DMS-0605166 . 1 In this paper all groups are assumed to be infinite unless stated otherwise. 1 (see Section 2 for the definition ). As abov e, we introdu ce the notation α # p ( G ) = α # L p ( G ) and α # ( G ) = α # 2 ( G ). Clearly α # X ( G ) ≤ α ∗ X ( G ). In the Hilbertia n case, w hen G is amenable we hav e α ∗ ( G ) = α # ( G ). T his was pro ved by by Aharoni, Maurey and Mityagi n [1] (see also Chapter 8 in [9]) when G is Abelian, and by Gromov fo r general amenable groups (see [14]). The modulus of unifo rm smoothness of a Banach space X is defined for τ > 0 as ρ X ( τ ) = sup ( k x + τ y k + k x − τ y k 2 − 1 : x , y ∈ X , k x k = k y k = 1 ) . (1) X is said to b e un iformly smooth if lim τ → 0 ρ X ( τ ) τ = 0. Furthermore , X is said to ha ve modulus of smoo thnes s of power type p if there exis ts a constant K such that ρ X ( τ ) ≤ K τ p for all τ > 0. It is straightf orward to check that in this case necessarily p ≤ 2. A deep theorem of Pisier [31] states that if X is unifo rmly smooth then there ex ists some 1 < p ≤ 2 such that X admits an equi v alent norm which has modulu s of smoothness of po wer type p . For concreten ess we note that L p has modul us of smoo thness of po wer type min { p , 2 } . See Section 2 for more informati on on this topic. Define β ∗ ( G ) to be the sup remum ov er all β ≥ 0 for which there exis ts a symmetric set of gene rators S of G and c > 0 such that for all t ∈ N , E  d G ( W t , e )  ≥ ct β , (2) where here, and in what follo ws, { W t } ∞ t = 0 is the canoni cal simple random walk on the Cayley graph of G determin ed by S , starting at the identity elemen t e . In [4] Austin, Naor and Peres used the method of Marko v type to sho w that if G is amenable and X has modulus of smoothness of power type p then α ∗ X ( G ) ≤ 1 p β ∗ ( G ) . (3) Our first result, which is prov ed in Sect ion 2, e stabli shes the same b ound as ( 3) for the equi v ariant compres - sion ex ponen t α # X ( G ), e ven whe n G is not necessari ly amenabl e. Theor em 1.1. Let X be a Banach space whic h has modulus of smoothn ess of power type p. Then α # X ( G ) ≤ 1 p β ∗ ( G ) . (4) Since when G is amenable α ∗ ( G ) = α # ( G ), Theorem 1.1 is a generali zation of (3) w hen X = L 2 . A theorem of Guentner and Kaminker [20 ] states that if α # ( G ) > 1 2 then G is amenable . Since for a non- amenable gro up G we h a ve β ∗ ( G ) = 1 (see [ 25, 43]), Theorem 1.1 impli es the Guen tner -K amink er theore m, while generalizin g it to non-Hilbert ian tar gets (whe n the tar get space X is a Hilbert space our met hod yiel ds a ver y simple ne w proof of the Guentner -K amink er theorem—see Remark 2.6). Note that both known proo fs of the Guentner -Kaminker theor em, namely the original proof in [20] and the ne w proof disco ve red by de Cornulie r , T essera and V alette in [14], rely crucially on the f act that X is a Hilbert space. It follo w s in particu lar from Theorem 1.1 that for 2 ≤ p < ∞ , if α # p ( G ) > 1 2 then G is amenable. This is sharp, since in Section 2 we sho w that for the free group on two generators F 2 , for ev ery 2 ≤ p < ∞ w e hav e α # p ( F 2 ) = 1 2 . This answers a questi on posed by T essera (see Question 1.6 in [37]). 2 Theorem 1.1 isolates a geometric property ( unifor m smooth ness) o f the targ et sp ace X which lies at t he heart of the phenomeno n discov ered by Guentner and Kaminker . Our proof is a modification of the martingale method de veloped by Naor , Peres, Schramm and She ffi eld in [28] for esti mating the speed of stationary re ve rsible M arko v chains in unifo rmly smooth Banach spaces . This method requi res se ver al adaptatio ns in the presen t setting since the rando m walk { W t } ∞ t = 0 is not statio nary—we refer to Section 2 for the deta ils. Giv en two grou ps G and H , the wreath product G ≀ H is the group of all pairs ( f , x ) w here f : H → G has finite suppor t (i.e. f ( z ) = e G for all bu t finitely many z ∈ H ) and x ∈ H , equipped with the product ( f , x )( g , y ) ≔  z 7→ f ( z ) g ( x − 1 z ) , xy  . If G is generate d by the set S ⊆ G and H is generated by the set T ⊆ H then G ≀ H is generated by the set { ( e G H , t ) : t ∈ T } ∪ { ( δ s , e H ) : s ∈ S } . Unless othe rwise stated we will always assume that G ≀ H is equipped with the word metric associated w ith this canoni cal set of generators (although in most cases our assertion s will be indepe ndent of the choi ce of gene rators) . The beha vior of the Hilbert compression expon ent unde r w reath product s was in vestiga ted in [3, 37, 36, 4]. In parti cular , S talder and V alette pro ved in [36] that α ∗ ( G ≀ Z ) ≥ α ∗ ( G ) α ∗ ( G ) + 1 . (5) Here we obtain the follo w ing improv ement of this bound: Theor em 1.2. F or ever y finitely gener ate d gr oup w e have , α ∗ ( G ) ≥ 1 2 = ⇒ α ∗ ( G ≀ Z ) ≥ 2 α ∗ ( G ) 2 α ∗ ( G ) + 1 , (6) and α ∗ ( G ) ≤ 1 2 = ⇒ α ∗ ( G ≀ Z ) = α ∗ ( G ) . (7) W e refer to Theorem 3.3 for an analogous bound for α p ( G ≀ Z ), as well as a more general estimate for α p ( G ≀ H ). In addit ion to impro ving (5), we will see belo w instances in which (6) is actual ly an equalit y . In fact , we conjecture that (6) holds as an equality for ev ery amenabl e group G . ` Ershler [17] (see also [34]) pro ved that β ∗ ( G ≀ Z ) ≥ 1 + β ∗ ( G ) 2 . More generally , in Section 6 we sho w that β ∗ ( G ≀ H ) ≥ ( 1 + β ∗ ( G ) 2 if H has linear growth, 1 otherwis e. (8) Since if G is amenable then G ≀ Z is also amenable (see e.g. [30, 24]) it follows that for an amenab le group G , α ∗ ( G ≀ Z ) ≤ 1 1 + β ∗ ( G ) . (9) Cor ollar y 1.3. If G is amenable and α ∗ ( G ) = 1 2 β ∗ ( G ) then α ∗ ( G ≀ Z ) = 1 2 β ∗ ( G ≀ Z ) = 2 α ∗ ( G ) 2 α ∗ ( G ) + 1 . 3 In parti cular , if we define iter ativel y G (1) ≔ G and G ( k + 1) ≔ G ( k ) ≀ Z , then for all k ≥ 1 , α ∗ ( G ( k ) ) = 2 k − 1 α ∗ ( G ) (2 k − 2) α ∗ ( G ) + 1 . Corollary 1.3 follo ws immedia tely fro m Theorem 1.2 and the bound (9). Additional re sults along t hese lines are obtaine d in Section 4; for exa mple (see Remark 3.4) we deduce that α ∗  Z ≀ Z 2  = 1 2 . For r ∈ N let J ( r ) be the smallest constant J > 0 such that for ev ery f : G → R which v anish es outside the ball B ( e , r ) ≔ { x ∈ G : d G ( x , e ) ≤ r } , we hav e        X x ∈ G f ( x ) 2        1 / 2 ≤ J ·        X x ∈ G X s ∈ S | f ( s x ) − f ( x ) | 2        1 / 2 . Let a ∗ ( G ) be the supremum over all a ≥ 0 for which there exists c > 0 such that for all r ∈ N we hav e J ( r ) ≥ cr a . T essera prov ed in [37] that α ∗ ( G ) ≥ a ∗ ( G ) and asked if it is true that α ∗ ( G ) = a ∗ ( G ) for ev ery amenable group G (s ee Question 1.4 in [37]). C orollar y 1.3 implies that the answer to this questi on is neg ati v e. Indeed, Corollary 1.3 implies that the amenable group ( Z ≀ Z ) ≀ Z satisfies α ∗  ( Z ≀ Z ) ≀ Z  = 4 7 yet a ∗  ( Z ≀ Z ) ≀ Z  ≤ 1 2 . (10) In f act, the r atio a ∗ ( G ) /α ∗ ( G ) can b e