  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, C'(1/6) small cancellation groups, random groups at
density d<1/6, and more generally all virtually compact special groups.
