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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 |