Cosets, Voltages, and Derived Embeddings

TL;DR

This paper introduces a method using cosets and voltage groups to analyze derived embeddings of graphs on surfaces, enabling topological insights without full construction of covering spaces.

Contribution

It defines special subgroups related to subgraphs in voltage graph embeddings and uses cosets to extract topological information about derived embeddings.

Findings

Cosets help determine symmetry properties of derived embeddings.

The method applies to cycles with disjoint cycle fibers and annular neighborhoods.

Infinite examples demonstrate the approach's effectiveness.

Abstract

An ordinary voltage graph embedding of a graph in a surface encodes a certain kind of highly symmetric covering space of that surface. Given an ordinary voltage graph embedding of a graph $G$ in a surface with voltage group $A$ and a connected subgraph $H$ of $G$, we define special subgroups of $A$ that depend on $H$ and show how cosets of these groups in $A$ can be used to find topological information concerning the derived embedding without constructing the whole covering space. Our strongest theorems treat the case that $H$ is a cycle and the fiber over $H$ is a disjoint union of cycles with annular neighborhoods, in which case we are able to deduce specific symmetry properties of the derived embeddings. We give infinite families of examples that demonstrate the usefulness of our results.