arbitr arily small, since if we denot e Z (1) ≔ Z and Z ( k + 1) ≔ Z ( k ) ≀ Z the n for k ≥ 2, α ∗ ( Z ( k ) ) = 1 2 − 2 1 − k yet a ∗ ( Z ( k ) ) ≤ 1 k − 1 . (11) T o pro ve (11), and hence also its special case (10), note that the assertion in (11 ) about α ∗ ( Z ( k ) ) is a conse- quenc e of Corollar y 1.3. T o pro ve the upper bound on a ∗ ( Z ( k ) ) in ( 11) we note that i f G is a finitely gen erated group such that the proba bility of retu rn of the sta ndard random walk { W t } ∞ t = 0 satisfies P [ W t = e ] ≤ exp  − C t γ  (12) for some C , γ ∈ (0 , 1 ) and all t ∈ N , then a ∗ ( G ) ≤ 1 − γ 2 γ . (13) This implies (11) since Pittet and Salo ff -Coste [32] prove d that for all k ≥ 2 there ex ists c , C > 0 such that for G = Z ( k ) we ha ve for all t ≥ 1 exp  − C t k − 1 k + 1  log t  2 k + 1  ≤ P [ W t = e ] ≤ exp  − ct k − 1 k + 1  log t  2 k + 1  . (14) The bound (13) is esse ntiall y kno wn. Indeed, assu me that J ( r ) ≥ cr a for e very r ≥ 1. Followin g th e no tation of Coulhon [12], for v ≥ 1 let Λ ( v ) denote the large st constant Λ ≥ 0 such that for all Ω ⊆ G with | Ω | ≤ v , e ver y f : G → R w hich v anish es outsi de Ω satisfies Λ · X x ∈ G f ( x ) 2 ≤ X x ∈ G X s ∈ S | f ( s x ) − f ( x ) | 2 . 4 Since for r ≥ 2 w e ha ve | B ( e , r ) | ≤ | S | r , it follo ws immediately from the definitions that J ( r ) 2 ≤ 1 Λ ( | S | r ) . Theorem 7.1 in [12] implies that there exis ts a constan t K > 0 such that if e K t γ ≥ | S | then , t ≥ Z e K t γ | S | dv v Λ ( v ) = Z K t γ log | S | 1 log | S | Λ ( | S | r ) dr ≥ log | S | Z K t γ log | S | 1 J ( r ) 2 dr ≥ c 2 log | S | Z K t γ log | S | 1 r 2 a dr = c 2 log | S | (2 a + 1)        K t γ log | S | ! 2 a + 1 − 1        . Letting t → ∞ it follows that (2 a + 1) γ ≤ 1, implying (13). Remark 1.4. In [37] T essera asserted that if the oppo site inequ ality to (12) holds true, i.e. if we hav e P [ W t = e ] ≥ exp ( − K t γ ) for some γ ∈ (0 , 1), K > 0, and e ver y t ≥ 1, then a ∗ ( G ) ≥ 1 − γ . U nfortu nately , this claim is fals e in general. 2 Indeed , if it were true, the n using (14) w e wo uld deduce that a ∗   Z ≀ Z  ≀ Z  ≀ Z  = a ∗ ( Z (4) ) ≥ 2 5 , b ut from (11) we know that a ∗ ( Z (4) ) ≤ 1 3 . O n inspection of the proof of Proposition 7.2 in [3 7] w e see that the ar gument gi ven there actually yield s the lower bound a ∗ ( G ) ≥ 1 − γ 2 (note the squares in the first equat ion of the proof of Proposition 7.2 in [ http:/ /arxiv.o rg/abs/ma th/0603138v3 ]). Thus, the original ar gument presen ted in [37] to estab lish the lo wer bound a ∗ ( Z ≀ Z ) ≥ 2 3 only pr ov es tha t a ∗ ( Z ≀ Z ) ≥ 1 3 . Nev erthe less, the lo wer bound of 2 3 , w hich was used crucial ly in [4], is correct, as follo ws from our T heorem 1.2. After the presen t paper was poste d and sent to T essera, he replace d the origina l ar gument in [37] for the lo wer bound α ∗ ( Z ≀ Z ) ≥ 2 3 by a correct ar gument, along the same lines as our proof of Theorem 1.2. ⊳ In Section 4 we sho w that the cycli c lamplighter gro up C 2 ≀ C n admits a bi-Lipsch itz embedd ing into L 1 with distortio n indepe ndent of n (here, and in what follo w s C n denote s the cycli c group of order n ). This answers a question posed in [26] and in [5 ]. In S ection 5 we use the notion of Hilbert space compression to sho w that Z ≀ Z has edge Marko v type p for any p < 4 3 , but it does not ha ve Enflo type p for an y p > 1. W e refer to Section 5 for the relev ant definitions. This result sho ws that there is no metric analogue of the well kno wn Banach space phenomen on “equal norm Rademacher type p implies Rademacher p ′ for ev ery p ′ < p ” ( see [38]). Finally , in Section 7 we present se ve ral open problems that arise from our work. 2 Equiv ariant compression an d random walks In what follo ws we will use ≍ and . , & to denote, respecti vely , equality or the correspond ing inequa lity up to some positi ve m ultipli cati v e const ant. Let X be a Banach space. W e denote the group of linear isometric automorphis ms of X by Isom ( X ). Fix a homomorph ism π : G → Isom ( X ), i.e. an action of G on X by linear isometri es. A function f : G → X is called a 1-coc ycle with respect to π if for e ve ry x , y ∈ G we hav e f ( xy ) = π ( x ) f ( y ) + f ( x ). The space of all 1-coc ycle s with respect to π is denoted Z 1 ( G , π ). Equiv alently , f ∈ Z 1 ( G , π ) if and only if v 7→ π ( x ) v + f ( x ) is an action of G on X by a ffi ne isometrie s. A function f : G → X is called a 1-coc ycle if there exists a 2 This remark concerns the version http://arxiv.org/ abs/math/ 0603138v3 of [37]; after we informed the author of t his mistake, it wa s corrected in later versions of [37] . 5 homomorph ism π : G → Isom( X ) such that f ∈ Z 1 ( G , π ). A mapping ψ : G → X is called equi v ari ant if it is giv en by the orbit of a vector v ∈ X under an a ffi ne iso metric actio n of G on X , or equiv alently ψ ( x ) = π ( x ) v + f ( x ) for some homomorphism π : G → Isom ( X ) and f ∈ Z 1 ( G , π ). N ote that since the functi on x 7→ π ( x ) v is bounded, the compre ssion expon ents of ψ and f coincide. Therefore in order to bound the equi v ariant compressio n exp onent of G in X it su ffi ces to stud y the growth rate of 1-co cyc les. Recall the definit ion (1) of the modul us of uniform smoothne ss ρ X ( τ ), and tha t X is said to ha v e modulus of smoothn ess of po w er type p if there ex ists a co nstant K such that ρ X ( τ ) ≤ K τ p for all τ > 0. By Proposit ion 7 in [8], X has modulus of smoothnes s of po wer type p if and only if there exists a con stant S > 0 such that for e very x , y ∈ X k x + y k p + k x − y k p ≤ 2 k x k p + 2 S p k y k p . (15) The infimum ov er all S for which (15) holds i s called th e p -smoot hness c onstan t of X , and is denoted S p ( X ). It was sho wn in [8] (see also [18]) that S 2 ( L p ) ≤ p p − 1 for 2 ≤ p < ∞ and S p ( L p ) ≤ 1 for 1 ≤ p ≤ 2 (the order of magnitud e of these consta nts was first calculate d in [21]). Our pro of of Theorem 1.1 is based on the fo llo wing inequ ality , which is of i ndepe ndent interest. Its proo f is a modification of the method that was used in [28] to study the Marko v type of uniformly smooth Banach spaces . Theor em 2.1. Let X be a Banach space with modul us of smoothness of power type p, and assume that f : G → X is a 1 -cocycle . Then for eve ry time t ∈ N , E  k f ( W t ) k p  ≤ C p ( X ) t · E  k f ( W 1 ) k p  , wher e C p ( X ) = 2 2 p S p ( X ) p 2 p − 1 − 1 . Theorem 2.1 sho ws that images of { W t } ∞ t = 0 under 1-co cyc les satisfy an in equali ty similar to the Mark ov t ype inequa lity (note that f ( W 0 ) = f ( e ) = f ( e · e ) = π ( e ) f ( e ) + f ( e ) = 2 f ( e ), whence f ( e ) = 0). W e stress that one cannot apply Markov type directly in this case because of the lack of stationar ity of the Marko v chain { f ( W t ) } ∞ t = 0 . W e overc ome this problem by crucially using the fact that f is a 1-coc ycle. Before pro ving Theorem 2.1 we sho w how it impl ies Theorem 1.1. Pr oof of Theor em 1.1. Observ e that (4) is trivial if α # X ( G ) ≤ 1 p (since β ∗ ( G ) ≤ 1). So, w e may assume that α # ( G ) > 1 p . Fix 1 p ≤ α < α # X ( G ) and 0 < β < β ∗ ( G ). Then there exists a 1-co cyc le f : G → X satisfy ing x , y ∈ G = ⇒ d G ( x , y ) α . k f ( x ) − f ( y ) k . d G ( x , y ) . In addit ion w e kno w that E [ d G ( W t , e )] & t β . An applic ation of Theorem 2.1 yields E  k f ( W t ) k p  . t E  k f ( W 1 ) k p  = t E  k f ( W 1 ) − f ( e ) k p  . t E  d G ( W 1 , e ) p  = t . (16) On the other hand, since p α ≥ 1 we may use Jensen ’ s inequ ality to dedu ce that E  k f ( W t ) k p  = E  k f ( W t ) − f ( e ) k p  & E  d G ( W t , e ) p α  ≥  E [ d G ( W t , e )]  p α & t p αβ . (17) Combining (16) and (17), and letting t → ∞ , implies that p αβ ≤ 1, as required.  