Pure infinitely braided Thompson groups (Q6636334)

The braided Thompson group \(BV\) was independently introduced by \textit{M. G. Brin} [J. Group Theory 10, No. 6, 757--788 (2007; Zbl 1169.20021)] and \textit{P. Dehornoy} [Adv. Math. 205, No. 2, 354--409 (2006; Zbl 1160.20027)] and the pure version of the group, denoted \(BF\), by \textit{T. Brady} et al. [Publ. Mat., Barc. 52, No. 1, 57--89 (2008; Zbl 1185.20043)]. Natural infinite families \(BV_n\) and \(BF_n\) are obtained by taking \(n\)-ary trees instead of binary trees. A generalization of \(BV_n\), denoted \(BV_n(H)\) for \(H\) a subgroup of the braid group \(B_n\), was introduced by \textit{J. Aroca} and \textit{M. Cumplido} [J. Algebra 607, 5--34 (2022; Zbl 1511.20146)], which allows for braids to be recursively nested, in a sense. The focus of the present paper is the pure version of this, \(BF_n(H)\), for \(H\) a subgroup of the pure braid group \(P_n\). One of the main results (Theorem~4) is that \(BF_n(H)\) is bi-orderable for any \(H\le P_n\). This generalizes work of \textit{T. Ishida} [Commun. Algebra 46, No. 9, 3806--3809 (2018; Zbl 1392.20034)] proving that \(BF=BF_2(\{1\})\) is bi-orderable. Another main result (Theorem~9) is to produce an explicit finite generating set for \(BF_n(H)\) for any finitely generated \(H\le P_n\).