Finiteness Properties of Some Groups of Local Similarities

Abstract

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∞. 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 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.

Röver's simple group is of type $F_\infty$

“…The three Thompson groups F , T , and V have type F ∞ [4,7], as do many of their variants such as the generalized groups F n,k , T n,k and V n,k [4], certain diagram groups [12] and picture groups [14], braided Thompson groups [8], higher-dimensional groups nV [16,20], and various other generalizations [1,10,11,15,21,22].…”

Some isomorphism results for Thompson-like groups V n (G)

“…Our first theorem is perhaps surprising in that many experts had the opinion that there would be many isomorphism types for groups V n (G), with G a semiregular subgroup of S n (e.g., see [8], where such an expectation is expressed at the end of Section 7, in the discussion following Example 7.24). Theorem 1.…”

“…Elizabeth Scott in her research [19,20,21] describes the first such family of groups which are developed further by Claas Röver, specifically including an extension of V with the Grigorchuk group Γ. The second family are the finite similarity structure groups of Hughes (FSS groups), which are the focus of study in [14,8].…”

“…The class of groups {V n (G)} is first explicitly studied in [8]. Farley and Hughes show that if G is not semiregular, then V n (G) is not isomorphic to V n and that the commutator subgroup V n (G)…”

“…We can mention the papers of [17,16,8,1,22] which all explore properties of such groups. Our own perspective has been heavily influenced by the dynamical methods which have arisen through the use of Rubin's theorem [15,18] and, as will be seen below, through the interaction between the theory of the extended Thompson type groups and the theory of groups of automata (c.f., [11]).…”