Stable actions and central extensions

TL;DR

This paper investigates the stability of group actions in the context of central extensions, showing that stability of the quotient implies stability of the original group, and characterizing when central extensions admit stable actions.

Contribution

It establishes that if a quotient group has a stable action, then the original group with a central subgroup also has a stable action, providing a new criterion for stability in central extensions.

Findings

Stability of a quotient group implies stability of the original group with a central subgroup.

Characterization of central extensions that admit stable actions.

Extension of previous results on stable actions in group theory.

Abstract

A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type II_1 under direct product. We show that for a countable group G and its central subgroup C, if G/C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.