Maximally symmetric cell decompositions of surfaces and their coverings

Abstract

Regular (maximally symmetric) cell decompositions of closed oriented 2-dimensional surfaces (that is, regular maps or regular abstract polyhedra) are considered. These objects are also known as maximally symmetric oriented atoms. An atom is reducible if it is a branched covering of another atom, with branching points at vertices of the decomposition and/or the centres of faces. The following two problems have arisen in the theory of integrable Hamiltonian systems: describe the irreducible maximally symmetric atoms; describe all the maximally symmetric atoms covering a fixed irreducible maximally symmetric atom. In this paper, these problems are solved in important cases. As applications, the following maximally symmetric atoms are listed: the atoms containing at most 30 edges; the atoms containing at most six faces; the atoms containing p or 2p, where p is a prime. Bibliography: 52 titles.