In this paper, we prove the rationality of the gluing relation of edge replacement systems, which were introduced for studying rearrangement groups of fractals. More precisely, we describe an algorithmic procedure for building a finite state automaton that recognizes pairs or equivalent sequences that are glued in the fractal. This fits in recent interest towards the rationality of gluing relations on totally disconnected compact metrizable spaces.

We study a family of Thompson-like groups built as rearrangement groups of fractals from [BF19], each acting on a Wazewski dendrite. Each of these is a finitely generated group that is dense in the full group of homeomorphisms of the dendrite (studied in [DM19]) and has infinite-index finitely generated simple commutator subgroup, with a single possible exception.

More properties are discussed, including finite subgroups, the conjugacy problem, invariable generation and existence of free subgroups. We discuss many possible generalizations, among which we find the Airplane rearrangement group TA.

Despite close connections with Thompson's group F, dendrite rearrangement groups seem to share many features with Thompson's group V.

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, Q̃V, QT, Q̃T 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.

A group G is invariably generated if there exists a subset S ⊆ G such that, for every choice gs ∈ G for s ∈ S, the group G is generated by { sgs ∣ s ∈ S }. In [GGJ16] Gelander, Golan and Juschenko showed that Thompson groups T and V are not invariably generated. Here we generalize this result to the larger setting of rearrangement groups, proving that any subgroup of a rearrangement group that has a certain transitive property is not invariably generated. 

We study the group TA of rearrangements of the Airplane limit space introduced by Belk and Forrest in [3]. We prove that TA is generated by a copy of Thompson's group F and a copy of Thompson's group T, hence it is finitely generated. Then we study the commutator subgroup [TA, TA], proving that the abelianization of TA is isomorphic to Z and that [TA, TA] is simple, finitely generated and acts 2-transitively on the so-called components of the Airplane limit space. Moreover, we show that TA is contained in T and contains a natural copy of the Basilica rearrangement group TB studied in [2].