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The braided Thompson’s groups are of type $$F_\infty$$. (English) Zbl 1397.20053
Summary: We prove that the braided Thompson’s groups $$V_{\mathrm{br}}$$ and $$F_{\mathrm{br}}$$ are of type $$\mathrm{F}_{\infty}$$, 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 H. Abels and S. Holz [J. Algebra 160, No. 2, 310–341 (1993; Zbl 0811.20036)].

##### MSC:
 20F65 Geometric group theory 20F36 Braid groups; Artin groups 20J05 Homological methods in group theory 57M07 Topological methods in group theory 20E32 Simple groups
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