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 PTn 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 PTn is free for n = 3, 4, and not free for n >= 6. We determine a generating set for PTn, and give an upper bound for its rank. We also construct a natural faithful representation of T-4 into Aut(F-7). In the end, we propose virtual and welded analogues of these groups and some directions for future work.