Free subgroups acting properly discontinuously

TL;DR

This paper provides a precise criterion for when a group acting on a topological space contains a free subgroup of rank 2 that acts properly discontinuously on some invariant subspace, extending understanding of free subgroup actions.

Contribution

It establishes a necessary and sufficient condition for the existence of free subgroups of rank 2 acting properly discontinuously within group actions on topological spaces.

Findings

Characterizes free subgroups of rank 2 acting properly discontinuously

Provides a criterion for free subgroup existence in topological group actions

Specializes to free subgroups acting freely on orbits in discrete cases

Abstract

Given an action of a group G on a topological space X, we establish a necessary and sufficient condition for the existence of a free subgroup F of rank 2 of G acting properly discontinuously on at least one nonempty, open, F-invariant subspace of X. In the case of a discrete topology (group action on a set X), the condition simply detects free subgroups of rank 2 acting freely on some orbit.