We also prove that certain additional groups, such as the Houghton groups Hn, and QAut(T2,c), lie in both classes.

Introduction

In [7], Hughes described a class of groups that act as homeomorphisms on compact ultrametric spaces. Fix a compact ultrametric space X. The essence of the idea was to associate to X a finite similarity structure, which is a function that associates to each ordered pair of balls B1,B2 ⊆ X a finite set SimX(B1,B2) of surjective similarities from B1 to B2. (A similarity is a map that stretches or contracts distances by a fixed constant.) The finite sets SimX(B1,B2) are assumed to have certain desirable closure properties (such as closure under composition). A homeomorphism h : X → X is said to be locally determined by SimX if each x ∈ X has a ball neighborhood B with the property that h(B) is a ball and the restriction of h to B agrees with one of the local similarities σ ∈ SimX(B, h(B)). The collection of all homeomorphisms that are locally determined by SimX forms a group under composition. We will call such a group an FSS group (finite similarity structure group) for short. Hughes [7] proved that each FSS group has the Haagerup property, and even acts properly on a CAT(0) cubical complex. In [5], the authors described a class of FSS groups that have type F∞. That class includes Thompson's group V, and the main theorem of [5] is best understood as a generalization of [2], where Brown originally showed that V has type F∞.