6 Remark 2.2. T heorem 1.1 is optimal for the class of L p spaces . Indeed let F 2 denote the free group on two genera tors. W e claim that for ev ery p ≥ 1, α # p ( F 2 ) = max ( 1 2 , 1 p ) . (18) Observ e tha t sinc e (tri via lly) β ∗ ( F 2 ) = 1, T heorem 1.1 implies that α # p ( F 2 ) ≤ max n 1 2 , 1 p o . In the rev erse directi on Guentner and Kaminker [20] gav e a simple constr uction of an equi v aria nt mapping f : F 2 → L p satisfy ing k f ( x ) − f ( y ) k p ≥ d F 2 ( x , y ) 1 / p for all x , y ∈ F 2 . This implies (18 ) for 1 ≤ p ≤ 2. T he case p ≥ 2 follo w s from Lemm a 2.3 be lo w . ⊳ Lemma 2.3. F or eve ry finitely gen era ted gr oup G and every p ≥ 1 we hav e α # p ( G ) ≥ α # 2 ( G ) . Pr oof. In what follo w s we denote the standard orthon ormal basis of ℓ 2 ( C ) by ( e j ) ∞ j = 1 . Let γ denote the standa rd Gaussian measu re on C . Consi der the countable prod uct Ω ≔ C ℵ 0 , equipped with the pro duct measure µ ≔ γ ℵ 0 . L et H deno te the subspace of L 2 ( Ω , µ ) consisti ng of all linear functi ons. Thus, if w e consid er the coo rdinat e functions g j : Ω → C gi ven by g ( z 1 , z 2 , . . . ) = z j then H is the space of all functions h : Ω → C of the fo rm h = P ∞ j = 1 a j g j , wher e the s equen ce ( a j ) ∞ j = 1 ⊆ C satisfies P ∞ j = 1 | a j | 2 < ∞ , i.e. ( a j ) ∞ j = 1 ∈ ℓ 2 ( C ). Note that we are using here the stand ard prob abilist ic fact (see [15]) that P ∞ j = 0 a j g j con ver ges almost e ver ywhere, and has the same distrib ution as  P ∞ i = 1 | a i | 2  1 / 2 g 1 (since { g j } ∞ j = 1 are i.i.d. standard comple x Gaussian random v ariab les). This fact also implie s that for ev ery unit ary opera tor U : ℓ 2 ( C ) → ℓ 2 ( C ), U z ≔         ∞ X k = 1 h U e k , e j i z j         ∞ k = 1 ∈ Ω , is well defined for almost z ∈ Ω , and therefore U can be though t of as a measure preser ving automorphi sm U : Ω → Ω (we are slightly abusi ng notat ion here, but this will not crea te any confusi on). Fix a unitar y representati on π : G → Isom  ℓ 2 ( C )  and a coc ycle f ∈ Z 1 ( G , π ) whic h satisfies x , y ∈ G = ⇒ d G ( x , y ) α . k f ( x ) − f ( y ) k ℓ 2 ( C ) . d G ( x , y ) . (19) For x ∈ G and h ∈ L p ( Ω , µ ) define e π ( x ) h ∈ L p ( Ω , µ ) by e π ( x ) h ( z ) = h ( π ( x ) z ). By the abo ve reasoning, since π ( x ) is a measure prese rving automorph ism of ( Ω , µ ), e π ( x ) is a linea r isometry of L p ( Ω , µ ), and hence e π : G → Isom  L p ( Ω , µ )  is a homomorphism. Note that since all the elements of H ha ve a Gaussia n distrib ution, all of their moments a re finite. Hence H ⊆ L p ( Ω , π ). W e can therefor e define e f : G → L p ( Ω , µ ) by e f ( x ) ≔ P ∞ j = 1 h f ( x ) , e j i g j ∈ H ⊆ L p ( Ω , µ ). It is immediate to check that e f ∈ Z 1  G , e π  and that for e ver y x , y ∈ G we hav e     e f ( x ) − e f ( y )     L p ( Ω ,µ ) = k g 1 k L p ( Ω ,µ ) · k f ( x ) − f ( y ) k ℓ 2 ( C ) . Hence e f satisfies (19) as well.  Remark 2.4. Lemma 2.3 actuall y establishe s the follo wing fact: there exists a measure space ( Ω , µ ) and a subspace H ⊆ T p ≥ 1 L p ( Ω , µ ) w hich is closed in L p ( Ω , µ ) for all 1 ≤ p < ∞ and such that the L p ( Ω , µ ) norm restri cted to H is proportio nal to the L 2 ( Ω , µ ) norm. For an y group G , any unitary representat ion π : G → Isom( H ) can be extende d to a homomorphi sm e π : G → Isom  L p ( Ω , µ )  . The space H is widely used in Banach space theory , and is known as the Gaussian Hilbert space . The abov e corollary about the ext ension of group action s was previo usly noted in [6] under the additi onal restriction that 1 < p < 2 Z , as a simple cor ollary of an a bstrac t extensio n theorem due to Hardi n [22] (a lternat i ve ly this is al so a co rollar y of 7 the clas sical Plotkin-Rudin theorem [33, 35]). L emma 2.3 sho ws that no restrictio n on p is necessar y , while the theorem of Hardin used in [6] does require the abov e restric tion on p . The ke y point here is the use of the p articu lar subspa ce H ⊆ L p ( Ω , µ ) for whi ch un itary o perato rs ha ve a simple explic it ex tensio n to a linea r isometric automor phism of L p ( Ω , µ ) fo r any 1 ≤ p < ∞ . ⊳ W e shall now pass to the proof of Theorem 2.1 . W e will use unifor m smoothness via the follo w ing famo us inequa lity due to Pisier [31] (for the exp licit constan t below see Theorem 4.2 in [28]). Theor em 2.5 (Pisier) . F ix 1 < p ≤ 2 and let { M k } n k = 0 ⊆ X be a martingale in X . Then E  k M n − M 0 k p  ≤ S p ( X ) p 2 p − 1 − 1 · n − 1 X k = 0 E  k M k + 1 − M k k p  . Pr oof of Theor em 2.1. By assumpt ion f ( x ) ∈ Z 1 ( G , π ) for some homomorphism π : G → Isom( X ). Let { σ k } ∞ k = 1 be i.i.d. random vari ables uniformly distrib uted ov er S . Then for t ≥ 1 W t has the same distrib ution as the rando m produ ct σ 1 · · · σ t . For e v ery t ≥ 1 the follo wing identity holds true: 2 f ( W t ) = t X j = 1 π  W j − 1  f  σ j  − t X j = 1 π  W j  f  σ − 1 j  . (20) W e shall prov e (20) by induction on t . Note that ev ery x ∈ G satisfies 0 = f ( e ) = f  x − 1 · x  = π ( x ) − 1 f ( x ) + f  x − 1  , i.e. f ( x ) = − π ( x ) f  x − 1  . This implies (20) when t = 1. Hence, assuming the v alidity of (20) for t we can use the ident ity 2 f ( xy ) = 2 f ( x ) + π ( x ) f ( y ) − π ( xy ) f  y − 1  to dedu ce that 2 f ( W t + 1 ) = 2 f ( W t σ t + 1 ) = 2 f ( W t ) + π ( W t ) f ( σ t + 1 ) − π ( W t + 1 ) f  σ − 1 t + 1  = t X j = 1 π  W j − 1  f  σ j  − t X j = 1 π  W j  f  σ − 1 j  + π ( W t ) f ( σ t + 1 ) − π ( W t + 1 ) f  σ − 1 t + 1  = t + 1 X j = 1 π  W j − 1  f  σ j  − t + 1 X j = 1 π  W j  f  σ − 1 j  , pro ving (20). Define M t ≔ t X j = 1 π  W j − 1   f  σ j  − v  = t X j = 1 π  σ 1 · · · σ j − 1   f  σ j  − v  , and N t ≔ t X j = 1 π  W − 1 t W j   f  σ − 1 j  − v  = t X j = 1 π  σ − 1 t · · · σ − 1 j + 1   f  σ − 1 j  − v  , 8 where v ≔ E  f ( W 1 )  ∈ X . Note that since S is symmetric, σ − 1 j has the same distrib ution a s σ j , and therefo re N t has the s ame dis trib uti on as M t . Moreov er , (20) implies that 2 f ( W t ) = M t − π ( W t ) N t − v + π ( W t ) v . Since π ( W t ) is an isometry , we deduc e that 2 p E  k f ( W t ) k p  ≤ 4 p − 1 E  k M t k p  + 4 p − 1 E  k N t k p  + 2 · 4 p − 1 k v k p = 2 · 4 p − 1 E  k M t k p  + 2 · 4 p − 1    E  f ( W 1 )     p ≤ 2 · 4 p − 1 E  k M t k p  + 2 · 4 p − 1 E  k f ( W 1 ) k p  . (21) Note that for e ve ry t ≥ 1, E h M t    σ 0 , . . . , σ t − 1 i = E          t X j = 1 π  σ 0 · · · σ j − 1   f  σ j  − v      σ 0 , . . . , σ t − 1          = M t − 1 + π ( σ 0 · · · σ t − 1 )  E h f  σ j i − v  = M t − 1 , Hence { M k } ∞ k = 0 is a marting ale with respect to the filtration induced by { σ k } ∞ k = 0 . By theorem 2.5, E  k M t k p  ≤ S p ( X ) p 2 p − 1 − 1 · t − 1 X k = 0 E  k M k + 1 − M k k p  = t − 1 X k = 0 E  k f ( σ k ) − v k p  ≤ S p ( X ) p 2 p − 1 − 1 · t 2 p − 1  E  k f ( W 1 ) k p  + k v k p  ≤ 2 p S p ( X ) p 2 p − 1 − 1 · t E  k f ( W 1 ) k p  . (22) Combining (21) and (22) complet es the pro of of Theorem 2.1.  