  We prove that the braided Thompson's groups $V_{\rm br}$ and $F_{\rm br}$ are
of type $F_\infty$, confirming a conjecture by John Meier. The proof involves
showing that matching complexes of arcs on surfaces are highly connected. In an
appendix, Zaremsky uses these connectivity results to exhibit families of
subgroups of the pure braid group that are highly generating, in the sense of
Abels and Holz.
