Geometric structures related to the braided Thompson groups

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type ${\text{F}}_{\mathrm{\infty }}$. 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 ${\mathrm{\Sigma }}^m(F_{\text{br}})$ of the pure braided Thompson group $F_{\text{br}}$. Only the first invariant ${\mathrm{\Sigma }}^1(F_{\text{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\text{br}}$ containing the commutator subgroup $[F_{\text{br}},F_{\text{br}}]$ is finitely presented, it is automatically of type ${\text{F}}_{\mathrm{\infty }}$.