The braided Thompson's groups are of type $F_\infty$

TL;DR

This paper proves that the braided Thompson's groups $V_{br}$ and $F_{br}$ have a highly complex algebraic structure, confirming a long-standing conjecture and employing topological methods involving surface arc complexes.

Contribution

It establishes that $V_{br}$ and $F_{br}$ are of type $F_$, using connectivity of matching complexes, and explores subgroup generation in pure braid groups.

Findings

$V_{br}$ and $F_{br}$ are of type $F_$

Matching complexes of arcs are highly connected

Subgroups of pure braid groups are highly generating

Abstract

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.