Non-orientable and orientable regular maps

Abstract

There do exist quite a lot of papers investigating regular maps on connected compact 2-manifolds, in the orientable case as well as in the non-orientable case (see the References). We want to bring about some new aspects, which also lead to interesting applications.

It is well known that the closed orientable surface of characteristic 2x is a regular two-fold cover of the closed non-orientable surface of characteristic x. Thus, every non-orientable regular (i.e. flag-transitive) map of genus h induces an orientable regular map of genus g=h − 1. But not every orientable map is induced, and different non-orientable regular maps can induce the same orientable map. We study this relation more closely and give several criteria by which one can easily get a classification of the non-orientable regular maps of genus h, if the list of the orientable ones of genus g is known. We construct all non-orientable regular maps having less than 6 faces.

Our and earlier results indicate that the orientable case seems to admit in a certain sense much more regular maps than the non-orientable case. This becomes plausible, if we take into consideration the crystallographic aspect. The group of the map (or a factor group) acts as a Euclidean crystallographic point group on a 2g- or (h − 1)-dimensional Z-lattice, if the genus is g or h, respectively. So the crystallographic restrictions are stronger in the non-orientable case.

Apart from the standard abelianizing procedure we can construct in the non-orientable case a normal subgroup N** of the group of the universal tessellation, yielding the Z-lattice and giving a new method for proving certain groups to be infinite. We illustrate this by showing that the infiniteness of the group (2,3,7;9) — which was proved by C.C. Sims [20] and J. Leech [16], [17] — is an immediate consequence of our results, as well as the new informations that (2,3,7;13) and (2,3,7;15) are infinite. (In fact: Using earlier findings of the second author, one can prove further new results on the groups G ^p,q,r and (l,m,n;q)). A more general and systematic exposition of our method will be given in a forthcoming publication [1].