Quotients by countable subgroups are hyperfinite

TL;DR

This paper proves that for any Polish group with a countable normal subgroup, the associated coset equivalence relation is hyperfinite, implying the outer automorphism group of any countable group is also hyperfinite.

Contribution

It establishes that coset equivalence relations by countable normal subgroups are hyperfinite, extending to the outer automorphism groups of countable groups.

Findings

Coset equivalence relations by countable normal subgroups are hyperfinite.

Outer automorphism groups of countable groups are hyperfinite.

The result applies broadly to Polish groups and their subgroups.

Abstract

We show that for any Polish group $G$ and any countable normal subgroup $\Gamma\triangleleft G$, the coset equivalence relation $G/\Gamma$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.