The wreath product of Z with Z has Hilbert compression exponent 2/3

The wreath product of Z with Z has Hilbert compression exponent 2 3 T im Austin UCLA Assaf Naor ∗ Courant Institute Y uval Peres † Microsoft Research and UC Berkele y Abstract Let G be a finitely g enerated gr oup, equipped with the word metric d associated with some finite set of generators. The Hilbert comp ression exponen t of G is the supremum over a ll α ≥ 0 such that there exists a Lipschitz mapping f : G → L 2 and a constant c > 0 suc h that for all x , y ∈ G we have k f ( x ) − f ( y ) k 2 ≥ cd ( x , y ) α . In [2] it was shown that th e Hilbert co mpression e xpon ent of the wreath produ ct Z ≀ Z is at most 3 4 , and in [12] was proved th at this e xpo nent is at leas t 2 3 . Here we show that 2 3 is the correct value. Ou r proof is based on an application of K. Ball’ s notio n of Markov typ e. 1 Introd uction Let G be a finitely genera ted group . Fix a finite set of generat ors S ⊆ G , w hich we will alway s assume to be symmetr ic (i.e. S − 1 = S ). Let d be the left-in v ariant word metric indu ced by S on G . The Hilbert compr ession exponent of G , which w e denote by α ∗ ( G ), is the supremum o ver all α ≥ 0 such th at the re exi sts a 1-Lipschitz mapping f : G → L 2 and a consta nt c > 0 such that for all x , y ∈ G we ha ve k f ( x ) − f ( y ) k 2 ≥ cd ( x , y ) α . Note that α ∗ ( G ) does not depend on the cho ice of the finite set of generators S , and is thus an algebraic in varia nt of the group G . This notio n was introduced by Guentner a nd Kamink er in [7 ] as a natu ral quant ita- ti ve measure of Hilbert sp ace embed dabililt y in situation s when bi-Lip schitz embedd ings do not exist (whe n bi-Lipsch itz emb eddings do ex ist the natura l m easure would be the Eucli dean distortio n ). More generall y , the compr ession fun ction of a 1-Lip schitz mapping f : G → L 2 is defined as ρ ( t ) ≔ inf d ( x , y ) ≥ t k f ( x ) − f ( y ) k 2 . The mapping f is called a coarse embeddin g if lim t →∞ ρ ( t ) = ∞ . C oarse embeddings of discr ete groups ha ve been studied extens iv ely in recen t year s. The Hilbert compressio n expo nents of v arious groups were in vestiga ted in [7, 2, 5, 16, 1]—we refer to these papers and the referen ces therein for group-t heoretic al moti v ation and applicatio ns. Consider the wreath prod uct Z ≀ Z , i.e. the group of all pairs ( f , x ), where x ∈ Z and f : Z → Z has fi nite suppo rt, eq uipped with the group law ( f , x )( g , y ) ≔ ( z 7→ f ( z ) + g ( z − x ) , x + y ). In this not e we pr ov e that ∗ Research supported in part by NSF grants CCF-0635078 and DMS-0528387 . † Research supported in part by NSF grant DMS-0605166 . 