Regular maps with nilpotent automorphism group

A 2-cell decomposition of a closed orientable surface is called a regular map if its automorphism group acts transitively on the set of all its darts (or arcs). It is well known that the group of all orientation-preserving automorphisms of such a map is a finite quotient of the free product . In this paper we investigate the situation where G is nilpotent and the underlying graph of the map is simple (with no multiple edges). By applying a theorem of Labute (Proc Amer Math Soc 66:197-201, 1977) on the ranks of the factors of the lower central series of (via the associated Lie algebra), we prove that the number of vertices of any such map is bounded by a function of the nilpotency class of the group G. Moreover, we show that for a fixed nilpotency class c there is exactly one such simple regular map attaining the bound, and that this map is universal, in the sense that every simple regular map for which is nilpotent of class at most c is a quotient of . In particular, there are finitely many such quotients for any given value of c, and every regular map , whether simple or non-simple, for which is nilpotent of class at most c, is a cyclic cover of exactly one of them.