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DOI: https://doi.org/10.1070/SM2008v199n09ABEH003962

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 $2p$ edges, where $p$ is a prime.
Bibliography: 52 titles.

Received: 28.02.2008

Russian version:
Matematicheskii Sbornik, 2008, Volume 199, Number 9, Pages 3–96
DOI: https://doi.org/10.4213/sm4529

Bibliographic databases:

Citation: E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko, “Maximally symmetric cell decompositions of surfaces and their coverings”, Mat. Sb., 199:9 (2008), 3–96; Sb. Math., 199:9 (2008), 1263–1353

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