1 α ∗ ( Z ≀ Z ) = 2 3 . The problem of co mputing α ∗ ( Z ≀ Z ) was raised exp licitly in [2, 16, 1]. In [2] Arzhan tse va , Guba and Sapir showed that α ∗ ( Z ≀ Z ) ∈ h 1 2 , 3 4 i . In [1 6] T esse ra claimed to impro ve the lo wer bound on α ∗ ( Z ≀ Z ) to α ∗ ( Z ≀ Z ) ≥ 2 3 , and conjectur ed tha t α ∗ ( Z ≀ Z ) = 2 3 . Unfortunat ely , T esse ra’ s proof is flawed, as exp lained in Remark 1.4 of [12]; his method on ly yields the bound α ∗ ( Z ≀ Z ) ≥ 1 3 . Howe v er , the inequality α ∗ ( Z ≀ Z ) ≥ 2 3 is corr ect, as sho wn by N aor an d Peres in [1 2] using a di ff er ent method. H ere we obtain the matching uppe r bou nd α ∗ ( Z ≀ Z ) ≤ 2 3 . For the sak e of completenes s, in Remark 2.2 belo w we also prese nt the embeddin gs of Naor and Peres [12] which estab lish the lower bound α ∗ ( Z ≀ Z ) ≥ 2 3 . Our proo f of the upper bou nd α ∗ ( Z ≀ Z ) ≤ 2 3 is a simple applicatio n of K. B all’ s notion of M arko v type , a metric in varia nt that has found sev eral applicatio ns in metric geometry in the past two decad es—see [3, 11, 9, 4, 13, 10]. Recall that a Marko v chain { Z t } ∞ t = 0 with trans ition probabil ities a i j ≔ Pr( Z t + 1 = j | Z t = i ) on the state space { 1 , . . . , n } is sta tionary if π i ≔ Pr( Z t = i ) does not depend on t and it is re ver sible if π i a i j = π j a ji for e ve ry i , j ∈ { 1 , . . . , n } . Giv en a m etric spac e ( X , d X ) and p ∈ [1 , ∞ ), we say that X has Marko v type p if t here ex ists a constant K > 0 such tha t for e very st ationary re vers ible Marko v chain { Z t } ∞ t = 0 on { 1 , . . . , n } , e ve ry mapping f : { 1 , . . . , n } → X and ev ery 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  . (1) The least such K is called the Mark ov type p consta nt of X , and is denoted M p ( X ). The fact th at L 2 has Ma rko v type 2 with con stant 1, first noted by K . Ball [3] , follo ws fro m a s imple sp ectral ar gument (se e als o ineq uality (8) in [13]). Since for p ∈ [1 , 2] the metric space  L p , k x − y k p / 2 2  embeds isometric ally into L 2 (see [17]), it follo ws that L p has Mark ov type p with consta nt 1. For p > 2 it was sho wn in [13] tha t L p has Mark ov type 2 with const ants O  √ p  . W e refer to [13] for a compu tation of the Marko v type of va rious additiona l classes of metric spaces. The no tion of Marko v type ha s been succe ssfully applied to v ario us embedding problems of finite metric spaces . In this no te we observ e that one can us e this in v ariant in the cont ext of infinite amen able gro ups as well. In a certain s ense, our ar gument simpl y amounts to using Mark ov ty pe asymp totically along n eighbor - hoods of Føln er sequences . For the rest of the pa per , L et G be an amenable gro up with a fixe d finite symmetri c set of gene rators S and the c orrespon ding left-i n v arian t word m etric d . Let e d enote the i dentity elemen t of G , and le t { W t } ∞ t = 0 be th e canon ical simple ran dom wa lk on the Cayley graph of G determin ed by S , sta rting at e . Our main result is: Pro position 1.1. Assume that that ther e e xist c , δ , β > 0 su ch that for all t ∈ N , Pr  d ( W t , e ) ≥ ct β  ≥ δ . (2) Let ( X , d X ) be a metric space with Marko v type p, and assume that f : G → X satisfies ρ ( d ( x , y )) ≤ d X ( f ( x ) , f ( y )) ≤ d ( x , y ) (3) for all x , y ∈ G , wher e ρ : R + → R + is non-d ecr easi ng. T hen for all t ∈ N , ρ  ct β  ≤ M p ( X ) δ 1 / p t 1 / p . In parti cular , α ∗ ( G ) ≤ 1 2 β . 