Regular maps with nilpotent automorphism group

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 $G = \mathrm{Aut}^+(\mathcal {M})$ of all orientation-preserving automorphisms of such a map $\mathcal {M}$ is a finite quotient of the free product $\Gamma = \mathbb {Z}* C_2$. 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 $\Gamma$ (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 $\mathcal {M}_c$ attaining the bound, and that this map is universal, in the sense that every simple regular map $\mathcal {M}$ for which $\mathrm{Aut}^+(\mathcal {M})$ is nilpotent of class at most c is a quotient of $\mathcal {M}_c$. In particular, there are finitely many such quotients for any given value of c, and every regular map $\mathcal {M}$, whether simple or non-simple, for which $\mathrm{Aut}^+(\mathcal {M})$ is nilpotent of class at most c, is a cyclic cover of exactly one of them.