Remark 2.6. W hen the targ et spac e X is Hilbert space on e can p rov e Theorem 1 .1 via th e follo wing s impler ar gument . Using the notation in the proof of Theorem 2.1 we see that for each t ∈ N the random v ari- ables W − 1 t = σ − 1 t · · · σ − 1 1 and W − 1 t W 2 t = σ t + 1 · · · σ 2 t are indep enden t and hav e the same distrib ution as W t . Therefore Y 1 ≔ f  W − 1 t  and Y 2 ≔ f  W − 1 t W 2 t  = π  W − 1 t  f ( W 2 t ) + f  W − 1 t  are i.i.d., and hence satisfy E h k f ( W 2 t ) k 2 i = E      π  W − 1 t  f ( W 2 t )     2  = E h k Y 1 − Y 2 k 2 i = E h k Y 1 k 2 − 2 h Y 1 , Y 2 i + k Y 2 k 2 i = 2 E h k f ( W t ) k 2 i − 2    E  f ( W t )     2 ≤ 2 E h k f ( W t ) k 2 i . By induct ion it follo ws that for e ver y k ∈ N , E     f  W 2 k     2  ≤ 2 k E h k f ( W 1 ) k 2 i . This implies Theorem 1.1, and hence also the Guentner -Kamink er theore m [20], by argu ing exactl y as in the conclu sion of the proof of Theorem 1.1. ⊳ 3 The beha vior of L p compression under wre ath pr oducts Giv en two gr oups G , H let L G ( H ) denote the wreath product G ≀ H where the set of ge nerato rs of G is taken to be G \ { e } (i.e. any two distinc t elements of G are at distan ce 1 from each oth er). Wi th this definition it is immediate to check (see for exampl e the proof of Lemma 2.1 in [5]) that ( f , i ) , ( g , j ) ∈ L G ( Z ) = ⇒ d L G ( Z )  ( f , i ) , ( g , j )  ≍ | i − j | + max  | k | + 1 : f ( k ) , g ( k )  . (23) The case G = C 2 corres ponds to the clas sical lamplighter group on H . 9 Lemma 3.1. F or eve ry gr oup G we have α ∗  L G ( Z )  = 1 . Pr oof. As shown by T essera in [37], α ∗ ( C 2 ≀ Z ) = 1 (we prov ide an alternati v e ex plicit emb edding exhibi ting this fac t in Section 4 belo w). Therefore for ev ery α ∈ (0 , 1) there is a mapping θ : C 2 ≀ Z → L 2 satisfy ing ( x , i ) , ( y , j ) ∈ C 2 ≀ Z = ⇒ d C 2 ≀ Z  ( x , i ) , ( y , j )  α . k θ ( x , i ) − θ ( y , j ) k 2 . d C 2 ≀ Z  ( x , i ) , ( y , j )  . (24) Let { ε z } z ∈ G be i.i.d. { 0 , 1 } v alued Bernoull i random var iables , defined on some probability space ( Ω , P ). For e ver y f : Z → G define a random mapping ε f : Z → C 2 by ε f ( k ) = ε f ( k ) . W e now define an embedding F : L G ( Z ) → L 2 ( Ω , L 2 ) by F ( f , i ) ≔ θ ( ε f , i ) . Fix ( f , i ) , ( g , j ) ∈ L G ( Z ) and let k max ∈ Z satisfy f ( k max ) , g ( k max ) and | k max | = max  | k | : f ( k ) , g ( k )  . Then k F ( f , i ) − F ( g , j ) k 2 L 2 ( Ω , L 2 ) = E h k θ ( ε f , i ) − θ ( ε g , j ) k 2 2 i (24) . E h d C 2 ≀ Z  ( ε f , i ) , ( ε g , j )  2 i (23) ≍ E   | i − j | + max  | k | + 1 : ε f ( k ) , ε g ( k )   2  ≤ h ( | i − j | + | k max | + 1 ) 2 i (23) ≍ d L G ( Z )  ( f , i ) , ( g , j )  2 . In the rev erse dir ection note that since f ( k max ) , g ( k max ) with probabi lity 1 2 we ha v e ε f ( k max ) , ε g ( k max ) . Therefore k F ( f , i ) − F ( g , j ) k 2 L 2 ( Ω , L 2 ) = E h k θ ( ε f , i ) − θ ( ε g , j ) k 2 2 i (24) & E h d C 2 ≀ Z  ( ε f , i ) , ( ε g , j )  2 α i (23) ≍ E   | i − j | + max  | k | + 1 : ε f ( k ) , ε g ( k )   2 α  & h ( | i − j | + | k max | + 1 ) 2 α i (23) ≍ d L G ( Z )  ( f , i ) , ( g , j )  2 α . This complet es the proof of Lemma 3.1.  Remark 3.2. In [37] T essera sho ws that if H has volume gro wth of order d then α ∗  L G ( H )  ≥ 1 d . ( 25) Note th at T essera mak es this assertio n for L F ( H ), where F is finite (see Section 5.1 in [3 7], an d specifica lly Remark 5.2 there ). But, it is immediate from the proof in [37] that the constan t factors in T essera’ s embed- ding do not depend on the cardinality of F , and therefore (25) holds in full generality . Observe that (25) is a generaliz ation of Lemma 3.1, but w e belie v e that the ar gumen t in Lemma 3.1 which reduces the probl em to the case G = C 2 is of indepe ndent intere st. The case H = Z 2 in (25) can be prov ed via the follo wing explicit embedd ing. For simplicity w e describe it when G = C 2 . F ix 0 < α < 1 2 and let n v y , r , g : y ∈ Z 2 , r ∈ N ∪ { 0 } , g : y + [ − r , r ] 2 → { 0 , 1 } , g . 0 o be an orthonormal system of v ectors in L 2 . For simplicity we also write v y , r , 0 = 0. define ψ : C 2 ≀ Z 2 → R 2 ⊕ L 2 by ψ ( f , x ) = x ⊕          X y ∈ Z 2 \{ x } ∞ X r = 0 max { 1 − 2 r / k x − y k ∞ , 0 } k x − y k 3 2 − 2 α ∞ v y , r , f ↾ y + [ − r , r ] 2          . An elementar y (though a little tediou s) case analysis shows that ψ is Lipschit z and has compression α . ⊳ 10 The follo wing theor em, in combination with Lemma 3.1, contains Theorem 1.2 as a special case (note that (7) follo ws from (26) since clearl y α ∗ ( G ≀ H ) ≤ α ∗ ( G )). Theor em 3.3. Let G , H be gr ou ps and p ≥ 1 . Then min n α ∗ p ( G ) , α ∗ p  L G ( H )  o ≥ 1 p = ⇒ α ∗ p ( G ≀ H ) ≥ p α ∗ p ( G ) α ∗ p ( L G ( H )) p α ∗ p ( G ) + p α ∗ p  L G ( H )  − 1 , and min n α ∗ p ( G ) , α ∗ p  L G ( H )  o ≤ 1 p = ⇒ α ∗ p ( G ≀ H ) ≥ min n α ∗ p ( G ) , α ∗ p  L G ( H )  o . (26) Pr oof. W e shall start with some useful prelimin ary observ atio ns. Let ( X , d X ) be a metric space, p ≥ 1, and let Ω be a set. W e denote by ℓ p ( Ω , X ) the metric space of all finitely suppo rted functions f : Ω → X , equipp ed with the metric d ℓ p ( Ω , X ) ( f , g ) ≔         X ω ∈ Ω d X  f ( ω ) , g ( ω )  p         1 / p . It is immediate to verif y that for ev ery ( f , x ) , ( g , y ) ∈ G ≀ H we hav e d G ≀ H  ( f , x ) , ( g , y )  ≍ d L G ( H )  ( f , x ) , ( g , y )  + d ℓ 1 ( H , G ) ( f , g ) . (27) Indeed , it su ffi ces to verify the equiv alence (27) when ( g , y ) is the identity element ( e , e ) of G ≀ H . In this case (27) simply says that in order to m ov e from ( e , e ) to ( f , x ) one needs to visit the locatio ns z ∈ H where f ( z ) , e , and in each of these locatio ns one must mov e within G from e to the appropriate group element f ( z ) ∈ G . Another basic fac t that we will use is that for ev ery ( f , x ) , ( g , y ) ∈ G ≀ H ,    { z ∈ H : f ( z ) , g ( z ) }    ≤ d L G ( H )  ( f , x ) , ( g , y )  . (28) Once more, this fac t is entirely obvio us: in order to mov e in L G ( H ) from ( f , x ) to ( g , y ) once m ust visit all the locatio ns where f and g di ff er . W e shall now proc eed to the proof of Theorem 3.3. F ix a < α ∗ p ( G ) and b < α ∗ p  L G ( H )  . T hen there exist s a functi on ψ : G → L p such that u , v ∈ G = ⇒ d G ( u , v ) a . k ψ ( u ) − ψ ( v ) k p . d G ( u , v ) . (29) W e also kno w that there exist s a function φ : L G ( H ) → L p which satisfies u , v ∈ L G ( H ) = ⇒ d L G ( H ) ( u , v ) b . k φ ( u ) − φ ( v ) k p . d L G ( H ) ( u , v ) . (30) Define a functi on F : G ≀ H → L p ⊕ ℓ p ( H , L p ) by F ( f , x ) ≔ φ ( f , x ) ⊕ ( ψ ◦ f ) . 