On Groups $G_{n}^{2}$ and Coxeter Groups

TL;DR

This paper proves that the group of free n-strand braids, $G_{n}^{2}$, is isomorphic to a subgroup of a semidirect product involving a specific Coxeter group and the symmetric group, revealing new structural insights.

Contribution

It establishes an isomorphism between $G_{n}^{2}$ and a subgroup of a semidirect product with a Coxeter group, linking braid groups to Coxeter group structures.

Findings

$G_{n}^{2}$ is isomorphic to a subgroup of a semidirect product involving $C(n,2)$ and $S_{n}$

Provides a new structural understanding of free braid groups

Connects braid groups with Coxeter group theory

Abstract

In the present paper, we prove that the group $G_{n}^{2}$ of free $n$-strand braids is isomorphic to a subgroup of a semidirect product of some Coxeter group that we denote by $C(n,2)$ and the symmetric group $S_{n}$.