A Thompson group for the basilica

Abstract

Abstract. We describe a Thompson-like group of homeomorphisms of the Basilica Julia set. Each element of this group acts as a piecewise-linear homeomorphism of the unit circle that preserves the invariant lamination for the Basilica. We develop an analogue of tree pair diagrams for this group which we call arc pair diagrams, and we use these diagrams to prove that the group is finitely generated. We also prove that the group is virtually simple.

The Basilica Thompson group is not finitely presented

“…J. Belk and B. Forrest [BF15a] introduced the Basilica Thompson group T B (defined in Definition 1.10 below). They showed that it is virtually simple, generated by four elements, and is a sub-as well as a supergroup of Thompson's group T .…”

Rearrangement groups of fractals

“…This group is also a special case of a rearrangement group, where the basilica is realized as a limit space of a sequence of graphs in an appropriate way. We conjectured in [1] that T B is not finitely presented, and this was recently proven by S. Witzel and M. Zaremsky using the CATp0q complex we describe here [19].…”

“…The rearrangement group T B for the basilica replacement system from Example 1.5 is called the basilica Thompson group. In [1] the authors proved the following facts about this group:…”

“…In [1], the authors described a group T B of homeomorphisms of the basilica Julia set, which were defined using piecewise-linear functions on the Böttcher coordinates. This group is also a special case of a rearrangement group, where the basilica is realized as a limit space of a sequence of graphs in an appropriate way.…”

“…Certain Julia sets also arise as limits of sequences of finite Schreier graphs in Bartholdi and Nekrashevych's construction of iterated monodromy groups [17]. However, with the exception of [1], it seems that groups of piecewise-similar homeomorphisms of such spaces have not previously been considered.…”