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Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. (English) Zbl 1274.43009

Summary: We characterize the possible asymptotic behaviors of the compression associated to a uniform embedding into some \(L^p\)-space, with \(1<p<\infty\), for a large class of groups including connected Lie groups with exponential growth and word-hyperbolic finitely generated groups. In particular, the Hilbert compression exponent of these groups is equal to 1. This also provides new and optimal estimates for the compression of a uniform embedding of the infinite 3-regular tree into some \(L^p\)-space. The main part of the paper is devoted to the explicit construction of affine isometric actions of amenable connected Lie groups on \(L^p\)-spaces whose compressions are asymptotically optimal. These constructions are based on an asymptotic lower bound of the \(L^p\)-isoperimetric profile inside balls. We compute the asymptotic behavior of this profile for all amenable connected Lie groups and for all \(1\leq p<\infty\), providing new geometric invariants of these groups. We also relate the Hilbert compression exponent with other asymptotic quantities such as volume growth and probability of return of random walks.

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E15 General properties and structure of real Lie groups
43A07 Means on groups, semigroups, etc.; amenable groups
20F67 Hyperbolic groups and nonpositively curved groups
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