Braided Diagram Groups and Local Similarity Groups

TL;DR

This paper establishes a precise correspondence between braided diagram groups over tree-like semigroup presentations and groups acting as local similarities on compact ultrametric spaces, unifying two previously separate classes.

Contribution

It proves that braided diagram groups over tree-like semigroup presentations are exactly the groups acting on ultrametric spaces via small similarity structures, generalizing known results about Thompson's group V.

Findings

Equivalence of braided diagram groups and local similarity groups over ultrametric spaces.

Inclusion of Houghton groups and quasi-automorphism groups in both classes.

Both classes act properly by isometries on CAT(0) cubical complexes.

Abstract

Hughes defined a class of groups that act as local similarities on compact ultrametric spaces. Guba and Sapir had previously defined braided diagram groups over semigroup presentations. The two classes of groups share some common characteristics: both act properly by isometries on CAT(0) cubical complexes, and certain groups in both classes have type F-infinity, for instance. Here we clarify the relationship between these families of groups: the braided diagram groups over tree-like semigroup presentations are precisely the groups that act on compact ultrametric spaces via small similarity structures. The proof can be considered a generalization of the proof that Thompson's group V is a braided diagram group over a tree-like semigroup presentation. We also prove that certain additional groups, such as the Houghton groups, and a certain group of quasi-automorphisms lie in both classes.