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 $${{\,\mathrm{F}\,}}_\infty $$ . The proof utilized certain contractible cube complexes, which in this paper we prove are $${{\,\mathrm{CAT}\,}}(0)$$ . We then use this fact to compute the geometric invariants $$\Sigma ^m(F_{{\text {br}}})$$ of the pure braided Thompson group $$F_{{\text {br}}}$$ . Only the first invariant $$\Sigma ^1(F_{{\text {br}}})$$ 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 $${{\,\mathrm{F}\,}}_\infty $$ .