Conjugacy classes and automorphisms of twin groups

Abstract

The twin group Tn is a right-angled Coxeter group generated by n - 1 involutions, and the pure twin group PTn is the kernel of the natural surjection from Tn onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in Tn, which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in Tn. We give a new proof of the structure of Aut (Tn) for n ≥ 3, and show that Tn is isomorphic to a subgroup of Aut (PTn) for n ≥ 4. Finally, we construct a representation of Tn to Aut (Fn) for n ≥ 2