Strength of convergence in non-free transformation groups

TL;DR

This paper explores how non-free transformation groups converge and accumulate measures, using stability subgroups and C^*-algebra representations to establish equivalences under specific conditions.

Contribution

It introduces new notions of convergence strength and measure accumulation for non-free group actions, linking them through representation theory.

Findings

Convergence notions are equivalent under certain conditions.

Stability subgroups are key to defining convergence strength.

Representation theory of crossed product C^*-algebras connects these notions.

Abstract

Let (G, X) be a transformation group where the group $G$ does not necessarily act freely on the space X. We investigate the extent to which the action of G may fail to be proper. Stability subgroups are used to define new notions of strength of convergence in the orbit space and of measure accumulation along orbits. By using the representation theory of the associated crossed product C^*-algebra, we show that these notions are equivalent under certain conditions.