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Embeddings of discrete groups and the speed of random walks. (English) Zbl 1163.46007

The main purpose of this paper is to use random walks on groups in order to obtain some results about the geometry of the groups in terms of certain quantitative characteristics of their embeddings into Banach spaces. The obtained results are also used to answer a question about two versions of nonlinear types of Banach spaces (edge Markov type and Enflo type).
The idea of using random walks for metric problems goes back to K.Ball [Geom.Funct.Anal.2, No.2, 137–172 (1992; Zbl 0788.46050)], it was further developed by A.Naor, Y.Peres, O.Schramm and S.Sheffield [Duke Math.J.134, No.1, 165–197 (2006; Zbl 1108.46012)], but several adaptations are needed to apply the method in the present setting.
Let \(G\) be a finitely generated group. A set \(S\) of generators of \(G\) is called symmetric if \(s\in S\Rightarrow s^{-1}\in S\). Let \(\beta^*(G)\) be the supremum of \(\beta\geq 0\) for which there exists a symmetric set \(S\) of generators of \(G\) and \(c > 0\) such that for all \(t\in{\mathbb N}\) the inequality \({\mathbb E}[d_G(W_t, e)]\geq ct^\beta\) holds, where \(\{W_t\}_{t=0}^\infty\) is the canonical simple random walk on the Cayley graph of \(G\) determined by \(S\), starting at the identity element \(e\).
This probabilistic characteristic \(\beta^*(G)\) is used to derive information about the metric behavior of the group \(G\). More precisely, let \(d_G\) be the left-invariant word metric induced on \(G\) by \(S\). Let \(X\) be a Banach space. Then \(\alpha_X^*(G)\) denotes the supremum of \(\alpha>0\) for which there exists a Lipschitz mapping \(f:G\to X\) and \(c > 0\) such that \(||f(x)-f(y)||\geq c\cdot d_G(x,y)^\alpha\) for all \(x, y\in G\). If \(X=L_p\) \((p\geq 1)\) and \(X=L_2\), the notations \(\alpha_p^*(G)\) and \(\alpha^*(G)\), respectively, are used. The parameter \(\alpha^*(G)\) is called the Hilbert compression exponent of \(G\), it was introduced by E.Guentner and J.Kaminker [J. Lond.Math.Soc., II.Ser.70, No.3, 703–718 (2004; Zbl 1082.46049)] and has been actively studied since then.
The authors also consider the equivariant compression exponents \(\alpha^\#_X(G)\), which are defined exactly as \(\alpha_X^*(G)\) with the additional requirement that the embedding \(f:G\to X\) is equivariant; with the natural meaning of \(\alpha_p^\#(G)\) and \(\alpha^\#(G)\).
One of the main results of the paper is Theorem 1.1: Let \(X\) be a Banach space which has modulus of smoothness of power type \(p\). Then \[ \alpha_X^\#(G)\leq\frac1{p\beta^*(G)}. \] A combination of this result with some known facts implies one of the results of E.Guentner and J.Kaminker [op.cit.]: if \(\alpha^\#(G)>\frac12\), then \(G\) is amenable.
Another result which we would like to mention: \(\alpha_p^\#({\mathbb F}_2)=\frac12\) for every \(2\leq p<\infty\), where \({\mathbb F}_2\) is the free group with two generators.
The paper is very rich in contents and this review reflects only a small part of it. Reading of the paper is strongly recommended to everyone interested in the topic.

MSC:

46B07 Local theory of Banach spaces
05C12 Distance in graphs
20F65 Geometric group theory
54E35 Metric spaces, metrizability
60G50 Sums of independent random variables; random walks
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