Parallel-product decomposition of edge-transitive maps

Abstract

The parallel product of two rooted maps was introduced by S. E. Wilson in 1994. The main question of this paper is whether for a given reflexible map $M$ one can decompose the map into a parallel product of two reflexible maps. This can be achieved if and only if the monodromy (or the automorphism) group of the map has at least two minimal normal subgroups. All reflexible maps up to 100 edges, which are not parallel-product decomposable, are calculated and presented. For this purpose, all degenerate and slightly-degenerate reflexible maps are classified. Three different quotients of rooted maps are considered in the paper and a characterizaton of morphisms of rooted maps similar to the first isomorphism theorem for groups is presented. The monodromy quotient of a map is introduced, having the property that all the automorphisms project. A theory of edge-transitive maps on non-orientable surfaces is developed. A concept of reduced regularty in the manner of Breda d'Azevedo is applied on edge-transitive maps. Using that, the concept of parallel-product decomposability is extended to edge-transitive maps, where a characterization in terms of minimal normal subgroups of the automorphism group is obtained. Additionally, using Petrie triality and the parallel-product decomposition, a new organization of edge-transitive maps is presented, providing a basis for future censuses.