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On coarse embeddings of amenable groups into hyperbolic graphs

Romain Tessera

Abstract

In this note we prove that if a finitely generated amenable group admits a regular map to a direct product of a hyperbolic space and a euclidean space, then it must be virtually nilpotent. We deduce that an amenable group regularly embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto. We describe an application to Lorentz geometry due to Charles Frances.

On coarse embeddings of amenable groups into hyperbolic graphs

Abstract

In this note we prove that if a finitely generated amenable group admits a regular map to a direct product of a hyperbolic space and a euclidean space, then it must be virtually nilpotent. We deduce that an amenable group regularly embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto. We describe an application to Lorentz geometry due to Charles Frances.

Paper Structure

This paper contains 3 sections, 10 theorems, 8 equations.

Table of Contents

  1. The result
  2. An application to Lorentz geometry (after Charles Frances)
  3. ... and its proof

Key Result

Theorem \oldthetheorem

Let $G$ be a unimodular amenable locally compact compactly generated group. Assume that there exists $n\geq 2$ and $d\in \mathbb{N}$ such that $G$ regularly maps to $\mathbb{H}^n\times P$, where $P$ is a locally compact, compactly generated group of polynomial growth of degree $d$. Then $G$ must hav

Theorems & Definitions (20)

  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • proof : Proof of Proposition \ref{['prop:easydirection']}
  • Corollary \oldthetheorem
  • proof : Proof of Corollary \ref{['cor:main']}
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 10 more