Hypermap operations of finite order

TL;DR

This paper investigates the finite order elements in the outer automorphism groups of hypermap groups, exploring the operations they induce beyond the well-known duality and chirality.

Contribution

It characterizes the elements of finite order in the groups ${ m Out} \Delta$ and ${ m Out} \Delta^+$ and analyzes the hypermap operations they generate.

Findings

Identifies finite order elements in ${ m Out} \\Delta$ and ${ m Out} \\Delta^+$

Describes the hypermap operations induced by these elements

Extends understanding of hypermap symmetries beyond order 2 operations

Abstract

Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups ${\rm Out} \Delta\cong PGL_2({\bf Z})$ and ${\rm Out} \Delta^+\cong GL_2({\bf Z})$ of the groups $\Delta = C_2*C_2*C_2$ and $\Delta^+ = F_2$. We will consider the elements of finite order in these two groups, and the operations they induce.