Hole operations on Hurwitz maps

TL;DR

This paper explores the structure of graphs formed by orientably regular maps with a given automorphism group, focusing on how hole operations and duality affect their connectivity, with examples from Hurwitz and other groups.

Contribution

It introduces a graph model for analyzing the effects of hole operations and duality on orientably regular maps, providing new insights into their connectivity properties.

Findings

Some groups produce connected graphs, e.g., AGL groups.

Other groups, like alternating and symmetric groups, yield infinitely many components.

Examples include several small Hurwitz groups.

Abstract

For a given group $G$ the orientably regular maps with orientation-preserving automorphism group $G$ are used as the vertices of a graph $\O(G)$, with undirected and directed edges showing the effect of duality and hole operations on these maps. Some examples of these graphs are given, including several for small Hurwitz groups. For some $G$, such as the affine groups ${\rm AGL}_1(2^e)$, the graph $\O(G)$ is connected, whereas for some other infinite families, such as the alternating and symmetric groups, the number of connected components is unbounded.