Embeddings of locally compact hyperbolic groups into Lp-spaces

TL;DR

This paper investigates how locally compact hyperbolic groups can be embedded into Lp-spaces, computing their equivariant Lp-compressions, and establishing conditions for proper affine actions, thus extending known results from discrete to locally compact groups.

Contribution

It generalizes the study of embeddability of hyperbolic groups to locally compact groups, computes equivariant Lp-compressions for specific examples, and answers a question on proper affine actions related to the conformal boundary.

Findings

Equivariant Lp-compression of locally compact compactly generated groups is minimal at p=2.

Calculated all equivariant Lp-compressions of SO(n,1).

Any locally compact hyperbolic group admits a proper affine isometric action on an Lp-space for p above the boundary's conformal dimension.

Abstract

In the last years, there has been a large amount of research on embeddability properties of finitely generated hyperbolic groups. In this paper, we elaborate on the more general class of locally compact hyperbolic groups. We compute the equivariant $L_p$-compression in a number of locally compact examples, such as the groups $SO(n,1)$: by proving that the equivariant $L_p$-compression of a locally compact compactly generated group is minimal for $p=2$, we calculate all equivariant $L_p$-compressions of $SO(n,1)$. Next, we show that although there are locally compact, non-discrete hyperbolic groups $G$ with Kazhdan's property ($T$), it is true that any locally compact hyperbolic group admits a proper affine isometric action on an $L_p$-space for $p$ larger than the Ahlfors regular conformal dimension of $\partial G$. This answers a question asked by Yves de Cornulier. Finally, we elaborate on the locally compact version of property $(A)$ and show that, as in the discrete case, a locally compact second countable group has property (A) if its non-equivariant compression is greater than 1/2.