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 $F$, $V$, $V_{\mathrm{br}}$ and $F_{\mathrm{br}}$ 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.