On some decompositions of the 3-strand Singular Braid Group

Let $SB_n$ be the singular braid group generated by braid generators $σ_i$ and singular braid generators $τ_i$, $1 \leq i \leq n-1$. Let $ST_n$ denote the group that is the kernel of the homomorphism that maps, for each $i$, $σ_i$ to the cyclic permutation $(i, i+1)$ and $τ_i$ 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 ${\mathbb Z}^2 \ast {\mathbb Z}^2$.

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