WoG 2024 Talk 4.1: Matteo Tarocchi - Rearrangement Groups of Fractals and their Conjugacy Problem

Speaker: Matteo Tarocchi
Institution: University of Milano-Bicocca
Website: sites.google.c...
Title: Rearrangement Groups of Fractals and their Conjugacy Problem
Abstract: In 2019 J. Belk and B. Forrest introduced the family of Rearrangement Groups. These are groups of certain "piecewise-canonical" homeomorphisms of many fractals that act by "canonically" permuting the self-similar pieces that make up the fractal. In particular, this family includes the famous trio Richard Thompson groups, which are groups of piecewise-linear homeomorphisms of the unit interval, the unit circle and the Cantor space, respectively. Despite being countable, rearrangement groups seem to often be dense in the group of all homeomorphisms of the fractal on which they act.
Known results about rearrangement groups include the simplicity of commutator subgroups in many examples, a general result about invariable generation, rationality of the fractal spaces on which they act and a method to tackle their conjugacy problem. This talk will introduce this family of groups and highlight some facts about them, focusing on the solution to the conjugacy problem.