Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

We characterize the possible asymptotic behaviors of the compression associated to a uniform embedding into some L^p-space, with1 < p < ∞, 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 ≤ p < ∞, 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.