Structural aspects of twin and pure twin groups

Abstract

The twin group Tn is a Coxeter group generated by n−1 involutions and the pure twin group PTn is the kernel of the natural surjection of Tn 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 Tn 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 T4 into Aut(F7). In the end, we propose virtual and welded analogues of these groups and some directions for future work.