2 As an immediate corolla ry we de duce t hat α ∗ ( Z ≀ Z ) ≤ 2 3 . Indeed, Z ≀ Z is amen able (see for e xample [ 8, 14]), and it was sho wn by R e velle in [15] that Z ≀ Z has a set of generators (name ly the ca nonical gene rators S = { (1 , 0) , ( − 1 , 0) , (0 , 1) , (0 , − 1) } ) w hich satisfies the assumption of Proposition 1.1 with β = 3 4 (see also [6] for the corresp onding bou nd on th e expectat ion of d ( W t , e )). 2 Pr oof of Proposition 1.1 Let { F n } ∞ n = 0 be a Føl ner sequ ence for G , i.e., for ev ery ε > 0 and any finite K ⊆ G , we hav e | F n △ ( F n K ) | ≤ ε | F n | for lar ge enough n . Fix an inte ger t > 0 and denote A n ≔ [ x ∈ F n B ( x , t ) ⊇ F n , where B ( x , t ) is the ball of radius t centered at x in the word metri c determined by S . For e ve ry ε > 0, the re exists n ∈ N su ch that ε | F n | ≥ | F n △ ( F n B ( e , t )) | = | A n \ F n | . (4) Let { Z t } ∞ t = 0 be the delaye d standard random walk restricted to A n . In other wo rds, Z 0 is un iformly distrib uted on A n , and for all j ≥ 0 and x ∈ A n , Pr  Z j + 1 = x    Z j = x  = 1 − | ( xS ) ∩ A n | | S | , and if s ∈ S is su ch th at x s ∈ A n then Pr  Z j + 1 = x s    Z j = x  = 1 | S | . It is straig htforwa rd to check that { Z t } ∞ t = 0 is a stationar y rev ersible Mark ov cha in. Hence, using the Marko v type p prop erty of X , and the fact that f is 1-Lipschitz, w e see tha t E  d X ( f ( Z t ) , f ( Z 0 )) p  (1) ≤ M p ( X ) p t E  d X ( f ( Z 1 ) , f ( Z 0 )) p  (3) ≤ M p ( X ) p t E  d ( Z 1 , Z 0 ) p  ≤ M p ( X ) p t . (5) Note that E  d X ( f ( Z t ) , f ( Z 0 )) p  (3) ≥ E  ρ ( d ( Z t , Z 0 )) p  ≥ 1 | A n | X x ∈ F n E h ρ ( d ( Z t , Z 0 ) ) p    Z 0 = x i , (6) since the omitte d summands correspon ding to x < F n are nonne gati v e. If x ∈ F n then B ( x , t ) ⊆ A n ; this implies th at cond itioned on th e e ven t { Z 0 = x } , th e rando m var iable d ( Z t , Z 0 ) has the same d istrib ution as th e random v ariable d ( W t , e ). The as sumption (2) yields that E  ρ ( d ( W t , e ) ) p  ≥ ρ  ct β  p · Pr  d ( W t , e ) ≥ ct β  ≥ ρ  ct β  p · δ . (7) In conju nction with (6), this gi v es that E  d X ( f ( Z t ) , f ( Z 0 )) p  ≥ | F n | | A n | · E  ρ ( d ( W t , e ) ) p  (7) ≥ | F n | | A n | · ρ  ct β  p · δ (4) ≥ δ 1 + ε · ρ  ct β  p . (8) Combining (5) and (8), and letting ε → 0, concludes the proof of Proposition 1.1.  