Braided Diagram Groups and Local Similarity Groups

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.