Relative amenability

TL;DR

This paper introduces the concept of relative amenability for subgroups of locally compact groups, establishes equivalent conditions, and explores its implications, including a solution to Reiter's problem and properties of the class X of groups.

Contribution

It defines relative amenability, proves its equivalence to amenability in certain classes, and solves Reiter's longstanding problem.

Findings

Relative amenability is weaker than amenability.

The class X includes all familiar groups and no known group lies outside it.

Chabauty limits of amenable subgroups are amenable within class X.

Abstract

We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability. It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem. We study the class X of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that X contains all familiar groups. Actually, no group is known to lie outside X. Since relative amenability is closed under Chabauty limits, it follows that any Chabauty limit of amenable subgroups remains amenable if the ambient group belongs to the vast class X.