Realizing finite edge‐transitive orientable maps

Abstract J.E. Graver and M.E. Watkins, Memoirs Am. Math. Soc. 126 (601) (1997) established that the automorphism group of an edge‐transitive, locally finite map manifests one of exactly 14 algebraically consistent combinations (called types ) of the kinds of stabilizers of its edges, its vertices, its faces, and its Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. H.S.M. Coxeter (Regular Polytopes, 2nd ed., McMillan, New York, 1963) had previously observed that the nine finite edge‐transitive planar maps realize three of the eight planar types. In the present work, we show that for each of the 14 types and each integer n  ≥ 11 such that n  ≡ 3,11 (mod 12), there exist finite, orientable, edge‐transitive maps whose various stabilizers conform to the given type and whose automorphism groups are (abstractly) isomorphic to the symmetric group Sym( n ). Exactly seven of these types (not a subset of the planar eight) are shown to admit infinite families of finite, edge‐transitive maps on the torus, and their automorphism groups are determined explicitly. Thus all finite, edge‐transitive toroidal maps are classified according to this schema. Finally, it is shown that exactly one of the 14 types can be realized as an abelian group of an edge‐transitive map, namely, as ℤ n  × ℤ 2 where n  ≡ 2 (mod 4). © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 1–34, 2001