Abstract:
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 G= Aut +(M) of all orientation-preserving automorphisms of such a map M is a finite quotient of the free product Γ = Z∗ C2. 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 Γ (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 Mc attaining the bound, and that this map is universal, in the sense that every simple regular map M for which Aut +(M) is nilpotent of class at most c is a quotient of Mc. In particular, there are finitely many such quotients for any given value of c, and every regular map M, whether simple or non-simple, for which Aut +(M) is nilpotent of class at most c, is a cyclic cover of exactly one of them.
