Property of rapid decay for extensions of compactly generated groups

TL;DR

This paper proves that property RD is preserved under certain group extensions and generalizes the unimodularity result for locally compact groups with property RD, expanding understanding of this property in group theory.

Contribution

It establishes the permanence of property RD under group extensions and extends the unimodularity result to groups with subexponentially distorted automorphisms.

Findings

Property RD is preserved under specific group extensions.

Locally compact groups with property RD are unimodular.

Automorphisms with subexponential distortion preserve Haar measure.

Abstract

In the article we settle down the problem of permanence of property RD under group extensions. We show that if $1\to N\to G\to Q\to 1$ is a short exact sequence of compactly generated groups such that $Q$ has property RD, and $N$ has property RD with respect to the restriction of a word-length on $G$, then $G$ has property RD. We also generalize the result of Ji and Schweitzer stating that locally compact groups with property RD are unimodular. Namely, we show that any automorphism of a locally compact group with property RD which distorts distances subexponentially, preserves the Haar measure.