On some decompositions of the 3-strand Singular Braid Group

TL;DR

This paper studies the structure of the group $ST_3$, showing it is isomorphic to the singular pure braid group on three strands and decomposes as a semi-direct product involving an HNN-extension.

Contribution

It provides a presentation for $ST_3$, proves its isomorphism to $SP_3$, and describes its semi-direct product decomposition, advancing understanding of singular braid groups.

Findings

$ST_3$ is isomorphic to $SP_3$

$ST_3$ decomposes as a semi-direct product

$H$ is an HNN-extension of ${f Z}^2 imes {f Z}^2$

Abstract

Let $SB_n$ be the singular braid group generated by braid generators $\sigma_i$ and singular braid generators $\tau_i$, $1 \leq i \leq n-1$. Let $ST_n$ denote the group that is the kernel of the homomorphism that maps, for each $i$, $\sigma_i$ to the cyclic permutation $(i, i+1)$ and $\tau_i$ to $1$. In this paper we investigate the group $ST_3$. We obtain a presentation for $ST_3$. We prove that $ST_3$ is isomorphic to the singular pure braid group $SP_3$ on $3$ strands. We also prove that the group $ST_3$ is semi-direct product of a subgroup $H$ and an infinite cyclic group, where the subgroup $H$ is an HNN-extension of ${\mathbb Z}^2 \ast {\mathbb Z}^2$.