Groups and Dynamics Seminar

Date: March 19, 2025

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Santiago Radi Severo, Texas A&M University

Title: On finite generation, the congruence subgroup property and just-infiniteness in groups of finite type

Abstract: Groups of finite type (also known as finitely constrained groups) are closed subgroups of Aut(T), the automorphism group of a regular rooted tree T, whose action locally around every vertex is determined by a finite group of allowed actions. They were introduced in 2005 by Grigorchuk, who proved that the closure of regular branch groups belongs to this class. In 2006, Sunic proved the converse. In the study of groups acting on rooted trees, three important notions play a significant role: the congruence subgroup property (CSP), just-infiniteness (j-oo) and topologically finitely generation (tfg). For instance, if CSP holds, then the group is isomorphic to its profinite completion.
In my talk, I will prove that these three notions are equivalent for groups of finite type that satisfy the so-called Property (E), a property that will be developed in the talk and seems to be true for any group of finite type. As a consequence of this result, it will be shown that the Hanoi tower group in 3 pegs, a group introduced in 2006 by Grigorchuk and Sunic, and known not to be just-infinite, unexpectedly has a just-infinite closure.