Thompson groups for systems of groups, and their finiteness properties

TL;DR

This paper introduces a method to construct generalized Thompson groups from families of groups with cloning systems, exploring their finiteness properties and providing new examples including matrix and braid groups.

Contribution

It extends the cloning system framework to new group families and analyzes how their finiteness properties influence the resulting generalized Thompson groups.

Findings

New examples of groups with cloning systems, such as upper triangular matrices and loop braid groups.

Finiteness length of the generalized Thompson group equals the limit inferior of the original groups' finiteness lengths.

Develops methods to determine finiteness properties for groups over rings of S-integers in global function fields.

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, Vbr and Fbr 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.