A note on relative amenability

TL;DR

This paper investigates the class of locally compact groups where relatively amenable subgroups are amenable, proposing a large subclass stable under group extensions to support the conjecture that the entire class is extension-closed.

Contribution

It introduces a sizable, extension-stable subclass of the class al{X} and generalizes elementary groups, providing new insights into groups outside al{X}.

Findings

Proposed a large subclass of al{X} stable under group extensions.

Generalized the class of elementary groups.

Provided information on groups outside al{X}.

Abstract

P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact groups for which relatively amenable closed subgroups are amenable. It is unknown if $\mathscr{X}$ is closed under group extension. In this note, we exhibit a large, group extension stable subclass of $\mathscr{X}$, which suggests $\mathscr{X}$ is indeed closed under group extension. Along the way, we produce generalizations of the class of elementary groups and obtain information on groups outside $\mathscr{X}$.