Regular maps on surfaces with large planar width

A map is a cell decomposition of a closed surface; it is regular if its automorphism group acts transitively on the Rags, mutually incident vertex-edge-face triples. The main purpose of this paper is to establish, by elementary methods, the following result: for each positive integer w and for each pair of integers p greater than or equal to 3 and q greater than or equal to 3 satisfying 1/p + 1/q less than or equal to 1/2, there is an orientable regular map with face-size p and valency q such that every non-contractible simple closed curve on the surface meets the 1-skeleton of the map in at least w points. This result has several interesting consequences concerning maps on surfaces, graphs and related concepts. For example. MacBeath's theorem about me existence of infinitely many Hurwitz groups, or Vince's theorem about regular maps of given type (p. q), or residual finiteness of triangle groups, all follow from our result. (C) 2001 Academic Press.