# Conjugacy classes and automorphisms of twin groups

## TL;DR

This paper explores the structure of twin groups, deriving formulas for conjugacy classes, analyzing automorphisms, and establishing connections with Fibonacci numbers and automorphism groups.

## Contribution

It provides new formulas for conjugacy classes, a proof of automorphism group structure, and a representation of twin groups into automorphisms of free groups.

## Key findings

- Number of conjugacy classes of involutions relates to Fibonacci sequence
- Recursive formula for z-classes of involutions in $T_n$
- Twin groups embed into automorphism groups of free groups

## Abstract

The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ 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 $n \geq 4$. Finally, we construct a representation of $T_n$ to $\Aut(F_n)$ for $n \ge 2$.