Finiteness properties for relatives of braided Higman--Thompson groups

Abstract

We study the finiteness properties of the braided Higman-Thompson group bV d,r (H) with labels in H ≤ B d , and bF d,r (H) and bT d,r (H) with labels in H ≤ 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 ≥ 2 and r ≥ 1, the group bV d,r (H) (resp. bT d,r (H) or bF d,r (H)) is of type Fn if and only if H is. Our result in particular confirms a recent conjecture of Aroca and Cumplido. We then generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman-Thompson groups, answering a question of Aramayona and Vlamis in the case of the Higman-Thompson groups. When the underlying surface is a disk, these new asymptotic mapping class groups can be identified with the ribbon Higman-Thompson groups. We use this model to prove that the ribbon Higman-Thompson groups satisfy homological stability, providing the first homological stability result for dense subgroups of big mapping class groups.

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

“…Are they of type F 8 ? Along these lines, recent independent work by Genevois-Lonjou-Urech [28] and Skipper-Wu [56] proves that some related families of groups are F 8 .…”

Braided Thompson groups with and without quasimorphisms

“…The first braided Thompson group, which we denote by bV and which has also been denoted by BV , V br , and brV in the literature, was introduced independently by Brin [14] and Dehornoy [24], as a braided version of Thompson's group V . Other braided Thompson groups include the "F -like" pure braided Thompson groups bF [12], various "T -like" braided Thompson groups [33,34,59], braided Higman-Thompson groups bV n [4,53], braided Brin-Thompson groups sV br [56], the "ribbon braided" Thompson group rV [57] and braided Röver-Nekrashevych groups brV d (G) [55]. Most relevant to our purposes here is a close relative bV of bV , which was also introduced by Brin in [14] (there denoted BV ), and realised up to isomorphism as a concrete subgroup of bV by Brady-Burillo-Cleary-Stein [12]; see also [18].…”

“…We will always stick to the "n = 2 case" to avoid getting bogged down in notation, but the reader should note that all of our results can be adapted to the braided Higman-Thompson groups bV n (as in [4,53]) and their analogous subgroups bV n , with appropriate small modifications to the arguments. It would be interesting to try and adapt our arguments to other, more complicated Thompson-like groups related to asymptotically rigid mapping class groups, e.g., for positive genus surfaces in [2], or for higher-dimensional manifolds in [1].…”