Finiteness properties for relatives of braided Higman--Thompson groups

TL;DR

This paper investigates the finiteness properties of certain braided Higman--Thompson groups, establishing conditions under which these groups are of type $F_n$, and confirming a recent conjecture in the field.

Contribution

It provides a comprehensive analysis of the finiteness properties of braided Higman--Thompson groups with labels in subgroups of braid groups, confirming a conjecture by Aroca and Cumplido.

Findings

$bV_{d,r}(H)$ is of type $F_n$ iff $H$ is of type $F_n$

Similarly for $bT_{d,r}(H)$ and $bF_{d,r}(H)$

The results confirm a recent conjecture by Aroca and Cumplido.

Abstract

We study the finiteness properties of the braided Higman--Thompson group $bV_{d,r}(H)$ with labels in $H\leq B_d$, and $bF_{d,r}(H)$ and $bT_{d,r}(H)$ with labels in $H\leq PB_d$ where $B_d$ is the braid group with $d$ strings and $PB_d$ is its pure braid subgroup. We show that for all $d\geq 2$ and $r\geq 1$, the group $bV_{d,r}(H)$ (resp. $bT_{d,r}(H)$ or $bF_{d,r}(H)$) is of type $F_n$ if and only if $H$ is. Our result in particular confirms a recent conjecture of Aroca and Cumplido.