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

Abstract

We prove that the braided Thompson's groups $V_{\mathrm{br}}$ and $F_{\mathrm{br}}$ 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.

arXiv comment51 pages, 24 figures, v3 including erratum

References in corpus (5)

• The Algebra of Strand Splitting. I. A Braided Version of Thompson's Group V
• The Algebra of Strand Splitting. II. A Presentation for the Braid Group on One Strand
• Conjugacy and Dynamics in Thompson's Groups
• The braided Ptolemy-Thompson group is finitely presented
• The Brin-Thompson groups sV are of type F_\infty

Cited by in corpus (12)

• Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups
• Twisted Brin-Thompson groups
• Almost-automorphisms of trees, cloning systems and finiteness properties
• Classifying spaces from Ore categories with Garside families
• Braided Thompson groups with and without quasimorphisms
• Between buildings and free factor complexes: A Cohen-Macaulay complex for Out(RAAGs)
• On the finiteness length of some soluble linear groups
• Braiding groups of automorphisms and almost-automorphisms of trees
• Divergence function of the braided Thompson group
• Thompson-like groups, Reidemeister numbers, and fixed points
• Higher generating subgroups and Cohen-Macaulay complexes
• Matching arc complexes: connectedness and hyperbolicity