Rearrangement groups of connected spaces

Abstract

We develop a combinatorial framework that assists in finding natural infinite “geometric” presentations for a large subclass of rearrangement groups of fractals – defined by Belk and Forrest, namely rearrangement groups acting on F-type topological spaces. In this framework, for a given fractal set with its group of “rearrangements”, the group generators have a natural one-to-one correspondence with the standard basis of the fractal set, and the relations are all conjugacy relations.

We use this framework to produce a presentation for Richard Thompson’s group F. This presentation has been mentioned before by Dehornoy, but a combinatorial method to find the length of an element in terms of the generating set of this presentation has been hitherto unknown. We provide algorithms that express an element of F in terms of our generating set and reduce a word representing the identity in F to the trivial word.

We conjecture that this framework can be used to find infinite presentations for all groups in the subclass of rearrangement groups acting on F-type topological spaces.