Relative amenability

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 $\mathcal{X}$ of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that $\mathcal{X}$ contains all familiar groups. Actually, no group is known to lie outside $\mathcal{X}$.

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 $\mathcal{X}$.