Groups and Dynamics Seminar

Date: March 27, 2024

Time: 3:00PM - 4:00PM

Location: BLOC 123

Speaker: Jorge Fariña Asategui, Lund University, Sweden

Title: On the Hausdorff dimension of self-similar and branch profinite groups

Abstract:

Groups acting on regular rooted trees provide easy examples of groups with exotic properties such as Burnside groups and groups of intermediate growth. Of particular interest are branch profinite groups as they constitute one of the two classes partitioning the class of just infinite profinite groups. Based on the work of Abercrombie, Barnea and Shalev started the study of the Hausdorff dimension on profinite groups. The Hausdorff dimension of self-similar profinite groups is still the object of several open problems.

The first part of this talk is devoted to introducing groups acting on regular rooted trees and the Hausdorff dimension of their closures. Then we introduce a new tool to compute the Hausdorff dimension of the closure of a self-similar group. Using this new tool we solve an open problem of Grigorchuk on the self-similar Hausdorff spectrum of the group of q-adic automorphisms. Indeed, we completely determine the Hausdorff spectra of the group of q-adic automorphisms restricted to different classes of closed subgroups. We also solve a well-known open problem of Boston on the Hausdorff dimension of just infinite branch pro-p groups. Lastly, if time permits, we will discuss some open problems on the finitely generated Hausdorff spectrum of branch profinite groups and some new results in this direction on an ongoing joint project with Garaialde Ocaña and Uria-Albizuri.