Fixed Points of Automorphisms of Torus Knot Groups

TL;DR

This paper classifies fixed point subgroups of automorphisms in torus knot groups, providing explicit generators and isomorphism types using Bass-Serre tree actions, advancing understanding of their automorphism structures.

Contribution

It offers a complete classification of fixed point subgroups in torus knot groups, including explicit generators and isomorphism types, using Bass-Serre tree actions.

Findings

Explicit generators for fixed point subgroups

Isomorphism types of fixed point subgroups

Extension of automorphism actions on Bass-Serre trees

Abstract

We completely classify fixed point subgroups in Torus Knot Groups, that is groups of the form $G_{p,q} = \langle x , y | x^p = y^q \rangle$. We not only give the isomorphism type, but also the explicit generators for the fixed point subgroup of each automorphism of $G_{p,q}$. Our main tool is an action of $Aut(G_{p,q})$ on the Bass-Serre tree of $G_{p,q}$ which is compatible with the original action, in the sense that it extends the original action of $G_{p,q}$ on its Bass-Serre tree.