Periodic Groups via Orbital Graphs

Abstract

In this work, we introduce a method of proving when an infinite group of homeomorphisms of a Cantor set is periodic using the geometry of its orbital graphs. In doing so, we expand a recent class of infinite finitely generated periodic groups introduced by Volodymyr Nekrashevych. In particular, we generalize his concept of fragmentation to arbitrary groups of homeomorphisms of a Cantor set, and give examples of finitely generated groups that can be fragmented to produce groups of Burnside type. Although some examples start with a group of isometries of the boundary of an infinite regular rooted tree, the fragmentations of such a group, in general, will not be a group of isometries. It turns out that there is a strong relationship between fragmentations that produce a periodic group and certain subdirect products of a finite product of finite groups. We describe this relationship and give some results on when these types of subdirect products exists. In order to study the orbital graphs of a group, we will realize the Cantor set as a space of infinite sequences, namely, as a space of infinite paths of a Bratelli diagram. Using partial actions of the group on finite paths, we can approximate certain connected infinite subgraphs of an orbital graph using finite graphs. There is a recursive procedure to building these approximating finite graphs described by the defining Bratteli diagram. We can then “paste" together some infinite subgraphs to form the orbital graph. In the best case scenario, a single such infinite subgraph will coincide with the entire orbital graph.