Conjugacy in Rearrangement Groups of Fractals

TL;DR

This paper introduces a new method for solving the conjugacy problem in a broad class of fractal rearrangement groups, extending previous techniques from Thompson groups and applying to various complex groups.

Contribution

It generalizes strand diagram methods to a wide class of fractal groups, providing solutions for conjugacy problems in numerous new and existing groups.

Findings

Solved conjugacy problem for multiple fractal rearrangement groups

Extended strand diagram techniques to new classes of groups

Introduced graph rewriting systems with confluence properties

Abstract

We describe a method for solving the conjugacy problem in a vast class of rearrangement groups of fractals, a family of Thompson-like groups introduced in 2019 by Belk and Forrest. We generalize the methods of Belk and Matucci for the solution of the conjugacy problem in Thompson groups $F$, $T$ and $V$ via strand diagrams. In particular, we solve the conjugacy problem for the Basilica, the Airplane, the Vicsek and the Bubble Bath rearrangement groups and for the groups $QV$ (also known as $QAut(\mathcal{T}_{2,c})$), $\tilde{Q}V$, $QT$, $\tilde{Q}T$ and $QF$, and we provide a new solution to the conjugacy problem for the Houghton groups and for the Higman-Thompson groups, where conjugacy was already known to be solvable. Our methods involve two distinct rewriting systems, one of which is an instance of a graph rewriting system, whose confluence in general is of interest in computer science.