11 Fix ( f , x ) , ( g , y ) ∈ G ≀ H and denote m ≔ d L G ( H )  ( f , x ) , ( g , y )  and n ≔ d ℓ 1 ( H , G ) ( f , g ). W e kno w from (27) that d G ≀ H  ( f , x ) , ( g , y )  ≍ m + n . N o w , k F ( f , x ) − F ( g , y ) k p =         k φ ( f , x ) − φ ( g , y ) k p p + X z ∈ H k ψ ( f ( z )) − ψ ( g ( z )) k p p         1 / p ≤ k φ ( f , x ) − φ ( g , y ) k p + X z ∈ H k ψ ( f ( z )) − ψ ( g ( z )) k p (29) ∧ (30) . m + n ≍ d G ≀ H  ( f , x ) , ( g , y )  . In the re v erse direction we ha ve the lo w er bound k F ( f , x ) − F ( g , y ) k p (29) ∧ (30) &         m b p + X z ∈ H d G ( f ( z ) , g ( z )) a p         1 / p . (31) If a p ≤ 1 then P z ∈ H d G ( f ( z ) , g ( z )) a p ≥ P z ∈ H d G ( f ( z ) , g ( z ))  a p = n a p and (31) implies that k F ( f , x ) − F ( g , y ) k p &  m b p + n a p  1 / p & ( m + n ) min { a , b } & d G ≀ H  ( f , x ) , ( g , y )  min { a , b } . (32) Assume that a p > 1. It follo ws from (28) that    { z ∈ H : f ( z ) , g ( z ) }    ≤ m . T hus, using H ¨ older’ s inequalit y , we see that X z ∈ H d G ( f ( z ) , g ( z )) a p ≥ 1 m a p − 1         X z ∈ H d G ( f ( z ) , g ( z ))         a p = n a p m a p − 1 . (33) Note that m b p + n a p m a p − 1 ≥ n ab p 2 a p + b p − 1 , which follo ws by considering the cases m ≥ n a p a p + b p − 1 and m ≤ n a p a p + b p − 1 separa tely . Hence, k F ( f , x ) − F ( g , y ) k p (31) ∧ (33) & m b p + n a p m a p − 1 ! 1 / p & max  m b , n ab p a p + b p − 1  & ( m + n ) min n b , ab p a p + b p − 1 o ≍ d G ≀ H  ( f , x ) , ( g , y )  min n b , ab p a p + b p − 1 o . (34) Note that when a p > 1, if b ≤ ab p a p + b p − 1 then b p ≤ 1. Therefore (32) and (34) imply Theorem 3.3.  Remark 3.4. Theorem 3.3, in combination with Remark 3.2 and the results of Section 6 belo w , imply that if G is amenable and H has quadra tic growth then α ∗ ( G ≀ H ) = min ( 1 2 , α ∗ ( G ) ) . (35) Thus, in particu lar , α ∗  C 2 ≀ Z 2  = α ∗  Z ≀ Z 2  = 1 2 . T o see (35) note that by T heorem 6.1 in S ection 6 we hav e β ∗ ( G ≀ H ) = 1. Using (3) w e deduce that α ∗ ( G ≀ H ) ≤ 1 2 , and the inequa lity α ∗ ( G ≀ H ) ≤ α ∗ ( G ) is obviou s. T he re verse ineq uality in (35) i s a corollary of Theorem 3.3 and Remark 3.2. ⊳ 12 4 Embedding the lamplighter grou p into L 1 In this section w e show that the lamplighter group on the n -cy cle, C 2 ≀ C n , embeds into L 1 with distortion indepe ndent of n . This implies via a standard limiting ar gument that also C 2 ≀ Z embeds bi-Lipschi tzly into L 1 . W e presen t two embeddin gs of C 2 ≀ C n into L 1 . O ur fi rst embeddin g is a var iant of the embedding method used in [5]. In [5] there is a detailed expla nation of ho w such embeddings can be disco ve red by looking at the irreducible represe ntatio ns of C 2 ≀ C n . The embedding belo w can be m oti v ate d analo gously , and w e refer the int erested reader to [5] for the d etails. H ere we ju st present the resu lting embedding , which is v ery simple. O ur second embedding is moti v ated by direct geometric reasonin g rather than the “dual” point of vie w in [5]. In what follows we slightl y abu se the notation by consideri ng elements ( x , i ) ∈ C 2 ≀ C n as an index i ∈ C n and a subset x ⊆ C n . For the sake of simplicity we will denote the metric on C 2 ≀ C n by ρ . The metric d C n will denot e the canonical m etric on the n -cy cle C n . It is easy to check (see Lemma 2.1 in [5]) that ( x , j ) , ( y , ℓ ) ∈ C 2 ≀ C n = ⇒ ρ  ( x , j ) , ( y , ℓ )  ≍ d C n ( j , k ) + max k ∈ x △ y ( d C n (0 , k ) + 1) . (36) First embedding of C 2 ≀ C n into L 1 . W e denote by α : C n → C n the shift α ( j ) = j + 1. Let us write I for the family of all arcs (i.e. con necte d subsets) of C n of length ⌊ n / 3 ⌋ (of which there are n ). W e define an embeddi ng f : C 2 ≀ C n → L I ∈I L A ⊆ I ℓ 1 ( C n ) by f ( x , j ) ≔ M I ∈I M A ⊆ I ( − 1) | A ∩ α k ( x ) | · 1 I ( k + j ) + n 1 C n \ I ( k + j ) n 2 2 n / 3 ! k ∈ C n . It is immediat e to check that the metric on C 2 ≀ C n gi ve n by k f ( x , j ) − f ( x ′ , j ′ ) k 1 is C 2 ≀ C n -in va riant. T herefor e it su ffi ces to sho w that k f ( x , j ) − f ( ∅ , 0) k 1 ≍ ρ  ( x , j ) , ( ∅ , 0)  for all ( x , j ) ∈ C 2 ≀ C n . No w , k f ( x , j ) − f ( ∅ , 0) k 1 ≍ X I ∈I X A ⊆ I                    { k ∈ C n : 1 I ( k ) + 1 I ( k + j ) = 1 }    n 2 n / 3 + X k ∈ C n | A ∩ α k ( x ) | odd 1 I ( k ) + n 1 C n \ I ( k ) n 2 2 n / 3                 ≍ d C n (0 , j ) + 1 n 2 2 n / 3 X I ∈I X k ∈ C n    { A ⊆ I : | A ∩ α k ( x ) | odd }    ·  1 I ( k ) + n 1 C n \ I ( k )  ≍ d C n (0 , j ) + 1 n 2 X I ∈I X k ∈ C n I ∩ α k ( x ) , ∅  1 I ( k ) + n 1 C n \ I ( k )  . (37) It su ffi ces to prov e the Lipschitz condi tion k f ( x , j ) − f ( ∅ , 0) k 1 . ρ  ( x , j ) , ( ∅ , 0)  for th e gener ators of C 2 ≀ C n , i.e. when ( x , j ) ∈  ( { 0 } , 0) , ( ∅ , 1)  . This follows immediately from (37) since when ( x , j ) = ( ∅ , 1) then the second summand in (37) is empty , and therefore k f ( ∅ , 1) − f ( ∅ , 0) k 1 ≍ 1 = ρ  ( ∅ , 1 ) , ( ∅ , 0)  , and k f ( { 0 } , 0) − f ( ∅ , 0) k 1 ≍ 1 n 2 X I ∈I X k ∈ I  1 I ( k ) + n 1 C n \ I ( k )  ≍ 1 . ρ  ( { 0 } , 0) , ( ∅ , 0)  . 13 T o p rov e the lo wer b ound k f ( x , j ) − f ( ∅ , 0) k 1 & ρ  ( x , j ) , ( ∅ , 0)  suppo se that ℓ ∈ x is a point of x at a maximal distan ce from 0 in C n . By consid ering only the terms in (37) for which α k ( ℓ ) ∈ I we see that k f ( x , j ) − f ( ∅ , 0) k 1 & d C n (0 , j ) + 1 n 2 X I ∈I X k ∈ α − ℓ ( I )  1 I ( k ) + n 1 C n \ I ( k )  ≍ d C n (0 , j ) + 1 n 2 X I ∈I    I ∩ α − ℓ ( I )    + 1 n X I ∈I    α − ℓ ( I ) \ I    & d C n (0 , j ) +  1 + d C n (0 , ℓ )  & ρ ( ( x , j ) , ( ∅ , 0) ) . This complet es the proof that f is bi-Lipsch itz with O (1) dis tortio n.  