Embeddings of Discrete Groups and the Speed of Random Walks

Abstract

Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric d G . For a Banach space X let α * X (G) (respectively α # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f : G → X and c > 0 such that for all x, y ∈ G we have f (We show that if X has modulus of smoothness of power type p,Here β * (G) is the largest β ≥ 0 for which there exists a set of generators S of G and c > 0 such that for all t ∈ N we have E d G (W t , e) ≥ ct β , where {W t } ∞ t=0 is the canonical simple random walk on the Cayley graph of G determined by S , starting at the identity element. This result is sharp when X = L p , generalizes a theorem of Guentner and Kaminker [20], and answers a question posed by Tessera [37]. We also show that if. This improves the previous bound due to Stalder and Valette [36]. We deduce that if we write2−2 1−k , and use this result to answer a question posed by Tessera in [37] on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C 2 ≀ C n embed into L 1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres in [26]. Finally, we use these results to show that edge Markov type need not imply Enflo type.

On the Bi-Lipschitz Geometry of Lamplighter Graphs

“…The variety of geometric features of wreath products of groups has also come to play an important role, sometimes quite unexpectedly, in metric geometry. For instance, the geometry of Z Z is closely connected to the extension of Lipschitz maps [17], and is also used in distinguishing bi-Lipschitz invariants, namely Enflo type and edge Markov type [16].…”

A characterization of superreflexivity through embeddings of lamplighter groups

“…Lamplighter groups are a very interesting class of groups which has been a rich source of important examples in geometric group theory. In 2008, Naor and Peres [17,Section 4] proved that the finite lamplighter groups Z 2 ≀ Z n , with metric defined as a word length with respect to natural sets of generators, are embeddable into L 1 with uniformly bounded distortions. The first goal of this paper is to strengthen this result and to prove their embeddability into an arbitrary nonsuperreflexive Banach space with uniformly bounded distortions.…”

“…Remark 1.4. In [17], when proving embeddability into L 1 , Naor and Peres considered Z 2 ≀Z n with the set of generators equal to {t, a} instead of {t, ta} as we do. However, it is easy to see that the metrics induced by these two generating sets are bilipschitz equivalent to each other with a distortion 4.…”

“…In [17] Naor and Peres constructed two embeddings into L 1 , one based on irreducible representations of finite lamplighter groups, and the other motivated by what they refer to as a 'direct geometric reasoning'. Our embedding technique is quite different from either of the embeddings in [17].…”

“…In [17] Naor and Peres constructed two embeddings into L 1 , one based on irreducible representations of finite lamplighter groups, and the other motivated by what they refer to as a 'direct geometric reasoning'. Our embedding technique is quite different from either of the embeddings in [17]. Our approach is inspired by the description of the Cayley graph of the infinite lamplighter group Z q ≀ Z, for q ∈ N, q ≥ 2, as a horocyclic product of two trees, introduced by Bartholdi and Woess [1,28], and by the analysis of the metric structure of horocyclic products of trees by Stein and Taback in [26] who proved, among other results, We refer the reader to [27] and [29] for very nice presentations of horocyclic products of trees and their applications to lamplighter groups.…”