This paper uses combinatorial group theory to help answer some longstanding questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to g − 1, where g is the genus, all orientably-regular maps of genus p + 1 for p prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and

more »

... t orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda-Nedela-Širáň classification of non-orientable regular maps of Euler characteristic −p where p is prime. Regular maps are generalizations of the platonic solids (viewed as tessellations of the sphere) to surfaces of higher genus. Their formal study was initiated by Brahana [5] in the 1920s and continued by Coxeter (see [10] ) and others decades later. Regular maps on the sphere and the torus and other orientable surfaces of small genus are now quite well understood, but until recently, the situation for surfaces of higher genus has been something of a mystery. In particular, some long-standing questions have remained open, about the genera of orientable surfaces carrying a regular map having no multiple edges, or a regular map that is chiral (admitting no reflectional symmetry). This paper takes a significant step towards answering these questions. Here, a map M is an embedding of a connected graph or multigraph into a closed surface, such that each component (or face) of the complement is simply connected. The genus and the Euler characteristic of the map M are defined as the genus and the Euler characteristic of the supporting surface. The topological dual of M (which is denoted by M * ) is obtained from M by interchanging the roles of vertices and faces in the usual way. An automorphism of a map M is any permutation of the edges of the underlying (multi)graph that preserves the embedding, or equivalently, any automorphism of the (multi)graph induced by a homeomorphism of the supporting surface. By connectedness, any automorphism is uniquely determined by its effect on any flag (which is an incident vertex-edge pair (v, e) taken together with a chosen side along the edge e). The automorphism group of M is denoted by Aut(M ). If the surface is orientable, then the subgroup of Aut(M ) of all orientation-preserving automorphisms is denoted Aut o (M ), and this has index at most 2 in Aut(M ); if M admits an orientationreversing automorphism (so that Aut o (M ) has index 2 in Aut(M )), then M is said to be reflexible, and otherwise M is chiral . If the surface is non-orientable, then there is no such distinction. An orientable map M is called regular (or orientably-regular , or sometimes rotary) if G = Aut o (M ) acts regularly on the set of oriented edges (or arcs) of M . The platonic solids give the most famous examples. If each face has size k and each vertex has valence m, then the map M is said to have type {k, m}, and M is regular if and only if there is a k-fold rotation X about the centre of a face f and an m-fold rotation Y about an incident vertex v, with product XY an involutory rotation around the midpoint of an edge e incident with v and f . By connectedness, X and Y generate G = Aut o (M ), which is therefore a quotient of the ordinary (k, m, 2) triangle group ∆ o (k, m, 2) = x, y, z | x k = y m = z 2 = xyz = 1 (under an epimorphism taking x to X and y to Y ). The dual M * is also regular, with the roles of X and Y interchanged, and the map M (or its dual M * ) is reflexible if and only if the group G = Aut o (M ) admits an automorphism of order 2 taking X to X −1 and Y to Y −1 , or equivalently (following conjugation by X), an automorphism of order 2 taking X to X −1 and XY to Y −1 X −1 = (XY ) −1 = XY . Conversely, given any epimorphism θ from ∆ o (k, m, 2) to a finite group G with torsion-free kernel, a map M can be constructed using (right) cosets of the images of x , y and z as vertices, faces and edges, with incidence given by non-empty