Mathematische Annalen, Volume 391, pages 4733–4789, (2025)
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.