Remark 4.1. F ix s ∈ (1 / 2 , 1) and cons ider the embedd ing f : C 2 ≀ C n → L I ∈I L A ⊆ I ℓ 2 ( C n ) gi ve n by f ( x , j ) ≔ M I ∈I M A ⊆ I         ( − 1) | A ∩ α k ( x ) | · 1 I ( k + j ) + √ n ·  d C n ( k + j , I )  s − 1 2 n 2 n / 6         k ∈ C n . Arg uing similarly to [5] (and the abov e) sho w s that ρ ( u , v ) s . k f ( u ) − f ( v ) k 2 . ρ ( u , v ) for all u , v ∈ C 2 ≀ C n , where the implied constants are independ ent o f n . By a standar d limiting ar gumen t it follo ws that α ∗ ( C 2 ≀ Z ) = 1. This fact was first pr ov ed by T essera in [37] via a di ff erent approa ch. ⊳ Second emb edding of C 2 ≀ C n into L 1 . Let J be the set of all arcs in C n . In what follo ws for J ∈ J we let J ◦ denote the interior of J . Let { v J , A : J ∈ J , A ⊆ J } be disjointly supp orted unit vec tors in L 1 . Define f : C 2 ≀ C n → C ⊕ L 1 by f ( x , j ) ≔  ne 2 π i j n  ⊕         1 n X J ∈J 1 { j < J ◦ } v J , x ∩ J         . As before, since the metric on C 2 ≀ C n gi ve n by k f ( x , j ) − f ( x ′ , j ′ ) k 1 is C 2 ≀ C n -in va riant, it su ffi ces to sho w that k f ( x , j ) − f ( ∅ , 0) k 1 ≍ ρ  ( x , j ) , ( ∅ , 0)  for all ( x , j ) ∈ C 2 ≀ C n . Now , k f ( x , j ) − f ( ∅ , 0) k 1 ≍ d C n (0 , j ) + 1 n X J ∈J    1 { j < J ◦ } v J , x ∩ J − 1 { 0 < J ◦ } v J , ∅    1 = d C n (0 , j ) + 1 n X J ∈J x ∩ J = ∅    1 { j < J ◦ } − 1 { 0 < J ◦ }    + 1 n X J ∈J x ∩ J , ∅  1 { j < J ◦ } + 1 { 0 < J ◦ }  . (38) W e check the Lipschitz conditi on for the genera tors ( ∅ , 1) and ( { 0 } , 0) as follo ws: k f ( ∅ , 1) − f ( ∅ , 0) k 1 (38) ≍ 1 + 1 n     n J ∈ J :    { 0 , 1 } ∩ J ◦    = 1 o     ≍ 1 = ρ  ( ∅ , 1 ) , ( ∅ , 0)  , and k f ( { 0 } , 0) − f ( ∅ , 0) k 1 (38) ≍ 1 n     J ∈ J : 0 ∈ J \ J ◦     ≍ 1 = ρ  ( { 0 } , 0) , ( ∅ , 0)  . Hence k f ( x , j ) − f ( ∅ , 0) k 1 . ρ  ( x , j ) , ( ∅ , 0)  for all ( x , j ) ∈ C 2 ≀ C n . 14 T o p rov e the lo wer b ound k f ( x , j ) − f ( ∅ , 0) k 1 & ρ  ( x , j ) , ( ∅ , 0)  suppo se that ℓ ∈ x is a point of x at a maximal distan ce from 0 in C n . Then k f ( x , j ) − f ( ∅ , 0) k 1 (38) & d C n (0 , j ) + 1 n X J ∈J ℓ ∈ J  1 { j < J ◦ } + 1 { 0 < J ◦ }  ≍ d C n (0 , j ) + 1 n     J ∈ J : ℓ ∈ J ∧ { 0 , j } \ J ◦ , ∅     & d C n (0 , j ) + ( ℓ + 1)( n − ℓ ) n ≍ d C n (0 , j ) + d C n (0 , ℓ ) + 1 ≍ ρ  ( x , j ) , ( ∅ , 0)  , (39) Where in (39) we used the fac t that the interva ls  [ a , b ] : a ∈ { 0 , . . . , ℓ } , b ∈ { ℓ, . . . , n − 1 }  do not contain 0 in their interio r , but do conta in ℓ .  Remark 4.2. A separab le m etric space embeds with distort ion D into L p if and only if all its finite subse ts do. T herefor e our embed dings for C 2 ≀ C n into L 1 imply t hat C 2 ≀ Z admits a bi-Lip schitz embeddin g into L 1 . This can also be seen via the exp licit embedding F ( x , j ) ≔ j ⊕  ψ ( x , j ) − ψ (0 , 0 )  , where F ( x , j ) ≔ X k ≥ j v [ k , ∞ ) , x ∩ [ k , ∞ ) + X k ≤ j v ( −∞ , k ] , x ∩ ( −∞ , k ] , and { v J , A : J ∈ { [ k , ∞ ) } k ∈ Z ∪ { ( −∞ , k ] } k ∈ Z , A ⊆ J } are disj ointly supported unit vectors in L 1 . ⊳ 5 Edge Mark ov type need not imply Enflo type A Mark o v chai n { Z t } ∞ t = 0 with tra nsitio n probabiliti es a i j ≔ P ( Z t + 1 = j | Z t = i ) on t he s tate space { 1 , . . . , n } is statio nary if π i ≔ P ( Z t = i ) does not depend on t and it is r eversi ble if π i a i j = π j a ji for ev ery i , j ∈ { 1 , . . . , n } . Giv en a metri c space ( X , d X ) and p ∈ [1 , ∞ ), we say that X has Marko v ty pe p if there e xist s a constant K > 0 such that for e ver y stationary re v ersib le Marko v chain { Z t } ∞ t = 0 on { 1 , . . . , n } , ev ery mapping f : { 1 , . . . , n } → X and e ver y time t ∈ N , E  d X ( f ( Z t ) , f ( Z 0 )) p  ≤ K p t E  d X ( f ( Z 1 ) , f ( Z 0 )) p  . (40) The least such K is called the Markov type p constant of X , and is denoted M p ( X ). Similarly , gi ve n D > 0 we let M ≤ D p ( X ) den ote the le ast constan t K satisfying (40) w ith the additio nal restrict ion tha t d X ( f ( Z 0 ) , f ( Z 1 ) ) ≤ D holds pointwise. W e call M ≤ D p ( X ) the D -bound ed increment Mark ov type p constant of X . Finally , if ( X , d X ) is an unweighte d graph equipp ed with the sho rtest path metric th en the e dge Ma rko v type p const ant of X , deno ted M edge p ( X ), is the least con stant K satisfying (40) wit h the additio nal restrictio n that f ( Z 0 ) f ( Z 1 ) is an edge (poin twise). The fact that L 2 has Ma rko v type 2 with const ant 1, first noted by K. Ball [7], f ollo ws from a simple spe ctral ar gument (see als o ineq uality (8) in [28]). Since for p ∈ [1 , 2] the metric space  L p , k x − y k p / 2 2  embeds isometric ally into L 2 (see [42]), it follo ws that L p has Mark ov type p with consta nt 1. For p > 2 it wa s sho wn in [28] that L p has Mark ov type 2 w ith const ants O  √ p  . W e refer to [28] for a computation of the Marko v type of v ariou s additional classes of metric spac es. A metric space ( X , d X ) is said to hav e Enflo type p if there exists a const ant K such that for ev ery n ∈ N and 15 e ver y f : { − 1 , 1 } n → X , E  d X ( f ( ε ) , f ( − ε )) p  ≤ T p n X j = 1 E h d X  f ( ε 1 , . . . , ε j − 1 , ε j , ε j + 1 , . . . , ε n ) , f ( ε 1 , . . . , ε j − 1 , − ε j , ε j + 1 , . . . , ε n )  p i , (41) where the expe ctatio n is with respect to the uniform measure on {− 1 , 1 } n . In [29] it was sho wn that Marko v type p implies Enflo type p . W e define analogou sly to the case of Marko v type the notion s of bou nded incremen t E nflo type and edge Enflo type. The notions of Enflo type and Marko v type were introduc ed as non-linear an alogue s of the fundament al Banach space notion of Rademac her type . W e refer to [16, 10, 7, 29, 27, 28] and the refere nces therein for backg round on this topic and many application s. In Banach space theory the notion analo gous to bounde d incremen t Mark ov type is kno wn as equal norm Rademach er type . It is well kno wn (see [38]) that for Banach spaces equal norm R ademache r type 2 implies R ademache r type 2 and that for 1 < p < 2 equal norm Rademach er type p implie s Rademacher type q for ev ery q < p (but is does not gen erally imply Rademache r type p ). It is na tural to ask whe ther the ana logou s phenomen on holds true for th e abo ve metric analog ues of Rademache r type. H ere we sho w that this is not the case. It follo ws from T heorem 1.2 that α ∗ ( Z ≀ Z ) ≥ 2 3 . Therefore for ev ery 0 < α < 2 3 there is a mappin g F : Z ≀ Z → L 2 such that x , y ∈ Z ≀ Z = ⇒ d Z ≀ Z ( x , y ) α . k F ( x ) − F ( y ) k 2 . d Z ≀ Z ( x , y ) . Fix a stationa ry re versible Marko v chain { Z t } ∞ t = 0 on { 1 , . . . , n } and a m appin g f : { 1 , . . . , n } → Z ≀ Z such that d Z ≀ Z ( f ( Z 0 ) , f ( Z 1 ) ) ≤ D holds pointwise. Using the fac t that L 2 has Marko v type 2 w ith const ant 1 we deduc e that E h d Z ≀ Z  f ( Z t ) , f ( Z 0 )  2 α i . E h k F ◦ f ( Z t ) − F ◦ f ( Z 0 ) k 2 2 i ≤ t E h k F ◦ f ( Z 1 ) − F ◦ f ( Z 0 ) k 2 2 i . t E h d Z ≀ Z  f ( Z 1 ) , f ( Z 0 )  2 i . D 2(1 − α ) t E h d Z ≀ Z  f ( Z 1 ) , f ( Z 0 )  2 α i . Thus M ≤ D 2 α ( Z ≀ Z ) . D 1 − α . In parti cular Z ≀ Z has D -bounded increment Markov typ e p and edge Marko v type p for e ver y p < 4 3 . On t he ot her h and we claim that Z ≀ Z does not ha ve Enflo type p for an y p > 1. This is seen via an ar gumen t that was use d by Arzhantse va, Guba and Sapir in [3]. Fix n ∈ N and define f : {− 1 , 1 } n → Z ≀ Z by f ( ε 1 , . . . , ε n ) ≔          2 n X j = n + 1 ε j − n n δ j , 0          , (42) where δ j is the delta functio n suppor ted at j . Then for ev ery ε ∈ { − 1 , 1 } n , d Z ≀ Z  f ( ε ) , f ( − ε )  ≍ n 2 (43) and for e very j ∈ { 1 , . . . , n } , d Z ≀ Z  f ( ε 1 , . . . , ε j − 1 , ε j , ε j + 1 , . . . , ε n ) , f ( ε 1 , . . . , ε j − 1 , − ε j , ε j + 1 , . . . , ε n )  ≍ n . (44) Therefore if Z ≀ Z has Enflo typ e p , i.e. if (41) ho lds tru e, then for e very n ∈ N we ha v e n 2 p . n p + 1 , impl ying that p ≤ 1.  