Braiding groups of automorphisms and almost-automorphisms of trees

TL;DR

This paper introduces braided versions of self-similar and R"over--Nekrashevych groups, demonstrating their finiteness properties and constructing new groups with advanced algebraic and topological techniques.

Contribution

It generalizes existing work by defining braided groups, uses $d$-ary cloning systems, and constructs a new braided R"over group of type $F__$, expanding the understanding of automorphism groups of trees.

Findings

Braided R"over group is of type $F__$.

Use of $d$-ary cloning systems to construct groups.

Analysis of disk complexes in surfaces to determine finiteness.

Abstract

We introduce "braided" versions of self-similar groups and R\"over--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 R\"over group, which we prove is of type $F_\infty$. Our techniques involve using so called $d$-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.