Realizing finite edge-transitive orientable maps (Q2725047)

It is known [\textit{J. E. Graver} and \textit{M. E. Watkins}, Mem. Am. Math. Soc. 601 (1997; Zbl 0901.05087)] that the automorphism group of an edge-transitive, locally finite map manifests one of exactly 14 types of the kinds of stabilizers of its edges, vertices, faces and Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. The nine finite edge-transitive planar maps realize three of the eight planar types [\textit{H. M. S. Coxeter}, Regular polytopes. 2nd ed. (The Macmillan Company, New York) (1963; Zbl 0118.35901)]. NEWLINENEWLINENEWLINEThe authors show that for each of the 14 types and each integer \(n \geq 11,\;n \equiv 3,11\;(\text{mod }12)\), there exist finite, orientable, edge-transitive maps whose stabilizers are conform to the given types and whose automorphism groups are isomorphic to the symmetric group \(\text{Sym}(n)\). Further, exactly seven of these types admit infinite families of finite, edge-transitive maps on the torus, and their automorphism groups are determined explicitly. Finally, they show that exactly one type can be realized as an abelian group of an edge-transitive map, namely, as \(\mathbb{Z}_n \times \mathbb{Z}_2\) for \(n \equiv 2\pmod 4\). The proofs involve special Cayley graphs associated with each type and their suitable embeddings.