On automorphisms of graphs and Riemann surfaces acting with fixed points
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Let X be a finite connected graph, possibly with multiple edges. We provide each edge of the graph by two possible orientations. An automorphism group of a graph acts harmonically if it acts freely on the set of directed edges of the graph. Following M. Baker and S. Norine define a genus g of the graph X to be the rank of the first homology group. A finite group acting harmonically on a graph of genus g is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus g. In the present paper, we give a sharp upper bound for the size of cyclic group acting harmonically on a graph of genus \(g\ge 2\) with a given number of fixed points. Similar results, for closed orientable surfaces, were obtained earlier by T. Szemberg, I. Farkas and H. M. Kra.
KeywordsCyclic group Fixed point Automorphism of graph Graph covering Genus of graph Graph of groups Riemann surface Orbifold
Mathematics Subject ClassificationPrimary 05C25 39A12 Secondary 30F10
The first author was supported by Polish National Sciences Center through the grant NCN 2015/17/B/ST1/03235. The results of the second and third authors presented in the first two Sections are supported by the Russian Foundation for Basic Research (Grants 18-01-00420 and 18-501-51021) while the results given in the next two Sections are supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (Contract No. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). All authors are very thankful to both of two anonymous referees for their critical but helpful remarks and suggestions.
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Conflict of interest
The authors declare that they have no conflict of interest.
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