Finiteness properties for relatives of braided Higman--Thompson groups

Abstract

We study the finiteness properties of the braided Higman--Thompson group $b{V}_{d,r}(H)$ with labels in $H\le B_d$, and $b{F}_{d,r}(H)$ and $b{T}_{d,r}(H)$ with labels in $H\le P{B}_d$ where $B_d$ is the braid group with $d$ strings and $P{B}_d$ is its pure braid subgroup. We show that for all $d\ge 2$ and $r\ge 1$, the group $b{V}_{d,r}(H)$ (resp. $b{T}_{d,r}(H)$ or $b{F}_{d,r}(H)$) is of type $F_n$ if and only if $H$ is. Our result in particular confirms a recent conjecture of Aroca and Cumplido.

25 pages;first part of arXiv:2103.14589v1, the second part to appear separately. v2, fixed a gap in Section 3.4, to appear in Groups Geom. Dyn..; v3, final version. arXiv admin note: text overlap with arXiv:1210.2931. text overlap with arXiv:2101.03462 by other authors

References in corpus (3)

• Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups
• The homology of Richard Thompson's group F
• Homological stability for the ribbon Higman--Thompson groups

Cited by in corpus (1)

• Homological stability for the ribbon Higman--Thompson groups