Damien Gaboriau

Dynamics on the Space of Subgroups

To every countable group G one can attach a curious and rather mysterious topological object: the space of all its subgroups. This space, denoted Sub(G), is compact and totally disconnected, a kind of “mathematical universe” in which every possible subgroup of G appears as a point. The group G then acts on this space by conjugation, giving rise to a natural dynamical system.

One can explore this system by peeling away the isolated points of Sub(G) step by step, a process that eventually reveals its perfect kernel—the irreducible core that cannot be broken down any further—and a numerical invariant known as the Cantor–Bendixson rank.

In this talk, I will describe what these objects mean, why they are interesting, and what kinds of dynamics one can observe: from orderly behavior to more surprising phenomena of chaos.

Along the way, we will look at concrete families of groups -- such as abelian groups, hyperbolic groups and the more subtle (and sometimes rather tricky) Baumslag—Solitar groups -- where these phenomena can be seen in action.

This relies on joint works with P. Azuelos, S. Bontemps, A. Carderi, F. Le Maître, and Y. Stalder.