Conjugacy in Rearrangement Groups of Fractals

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.