Dense and comeager conjugacy classes in zero-dimensional dynamics

TL;DR

This paper systematically studies the structure of minimal and topologically transitive actions of countable groups on the Cantor space, focusing on the existence of dense and comeager conjugacy classes.

Contribution

It introduces a model-theoretic framework for analyzing conjugacy classes in group actions, with new results on free groups, amenable groups, and hyperbolic groups.

Findings

Comeager conjugacy class exists for free groups in minimal actions.

No comeager conjugacy class for non-finitely generated amenable groups.

Existence of dense conjugacy class in hyperbolic groups iff the group is virtually cyclic.

Abstract

Given a countable group $G$, we initiate a systematic study of the Polish spaces of all minimal and topologically transitive actions of $G$ on the Cantor space by homeomorphisms, with a focus on the existence of comeager conjugacy classes in these spaces. We develop a general model-theoretic framework to study this and related questions, recovering on the way many existing results from the literature. A substantial part of the paper is devoted to actions of free groups. We show that in that case, there is a comeager conjugacy class in the space of minimal actions, as well as in the space of minimal, probability measure-preserving actions. The first one is the Fra\"iss\'e limit of all sofic minimal subshifts and the second, the universal profinite action. The case of the integers was already treated by Hochman and there the two actions coincide with the universal odometer. In the non-abelian case, they are substantially different and new techniques are required. In the opposite direction, if $G$ is an amenable group which is not finitely generated, we show that there is no comeager conjugacy class in the space of all actions, and if $G$ is locally finite, also in the space of minimal actions. Finally, we study the question of existence of a dense conjugacy class in the space of topologically transitive actions. We show that if $G$ is hyperbolic or virtually polycyclic, then such a dense conjugacy class exists iff $G$ is virtually cyclic, suggesting that the case of the integers may be exceptional.