Relative property A and relative amenability for countable groups

TL;DR

This paper introduces the concepts of relative property A and relative amenability for countable groups, providing characterizations and conditions under which these properties are inherited or extended, leading to new classes of property A groups.

Contribution

It defines relative property A and relative amenability, offers characterizations, and establishes inheritance results, expanding understanding of property A in group theory.

Findings

Groups with property A relative to subgroups with property A also have property A.

Groups acting cocompactly on certain spaces with property A have property A.

Analogues of relative property A and amenability are introduced with similar inheritance results.

Abstract

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a group has property A relative to a family of subgroups, each of which has property A, then the group has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with at least one stabilizer of property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained.