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 PT _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 PT _n is free for n= 3 , 4 , and not free for n≥ 6. We determine a generating set for PT _n , 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.