Finiteness properties of some groups of local similarities

TL;DR

This paper studies FSS groups acting on ultrametric spaces, proving many are of type F-infinity, distinguishing their isomorphism types, and connecting them to braided diagram groups, thus advancing understanding of their algebraic and geometric properties.

Contribution

It establishes that a class of FSS groups are of type F-infinity, generalizes previous work on Thompson's group V, and links FSS groups to braided diagram groups over tree-like semigroup presentations.

Findings

A class of FSS groups are of type F-infinity.

All V_{d}(H) groups have simple subgroups of finite index.

FSS groups with small Sim-structures are braided diagram groups.

Abstract

Hughes has defined a class of groups, which we call FSS (finite similarity structure) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson's group V. Guided by previous work on Thompson's group V, we establish a number of new results about FSS groups. Our main result is that a class of FSS groups are of type F-infinity. This generalizes work of Ken Brown from the 1980s. Next, we develop methods for distinguishing between isomorphism types of some of the Nekrashevych-R\"over groups V_{d}(H), where H is a finite group, and show that all such groups V_{d}(H) have simple subgroups of finite index. Lastly, we show that FSS groups defined by small Sim-structures are braided diagram groups over tree-like semigroup presentations. This generalizes a result of Guba and Sapir, who first showed that Thompson's group V is a braided diagram group.