Structural aspects of twin and pure twin groups

The twin group $T_n$ is a Coxeter group generated by $n-1$ involutions and the pure twin group $P{T}_n$ is the kernel of the natural surjection of $T_n$ onto the symmetric group on $n$ letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group $T_n$ decomposes into a free product with amalgamation for $n>4$. It is shown that the pure twin group $P{T}_n$ is free for $n=3,4$, and not free for $n\ge 6$. We determine a generating set for $P{T}_n$, and give an upper bound for its rank. We also construct a natural faithful representation of $T_4$ into $\mathrm{Aut}(F_7)$. In the end, we propose virtual and welded analogues of these groups and some directions for future work.