16 6 A lower bound on β ∗ ( G ≀ H ) In this section we shall prov e (8), which is a generalizat ion of ` Ershler’ s work [17]. Namely , we will prov e the follo wing theore m: Theor em 6.1. Let G and H be finitely gener ate d gr oups. If H has linear gr owth (or equivale ntly , by Gr o- mov’ s theor em [19], H has a subgr oup of finite inde x isomorp hic to Z ) then β ∗ ( G ≀ H ) ≥ 1 + β ∗ ( G ) 2 . F or all other finitely gen era ted gr oups H we have β ∗ ( G ≀ H ) = 1 . Assume that G is generate d by a finite symmetric set S G ⊆ G and H is genera ted by a finite symmetric set S H ⊆ H . W e also let e G , e H denote the identity elements of G and H , respecti ve ly . Giv en g 1 , g 2 ∈ G and h ∈ H define a mapping f h g 1 , g 2 : H → G by f h g 1 , g 2 ( x ) ≔          g 1 if x = e H , g 2 if x = h , e G otherwis e . It is immediate to check that the set S G ≀ H ≔ n f h g 1 , g 2 : g 1 , g 2 ∈ S G and h ∈ S H o is symmetric and genera tes G ≀ H . From no w on, we w ill assume that the metrics on G , H and G ≀ H are induc ed by S G , S H and S G ≀ H , respec ti ve ly . Analog ously we shall denote by n W G k o ∞ k = 0 , n W H k o ∞ k = 0 and n W G ≀ H k o ∞ k = 0 the corre spond ing random walks , starting at the correspon ding identity elements. Theor em 6.2. Assume that for some β ∈ [0 , 1] we have E h d G  W G n , e G i & n β , (45) wher e the implied consta nt may depe nd on S G . If H has linear gr owth then E h d G ≀ H  W G ≀ H n , e G ≀ H i & n 1 + β 2 . (46) If H has quadra tic gr owth then E h d G ≀ H  W G ≀ H n , e G ≀ H i & n (1 + log n ) 1 − β . (47) If the ran dom walk n W H n o ∞ n = 0 is tra nsien t then E h d G ≀ H  W G ≀ H n , e G ≀ H i & n . (48) The implied constan ts in (46) , (47) and (48) may depe nd on S G and S H . Theorem 6.1 is a conseq uence of Theore m 6.2 since by V arop oulos ’ celebrated result [39, 41] (which relies on Gromov’ s gro wth the orem [19]. See [24] and [43] for a detail ed discussio n), the three possibiliti es in Theorem 6.2 are exhaus ti ve for infinite fi nitely generate d groups H . In the case when the random walk on H is transient, Theorem 6.2 was p re viou sly proved by Ka ˘ ımano vich and V ershik in [24]. The follo w ing lemma will be used in the proof of Theorem 6.2. 17 Lemma 6.3. D efine for n ∈ N , ψ H ( n ) ≔          √ n if H has linear gr owth, 1 + log n if H has quadr atic gr owth, 1 otherwise . Then E      n 0 ≤ k ≤ n : W H k = e H o     β  & ψ H ( n ) β , (49) and E h    W H [0 , n ]    i & n ψ H ( n ) , (50) wher e W H [0 , n ] ≔ n W H 0 , . . . , W H n o . Pr oof. By a theorem of V arapoulos [40, 41] (see also [23] and T heorem 4.1 in [43]) for e v ery k ≥ 0, P h W H k = e H i + P h W H k + 1 = e H i ≍        1 √ k + 1 if H has linear gro wth, 1 k + 1 if H has quadratic growth, (51) and if H has super -quad ratic gro wth then P ∞ k = 1 P h W H k = e H i < ∞ . Hence, if we denote X n ≔     n 0 ≤ k ≤ n : W H k = e H o     = n X k = 0 1 { W H k = e H } then it follo ws that E [ X n ] = n X k = 0 P h W H k = e H i (51) ≍ ψ H ( n ) . (52) T o prov e (49) note that E h X 2 n i = n X i , j = 0 P h W H i = e H ∧ W H j = e H i ≤ 2 n X i = 0 n − i X k = 0 P h W H i = e H i · P h W H k = e H i ≤ 2 ( E [ X n ] ) 2 (52) ≍ ψ H ( n ) 2 . Using H ¨ older’ s inequality we deduce that ψ H ( n ) ≍ E [ X n ] = E " X β 2 − β n · X 2 − 2 β 2 − β n # ≤  E h X β n i 1 2 − β  E h X 2 n i 1 − β 2 − β .  E h X β n i 1 2 − β ψ H ( n ) 2 − 2 β 2 − β . This simplifies to E h X β n i & ψ H ( n ) β , which is precise ly (49). W e no w pass to the p roof of (50). For e very k ∈ { 1 , . . . , n } denote by V 1 , . . . , V k the first k elemen ts of H that were visite d by the walk n W H j o ∞ j = 0 . Write Y k ≔     n 0 ≤ j ≤ n : W H j ∈ { V 1 , . . . , V k } o     . 18 Then E [ Y k ] = k X j = 1 E      n 0 ≤ j ≤ n : W H j = V j o      ≤ k n X r = 0 P h W H r = e H i (51) ≍ k ψ H ( n ) . Therefore for e ver y k ∈ N , P h    W H [0 , n ]    ≤ k i ≤ P [ Y k ≥ n ] ≤ E [ Y k ] n . k ψ H ( n ) n . Hence we can choose k ≍ n ψ H ( n ) for which P      W H [0 , n ]     ≥ k  ≥ 1 2 , implying (50).  Pr oof of Theor em 6.2. W e may assume tha t n ≥ 4. L et Q H : G ≀ H → H be the nat ural pro jection , i.e. Q H ( f , x ) ≔ x . Also, for e ve ry x ∈ H let Q x G : G ≀ H → G be the proj ection Q x G ( f , y ) ≔ f ( x ). Fix n ∈ N . For e v ery h ∈ H denote T h ≔     n 0 ≤ k ≤ n : Q H  W G ≀ H k  = h o     . The set of genera tors S G ≀ H was constructed so that the random walk on G ≀ H can be informally described as follo ws: at each step the “ H coordinate ” is m ultipl ied by a random element h ∈ S H . The “ G coord inate” is multipl ied by a ra ndom element g 1 ∈ S G at the ori ginal H coordi nate of the walk er , and also by a ra ndom element g 2 ∈ S G (which is indep enden t of g 1 ) at the ne w H coord inate of the walker . This immediately implies that the projection n Q H  W G ≀ H k  o ∞ k = 0 has the same dis trib uti on as n W H k o ∞ k = 0 . Moreov er , conditio ned on { T h } h ∈ H and on Q H  W G ≀ H n  , if h ∈ H \ n e H , Q H  W G ≀ H n o then the element Q h G  W G ≀ H n  ∈ G has the same distrib ution as W G 2 T h . If h ∈ n e H , Q H  W G ≀ H n o and e H , Q H  W G ≀ H n  then Q h G  W G ≀ H n  has the same distrib ution as W G max { 2 T h − 1 , 0 } , and if e H = Q H  W G ≀ H n  then Q h G  W G ≀ H n  has the same distri b ution as W G 2 T h . These observ ations impl y , using (45), that for e ve ry h ∈ H we ha ve E h d G  Q h G  W G ≀ H n  , e G i & E h T β h i . Writing A ℓ ≔ n h = W H ℓ ∧ h < W H [0 ,ℓ − 1] o we see that E h T β h i ≥ ⌊ n / 2 ⌋ X ℓ = 0 P ( A ℓ ) · E h T β h    A ℓ i (49) ≥ ⌊ n / 2 ⌋ X ℓ = 0 P ( A ℓ ) · ψ H ( n / 2) β = P h h ∈ W H [0 , ⌊ n / 2 ⌋ ] i ψ H ( n / 2) β . Hence, E h d G ≀ H  W G ≀ H n , e G ≀ H i & X h ∈ H E h d G  Q h G  W G ≀ H n  , e G i & X h ∈ H E h T β h i & ψ H ( n ) β X h ∈ H P h h ∈ W H [0 , ⌊ n / 2 ⌋ ] i = ψ H ( n ) β · E h    W H [0 , ⌊ n / 2 ⌋ ]    i (50) & n ψ H ( n ) 1 − β . This is precise ly the asse rtion of T heorem 6.2.  Remark 6.4. In [13] de Cornulier , Stalder and V alett e show that if G is a finite group then for e ve ry p ≥ 1 we ha v e α # p ( G ≀ F n ) ≥ 1 p , where F n denote s the free g roup on n ≥ 2 generator s. Note that in combina tion with Lemma 2.3 this implies that w e actually α # p ( G ≀ F n ) ≥ max n 1 p , 1 2 o . This bound is sharp due to Theorem 1.1 and the fact that β ∗ ( G ≀ F n ) = 1. 