Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

By Romain Tessera. 2006

75 citations recorded by the source corpus.

We characterize the possible asymptotic behaviors of the compression associated to a uniform embedding into some Lp-space, with 1 < 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 rate 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 Lp-space. The main part of the paper is devoted to the explicit construction of affine isometric actions of amenable connected Lie groups on Lp-spaces whose compressions are asymptotically optimal. These constructions are based on an asymptotic lower bound of the Lp-isoperimetric profile inside balls. We compute the asymptotic 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 rate with other asymptotic quantities such as volume growth and probability of return of random walks.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6240d99770cb56ff4b1bdf6543d556f6d2339599

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Isometric Group Actions on Hilbert Spaces: Growth of Cocycles (2005, background): “In the case of the regular representation, some results can be found in [Tes1], related to isoperimetric properties of the group.”
Lp compression, traveling salesmen, and stable walks (2009, methodology): “(f); x,x) d+1−γp dp & dC 2≀Zd ((f,x),(0,0)) d+1−γp dp . (40) Letting γtend from above to d+1−p p in (40) we get the lower bound α∗ p C2 ≀Zd ≥ 1 d . (41) While (41) reproduces the result of [54], it yields the sharp bound α∗ p C2 ≀Zd = 1 p only when p = d = 2, in which case the above embeddi”