Rearrangement Groups of Fractals

The Basilica Thompson group is not finitely presented

“…It is easy to see (and explained in [BF15b]) that µ extends to a Morse function on the cube complex K(G 0 , e → R) (for any graph G 0 and replacement system e → R). We review the relevant Morse theoretic concepts in the next subsection.…”

“…The Basilica Thompson group is an example of a rearrangement group of a fractal as defined by Belk and Forrest in [BF15b]. These groups arise from edge replacement systems and act naturally on self-similar spaces, for example Julia sets.…”

“…In this section we recall the necessary background material from Sections 1, 2 and 3 of [BF15b] on edge replacement systems, and the resulting groups and complexes. At the end we also recall some background on discrete Morse theory, and on CAT(0) cube complexes.…”

“…Given a base graph G 0 , we will denote the cube complex by K(G 0 , e → R). In [BF15b] it was denoted K(Γ), but for our purposes it is important to keep track of the base graph G 0 used, and less important to keep track of the group Γ. For any choice of base graph G 0 , denote by X(G 0 ) the limit space of the replacement system (G 0 , e → R).…”

The infinite simple group V of Richard J. Thompson: presentations by permutations

“…In many specific cases, F ∞ arguments already exist for individual groups and for classes of groups in these families. See, for example, [1,8,9,17].) The ideas of this paper ought to apply to all of these groups of "Thompson type" in aiding in the discovery of natural and small presentations.…”