A Geometric Interpretation of the Normal Closure of the Braid Group $B_n$ in the braid group of the torus $B_n(T)$

TL;DR

This paper explores the geometric structure of the normal closure of the full braid group of the disk within the braid group of the torus, extending previous results on pure braid groups.

Contribution

It provides a new geometric interpretation of the normal closure of the full braid group of the disk in the braid group of the torus.

Findings

Normal closure of $B_n(D)$ in $B_n(T)$ has an interesting geometric description.

Builds on previous work on pure braid groups to analyze full braid groups.

Extends the understanding of braid group structures on surfaces.

Abstract

Combining the results by Birman and Goldberg, it was proved the normal closure of the pure braid group of the disk $P_n(D)$ in the pure braid group of the torus $P_n(T)$ is the commutator subgroup $[P_n(T),P_n(T)]$. In this paper we are going to study the case for full braid groups: i.e. the normal closure of $B_n(D)$ in $B_n(T)$, which turns out to have an interesting geometric description.