Braiding groups of automorphisms and almost-automorphisms of trees

Abstract

We introduce "braided" versions of self-similar groups and Röver-Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call "self-identical". In particular we use a braided version of the Grigorchuk group to construct a new group called the braided Röver group, which we prove is of type F ∞ . Our techniques involve using so called -ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.