3 Remark 2.1. Giv en two grou ps G and H , the wreat h produc t G ≀ H is the group of all pairs ( f , x ) where f : H → G ha s finite support (i.e. f ( z ) is the identit y of G for all but finitely man y z ∈ H ) and x ∈ H , equipp ed with th e product ( f , x )( g , y ) ≔  z 7→ f ( z ) g ( x − 1 z ) , xy  . Consider the iterated wreath product s Z ( k ) , where Z (1) = Z an d Z ( k + 1) ≔ Z ( k ) ≀ Z . In [15] it is sho wn that Z ( k ) has a finite symmetric set of genera tors which s atisfies the assumpti on of Propositio n 1.1 with β = 1 − 2 − k . Thus α ∗ ( Z ( k ) ) ≤ 1 2 − 2 1 − k . In fact, as sh own in [12], α ∗ ( Z ( k ) ) = 1 2 − 2 1 − k . ⊳ Remark 2.2. In [12] the lower bound α ∗ ( Z ≀ Z ) ≥ 2 3 is a pa rticular case of a more ge neral resu lt. For the reader s’ con ve nience we pres ent the re sulting embeddings in the ca se of the group Z ≀ Z . In w hat follo ws . and & d enote the corresp onding in equality up to a uni versa l constan t. Fix α ∈ (0 , 1 / 2 ) and let n v g : g : A → Z finitely suppor ted , A ∈ { Z ∩ [ n , ∞ ) } n ∈ Z ∪ { Z ∩ ( −∞ , n ] } n ∈ Z o be disjoin tly supp orted unit vectors in L 2 ( R ). For ( f , k ) ∈ Z ≀ Z define a function φ α ( f , k ) : R → R by φ α ( f , k ) ≔ X n > k ( n − k ) α · v f ↾ [ n , ∞ ) + X n < k ( k − n ) α · v f ↾ ( −∞ , n ] . Observ e that φ α ( f , k ) − φ α (0 , 0) ∈ L 2 ( R ). Indeed, if f is suppo rted on [ − m , m ] then k φ α ( f , k ) − φ α (0 , 0) k 2 2 . m  m 2 α + | k | 2 α  + X n ∈ Z  | n | α − | n − k | α  2 . m  m 2 α + | k | 2 α  + ∞ X j = 1 k 2 j 2(1 − α ) < ∞ . W e can ther efore define F α : Z ≀ Z → R ⊕ ℓ 2 ( Z ) ⊕ L 2 ( R ) by F α ( f , k ) ≔ k ⊕ f ⊕  φ α ( f , k ) − φ α (0 , 0)  . W e clai m that for ev ery ( f , k ) ∈ Z ≀ Z we ha ve d Z ≀ Z  ( f , k ) , (0 , 0)  2 α + 1 2 α + 2 . k F α ( f , k ) k 2 . 1 √ 1 − 2 α · d Z ≀ Z  ( f , k ) , (0 , 0)  , (9) Since the metric k F α ( f 1 , k 1 ) − F α ( f 2 , k 2 ) k 2 is Z ≀ Z -in v ariant, and F α (0 , 0) = 0, the inequa lities in (9 ) imply that Z ≀ Z has Hilbert compres sion e xponen t at least 2 α + 1 2 α + 2 . L etting α ↑ 1 2 sho ws that α ∗ ( Z ≀ Z ) ≥ 2 3 . It su ffi ces to check the up per bound in (9) (i.e. the Lipschitz condition for F α ) when ( f , k ) is on e of the genera tors of Z ≀ Z , i.e. ( f , k ) = (0 , 1) or ( f , k ) = ( δ 0 , 0). Observe that k F α ( δ 0 , 0) k 2 = 1 an d k F α (0 , 1) k 2 2 . ∞ X n = 1  n α − ( n − 1) α  2 . 1 1 − 2 α , implying the upper bound in (9 ). T o pro ve the lo wer bound in (9) ass ume tha t m ∈ N is th e minima l integer such that f is supported on [ k − m , k + m ]. Then, k F α ( f , k ) k 2 2 & k 2 + k + m X j = k − m f ( j ) 2 + m X ℓ = 1 ℓ 2 α & k 2 + 1 m          X j ∈ Z | f ( j ) |          2 + m 2 α + 1 &          k + m + X j ∈ Z | f ( j ) |          4 α + 2 2 α + 2 & d Z ≀ Z  ( f , k ) , (0 , 0)  4 α + 2 2 α + 2 , where the penult imate inequalit y follo ws by cons idering the ca ses k f k 1 ≥ m α + 1 and k f k 1 ≤ m α + 1 sepa- rately . ⊳ 4 Refer ences [1] G. Arz hantse v a, C. Dru tu, and M. Sapir . 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