Topology and dynamics on the space of subgroups

The space of subgroups of a countable group is a compact topological space which encodes many of the properties of its non-free actions. We will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees. We will also give some conditions under which the conjugation action on the perfect kernel is highly topologically transitive and see how this can be applied to find new examples of groups (including all virtually compact special groups) which admit faithful transitive amenable actions. This is joint work with Damien Gaboriau.