On 3-strand singular pure braid group

TL;DR

This paper investigates the algebraic structure of the singular pure braid group for n=2,3, providing generators, relations, and a detailed decomposition of SP_3 as a semi-direct product with insights into its center.

Contribution

It explicitly describes the structure of SP_3 as a semi-direct product involving an HNN-extension and analyzes its center, advancing understanding of singular pure braid groups.

Findings

SP_3 is a semi-direct product of an HNN-extension and Z.

The center of SP_3 is a direct factor.

Generators and relations for SP_2 and SP_3 are established.

Abstract

In the present paper we study the singular pure braid group $SP_{n}$ for $n=2, 3$. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that $SP_{3}$ is a semi-direct product $SP_{3} = \widetilde{V}_3 \leftthreetimes \mathbb{Z}$, where $\widetilde{V}_3$ is an HNN-extension with base group $\mathbb{Z}^2 * \mathbb{Z}^2$ and cyclic associated subgroups. We prove that the center $Z(SP_3)$ of $SP_3$ is a direct factor in $SP_3$.