## 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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