Finiteness Properties of Some Groups of Local Similarities

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∞. This generalizes work of Ken Brown from the 1980s. Next, we develop methods for distinguishing between isomorphism types of some of the Nekrashevych-Röver groups Vd(H), where H is a finite group, and show that all such groups Vd(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.