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.


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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