On automorphisms of graphs and Riemann surfaces acting with fixed points

Research output: Contribution to journalArticle

Abstract

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≥ 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.

Original languageEnglish
Pages (from-to)2021-2031
Number of pages11
JournalAnalysis and Mathematical Physics
Volume9
Issue number4
DOIs
Publication statusPublished - 1 Dec 2019

Keywords

  • Automorphism of graph
  • Cyclic group
  • Fixed point
  • Genus of graph
  • Graph covering
  • Graph of groups
  • Orbifold
  • Riemann surface

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