Perfect kernel and dynamics: from Bass-Serre theory to hyperbolic groups

TL;DR

This paper explores the topological and dynamical properties of subgroup spaces in various countable groups, revealing new insights into their structure, perfect kernels, and actions, with applications to groups admitting transitive amenable actions.

Contribution

It introduces new methods to analyze the Cantor-Bendixson decomposition and dynamics on subgroup spaces for a wide class of groups, including hyperbolic and graph of groups.

Findings

Determined the perfect kernel and Cantor-Bendixson rank for many new groups.

Established conditions for topological transitivity of conjugation actions.

Identified numerous groups in class A with faithful transitive amenable actions.

Abstract

We introduce several approaches to studying the Cantor-Bendixson decomposition of and the dynamics on the (topological) space of subgroups for various families of countable groups. In particular, we uncover the perfect kernel and the Cantor-Bendixson rank of the space of subgroups of many new groups, including for instance infinitely ended groups, limit groups, hyperbolic 3-manifold groups and many graphs of groups. We also study the topological dynamics of the conjugation action on the perfect kernel, establishing the conditions for topological transitivity and higher topological transitivity. As an application, we obtain many new examples of groups in the class A of Glasner and Monod, i.e. admitting faithful transitive amenable actions. This includes for example right-angled Artin groups, limit groups, finitely presented C'(1/6) small cancellation groups, random groups at density d<1/6, and more generally all virtually compact special groups.