Conjugacy classes and automorphisms of twin groups

Abstract

The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and the pure twin group $P{T}_n$ is the kernel of the natural surjection from $T_n$ 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 $T_n$, 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 $T_n$. We give a new proof of the structure of $\Aut(T_n)$ for $n\ge 3$, and show that $T_n$ is isomorphic to a subgroup of $\Aut(PT_n)$ for $n\ge 4$. Finally, we construct a representation of $T_n$ to $\Aut(F_n)$ for $n\ge 2$.

18 pages, title changed, to appear in Forum Mathematicum

References in corpus (2)

• Structural aspects of twin and pure twin groups
• Planar pure braids on six strands

Cited by in corpus (7)

• Topological Exchange Statistics in One Dimension
• Structure and automorphisms of pure virtual twin groups
• Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension
• Automorphisms of odd Coxeter groups
• Virtual planar braid groups and permutations
• Congruence subgroups and crystallographic quotients of small Coxeter groups
• Flat braid groups, right-angled Artin groups, and commensurability