Free orbits for minimal actions on the circle

TL;DR

This paper proves that certain countable groups acting minimally and faithfully on the circle must have a free orbit, highlighting a distinction from actions on the line, with implications for group actions and dynamics.

Contribution

It establishes the existence of free orbits for a class of groups acting on the circle, extending understanding of group dynamics and minimal actions.

Findings

Groups without Z^2 subgroups have free orbits on the circle.

The result does not extend to actions on the line.

Examples show the limitation of the theorem for line actions.

Abstract

We prove that if $\Gamma$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line.