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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2017 Volume 23, Number 4, Pages 243–252 (Mi timm1483)

On the Oikawa and Arakawa theorems for graphs

A. D. Mednykha, I. A. Mednykha, R. Nedelyabc

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosbirsk, 630090 Russia
b University of West Bohemia, NTIS FAV, Universitni 8, Pilsen, Czech Republic
c Matej Bel University, Tajovskeho 40, Banska Bystrica, Slovakia

Abstract: The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine at the beginning of the century. This theory considers finite graphs as analogs of compact Riemann surfaces and branched coverings of graphs as holomorphic maps. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of arcs. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus $g>1$ does not exceed $84(g-1)$. Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a graph-theoretic version of the E. Bujalance and G. Gromadzki result improving the Arakawa theorem is obtained.

Keywords: Riemann surface, Riemann–Hurwitz formula, graph, automorphism group, harmonic map.

UDC: 519.177+517.545

MSC: 05C10, 57M12

Received: 14.06.2017

DOI: 10.21538/0134-4889-2017-23-4-243-252


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2019, 304, suppl. 1, S133–S140

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