Thompson groups for systems of groups, and their finiteness properties

Abstract

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F , V , V br and F br arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups.We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of S-integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.

Almost-automorphisms of trees, cloning systems and finiteness properties

“…In [WZ14], the framework of cloning systems was established to encode certain families of groups into a Thompson-like group. Examples include the classical Thompson groups F and V , the braided Thompson groups of Brin and Dehornoy [Bri07,Deh06], Thompson groups for upper triangular matrix groups [WZ14] and the generalized Thompson groups of Tanushevski [Tan16]. In this section we establish d-ary analogs, which we call d-ary cloning systems, and show that AAut(T d ) and any Nekrashevych group V d (G) arise as Thompson-like groups of d-ary cloning systems.…”

“…Usually the groups are of type F ∞ , e.g., see [BM16, Bro87, BFM + 16, FH15, FMWZ13, MPMN16, NSJG14, Thu17], though not always, e.g., Belk-Forrest's basilica Thompson group T B [BF15b] is type F 1 but not F 2 [WZ16]. Also, it is possible to build ad hoc Thompson-like groups with arbitrary finiteness properties, using "cloning systems" [WZ14].…”

“…In this paper we prove that AAut(T d ) arises naturally from a "d-ary cloning system" on the wreath products S n ≀ Aut(T d ). A d-ary cloning system is a natural generalization of the notion of cloning system from [WZ14], which is the d = 2 case. Similarly, for any self-similar G ≤ Aut(T d ) the Nekrashevych group V d (G) arises from a d-ary cloning system on the groups S n ≀ G. The results are as follows (with the notation T (−) explained in Subsection 2.1).…”

“…It is still an open problem to find examples in the world of Thompson groups of simple groups of type F n but not F n+1 , for n ≥ 2. Since there is some machinery in place to determine precise finiteness properties of Thompson groups arising from cloning systems (see for example [WZ14,Theorem 8.28]), it seems possible that certain V d (G) could provide such examples in the future. This paper is organized as follows.…”

Simple groups separated by finiteness properties

“…Often, when a group is of type F n−1 but not of type F n it has a natural action on an n-dimensional space that can be utilized, but no such action is available to us. In the few known examples of generalized Thompson groups that are not of type F ∞ , individually tailored arguments have been used; see for example [WZb,Section 8.2] and [WZa]. Here we employ a more robust strategy using quasi-retracts.…”