Groups acting distally and minimally on $\mathbb S^2$ and $\mathbb {RP}^2$

TL;DR

This paper proves that any finitely generated group acting minimally and distally on the 2-sphere or real projective plane must contain a nonabelian free subgroup, revealing structural constraints on such group actions.

Contribution

It establishes a new link between minimal distal actions and the algebraic structure of the acting group on $\

Findings

Finitely generated groups acting minimally and distally on $\

Such groups necessarily contain a nonabelian free subgroup.

Abstract

Let $X$ be the $2$-sphere $\mathbb S^2$ or the real projective plane $\mathbb {RP}^2$. We show that if $\Gamma$ is a finitely generated group acting minimally and distally on $X$, then $\Gamma$ contains a nonabelian free subgroup.