Automorphism groups of edge-transitive maps

TL;DR

This paper characterizes automorphism groups of edge-transitive maps across 14 classes, identifying which groups can occur and extending previous results to finite simple groups, infinite groups, and embeddings of complete graphs.

Contribution

It provides necessary and sufficient conditions for groups to be automorphism groups of edge-transitive maps in all classes, extending prior work to include finite simple groups and infinite groups.

Findings

Characterizes automorphism groups for all 14 classes of edge-transitive maps.

Determines which symmetric groups can be automorphism groups in each class.

Shows all sufficiently large nilpotence classes and derived lengths are realizable.

Abstract

For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class. Extending earlier results of Siran, Tucker and Watkins, these are used to determine which symmetric groups $S_n$ can arise in this way for each class. Similar results are obtained for all finite simple groups, building on work of Leemans and Liebeck, Nuzhin and others on generating sets for such groups. It is also shown that each edge-transitive class realises finite groups of every sufficiently large nilpotence class or derived length, and also realises uncountably many non-isomorphic infinite groups. Edge-transitive embeddings of complete graphs are classified, and there is a detailed discussion of edge-transitive maps with boundary.