Schedule for: 23w5044 - Group Actions on Cantor Sets

A just infinite group is an infinite group without infinite proper quotients. A group is said to be hereditarily just infinite (HJI) if all of its finite index subgroups are just infinite. A classical classification theorem of Grigorchuk-Wilson states that a residually finite just infinite group is either: (a) a branch group or (b) an HJI group. Branch groups have been extensively studied (e.g. Grigorchuk group), but HJI groups remain a very mysterious class. In this talk, I will report on some recent work on the search for new HJI subgroups of iterated wreath products in product action.

In the first part of the talk I will focus on the maximal, weakly maximal and finitely generated subgroups of branch groups. A Cantor set in this case is the boundary of a rooted tree on which group acts by automorphisms. In the second part I will talk about maximal subgroups of TFG's (Topological Full Groups). Here the Cantor set is a cornerstone of the definition of TFG. The results presented in the first part are based on joint articles of the speaker with L.Bartholdi, D.Francoeur, P-H.Leemann and T.Nagnibeda. The second part is based on a joint work with Y.Vorobets.

We will be interested in self-similar groups acting on the boundary of an infinite regular tree. Schreier graphs of these actions constitute interesting families of graphs from the viewpoint of spectral graph theory providing xamples of co-spectral graphs, graphs admitting exotic types of spectra and spectral measure, etc. Their self-similar nature provides keys to study them, from approximation by finite graphs to renormalization.