On 3-strand singular pure braid group

Abstract

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

18 pages, 1 figure