19 In fact, we ha v e the follo wing stronger result : if X is a Banach space w ith modulus of smoothnes s o f po wer type p , G is a nontri vial group, and H is a group whose volu me gro wth is at least quadratic , then α ∗ X ( G ≀ H ) ≤ 1 p . In particular α ∗ p ( G ≀ F 2 ) = max n 1 p , 1 2 o . T o prov e the above assertion note that it is enough to deal with the case G = C 2 . If H is amenab le then by Theor em 6.1 we ha ve β ∗ ( C 2 ≀ H ) = 1, so tha t the requir ed result follo ws from the result of [4] and the fact that X has Mark ov type p [28]. If H is nonamen able then it has exponen tial growth (see [30]). Thu s γ ≔ lim r →∞ | B ( e H , r ) | 1 / r > 1, w here B ( x , r ) denote s the ball of radius r cent ered at x in the word metr ic on H (note that the e xiste nce of the limit follo ws from submultiplic ati vity ). Fix δ ∈ (0 , 1) such that η ≔ (1 − δ ) 2 γ 1 + δ > 1 and let k 0 ∈ N be such that for all k ≥ k 0 we hav e [(1 − δ ) γ ] k ≤ | B ( e H , k ) | ≤ [(1 + δ ) γ ] k . Fo r k ≥ k 0 let { x 1 , . . . , x N } be a maximal subset of B ( e H , 2 k ) such that the balls { B ( x i , k / 2) } N i = 1 are disjoint. Maximalit y implies that the balls { B ( x i , k ) } N i = 1 cov er B ( x , 2 k ), so that [(1 + δ ) γ ] k N ≥ N | B ( e H , k ) | ≥        N [ i = 1 B ( x i , k )        ≥ | B ( e H , 2 k ) | ≥ [(1 − δ ) γ ] 2 k , which simplifies to gi ve t he lo wer bound N ≥ η k . Thus k . log N . Fix α ∈ [0 , 1] and assume that F : C 2 ≀ H → X satisfies x , y ∈ C 2 ≀ H = ⇒ d C 2 ≀ H ( x , y ) α . k F ( x ) − F ( y ) k . d C 2 ≀ H ( x , y ) . Our goal is to pro v e that α ≤ 1 p . For e very ε = ( ε 1 , . . . , ε N ) ∈ { − 1 , 1 } N define ψ ε : H → C 2 by ψ ε ( x i ) = 1 + ε i 2 , and ψ ε ( x ) = 0 if x < { x 1 , . . . , x N } . Let f : {− 1 , 1 } N → C 2 ≀ H be gi v en by f ( ε ) = ( f ε , e H ). It is immediate to check tha t for all ε, ε ′ ∈ {− 1 , 1 } N we hav e k 2 k ε − ε ′ k 1 ≤ k f ( ε ) − f ( ε ′ ) k ≤ 4 k k ε − ε ′ k 1 . Metric spac es with Marko v type p also ha v e Enflo type p [29], i.e. the y satisfy (41). T hus we can apply the Enflo type inequa lity (41) to the m apping F ◦ f : {− 1 , 1 } N → X and deduce that ( N k ) α p . N k p . Consequen tly , N α p . N k p . N (log N ) p . Since the last inequality holds for arbitrarily larg e N , we inf er that α p ≤ 1. ⊳ 7 Discussion and further questions In this sectio n we discuss some natural questions that arise from the results obtain ed in this paper . W e start with the follo w ing potential con verse to (3): Question 7.1. Is it true that for every finitely gen era ted amenable gr oup G, α ∗ ( G ) = 1 2 β ∗ ( G ) ? If true, Questio n 7.1, in combination with Corollary 1.3, would imply a positi ve solutio n to the follo wing questi on: Question 7.2. Is it true that for every finitely gen era ted amenable gr oup G, α ∗ ( G ≀ Z ) = 2 α ∗ ( G ) 2 α ∗ ( G ) + 1 ? Addition ally , since β ∗ ( G ) ≤ 1, a positi v e solution to Ques tion 7.1 would imply a positi v e solu tion to the follo w ing question: 20 Question 7.3. Is it true that for every finitely gen era ted amenable gr oup G, α ∗ ( G ) ≥ 1 2 ? Using (27), a nd ar gu ing analog ously to Lemma 3.1 while using the L 1 embeddi ng of C 2 ≀ Z in Section 4, we ha v e the follo wing fact: Lemma 7.4. If a finitely gener ated gr oup G admits a bi-Lipsc hitz embedding into L 1 then so does G ≀ Z . Question 7.5. Is it true that for every finitely gen era ted amenable gr oup G we have α ∗ 1 ( G ) = 1 ? Since the metric space  L 1 , p k x − y k 1  embeds isometrica lly into L 2 (see [42]), a positi v e solution to Q ues- tion 7.5 would imply a posi ti ve solutio n to Q uestion 7.3. Our repertoir e of groups G for which we kno w the ex act valu e of α ∗ ( G ) is currentl y very limited. In parti c- ular , w e do not kno w the answer to the follo wing questio n: Question 7.6. Does ther e exist a finitely gen era ted amenable gr oup G for which α ∗ ( G ) is irra tiona l? D oes ther e e xist a finitely gen era ted amenable gr oup G for which 2 3 < α ∗ ( G ) < 1 ? In [44] Y u prov ed that for ev ery finitely generate d hyperbolic group G there exists a lar ge p > 2 for which α # p ( G ) ≥ 1 p . In vie w of Theorem 1.1 it is natural to ask: Question 7.7. Is it tru e that for eve ry finitely gene rat ed hyperbolic gr oup G ther e exis ts some p ≥ 1 for which α # p ( G ) ≥ 1 2 ? W e do not kno w the v alue of α ∗ p ( Z ≀ Z ) for 1 < p < 2. The follo wing lemma contains some bound s for this number: Lemma 7.8. F or eve ry 1 < p < 2 , p 2 p − 1 ≤ α ∗ p ( Z ≀ Z ) ≤ m in ( p + 1 2 p , 4 3 p ) . (53 ) Pr oof. The lower bound in (53 ) is an immediate corollar y of Theorem 3.3. S ince β ∗ ( Z ≀ Z ) ≥ 3 4 , the upper bound α ∗ p ( Z ≀ Z ) ≤ 4 3 p follo w s immediately from the resul ts of [4] (or alternati v ely Theorem 1.1), using the fact t hat L p , 1 < p < 2, has Mark o v type p . The remainin g upper bound is an applica tion of the fa ct that L p , 1 < p < 2, has Enflo type p , which is similar to an ar gument in [3]. Inde ed, fix a mapping F : Z ≀ Z → L p such that x , y ∈ Z ≀ Z = ⇒ d Z ≀ Z ( x , y ) α . k F ( x ) − F ( y ) k p . d Z ≀ Z ( x , y ) . Let f : {− 1 , 1 } n → Z ≀ Z be as in (42). Plugging the boun ds in (43) and (44) into the E nflo type p inequa l- ity (41) f or the mapp ing F ◦ f : {− 1 , 1 } n → L p , we see that for all n ∈ N we ha ve n 2 p α . n p + 1 , implyin g that α ≤ p + 1 2 p .  Question 7.9. Evaluate α ∗ p ( Z ≀ Z ) for 1 < p < 2 . W e end with the follo wing question which arises natural ly from the discu ssion in Section 5: 21 Question 7.10. Does ther e exist a finitely gener ated gr oup G which has edge Marko v type 2 bu t does not have Enflo type p for any p > 1 ? W e do not ev en know whether there exists a finitely generat ed group G which has edge Markov type 2 b ut does not h a ve Marko v type 2 . N ote that th e results o f Section 5 imply that i f 1 < p < 4 3 then the metr ic space  Z ≀ Z , d p / 2 Z ≀ Z  has bou nded increment Mark o v type 2, b ut does not ha ve Enflo type q fo r an y q > 2 p . Howe v er , this metric is not a graph metric. Ackno wledgements W e are gratefu l to Laurent Salo ff -Coste for helpful comments. Refer ences [1] I. Aharon i, B. Maur ey , and B. S. Mitya gin. Uniform embedding s of metric spaces and of Banach spaces into Hilbert space s. Israel J. Math. , 52(3):25 1–265 , 1985. [2] G. Arzhants e v a, C. Drutu, an d M. S apir . C ompressi on func tions of unif orm embe ddings of gro ups int o Hilbert and Banach spaces. Prepri nt, 2006. A va ilable at http://arx iv.org/a bs/math/0612378 . [3] G. N. Arzhantse va, V . S. Guba, and M. V . Sapir . Metrics on diagram groups and unifo rm embedding s in a Hilbert space. Comment. Math. Helv . , 81(4): 911–9 29, 2006. [4] T . A ustin, A. Naor , and Y . Peres. 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