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.