Self-dual, self-Petrie-dual and M\"obius regular maps on linear fractional groups

TL;DR

This paper investigates special symmetries of regular maps on linear fractional groups, providing conditions for self-duality, self-Petrie-duality, and M"obius regularity, along with enumeration results for small cases.

Contribution

It establishes necessary and sufficient conditions for these properties in regular maps on $PSL(2,q)$ and $PGL(2,q)$, including detailed enumeration for small values of q.

Findings

Conditions for self-duality, self-Petrie-duality, and M"obius regularity are derived.

Enumeration of $PSL(2,q)$ maps with these properties for $q \\le 81$ and $q \\le 49$.

Special case analysis for maps of type (5,5).

Abstract

Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been studied for many years and the theory is well-developed, including generating sets for the asscoiated groups. This paper studies the properties of self-duality, self-Petrie-duality and M\"obius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5,5). The final section includes an enumeration of the $PSL(2,q)$ maps for $q\le81$ and a list of all the $PSL(2,q)$ maps which have any of these special properties for $q\le49$.