Geometric structures related to the braided Thompson groups

TL;DR

This paper proves that certain geometric complexes related to braided Thompson groups are CAT(0), enabling the computation of their geometric invariants and revealing new finiteness properties of their subgroups.

Contribution

It establishes that the complexes are CAT(0) and computes the full set of geometric invariants for the pure braided Thompson group.

Findings

The complexes are CAT(0).

Computed all geometric invariants $ ext{Sigma}^m(F_{br})$.

Subgroups containing the commutator subgroup are of type $ ext{F}_ extinfty$ if finitely presented.

Abstract

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $\Sigma^m(F_{\textrm{br}})$ of the pure braided Thompson group $F_{\textrm{br}}$. Only the first invariant $\Sigma^1(F_{\textrm{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\textrm{br}}$ containing the commutator subgroup $[F_{\textrm{br}},F_{\textrm{br}}]$ is finitely presented, it is automatically of type $\textrm{F}_\infty$.