A new family of infinitely braided Thompson's groups

TL;DR

This paper introduces a broad family of braided Thompson's groups using recursive braids, providing a new diagrammatic approach and establishing finite generation under certain conditions.

Contribution

It generalizes the Dehornoy-Brin braided Thompson group by defining new groups $BV_{n,r}(H)$ with recursive braids and strand diagrams, expanding the theoretical framework.

Findings

$BV_{n,r}(H)$ is finitely generated if $H$ is finitely generated

Introduces a new strand diagram approach for braided Thompson groups

Defines a broad family of groups using recursive braids

Abstract

We present a generalization of the Dehornoy-Brin braided Thompson group $BV_2$ that uses recursive braids. Our new groups are denoted by $BV_{n,r}(H)$, for all $n\geq 2,r\geq 1$ and $H \leq \mathcal{B}_n$, where $\mathcal{B}_n$ is the braid group on $n$ strands. We give a new approach to deal with braided Thompson groups by using strand diagrams. We show that $BV_{n,r}(H)$ is finitely generated if $H$ is finitely generated.