On automorphisms of graphs and Riemann surfaces acting with fixed points

Gromadzki, Grzegorz; Mednykh, Alexander; Mednykh, I. A.

doi:10.1007/s13324-019-00298-7

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2021

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“…In paper [6] a discrete analogue of the Wiman theorem has been established. More precisely, it was shown that the order of a harmonic automorphism of a graph X of genus g ≥ 2 does not exceed 2g + 2 and this bound is achieved for any even g. The size of cyclic group acting harmonically on X with given number of xed points was estimated from the above in [4].…”

Section: Introductionmentioning

confidence: 99%

Fixed points of cyclic groups acting purely harmonically on a graph

Mednykh¹

2021

Sib. Elektron. Mat. Izv.

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Let X be a nite connected graph, possibly with loops and multiple edges. An automorphism group of X acts purely harmonically if it acts freely on the set of directed edges of X and has no invertible edges. Dene a genus g of the graph X to be the rank of the rst homology group. A discrete version of the Wiman theorem states that the order of a cyclic group Zn acting purely harmonically on a graph X of genus g > 1 is bounded from above by 2g + 2. In the present paper, we investigate how many xed points has an automorphism generating a ¾large¿ cyclic group Zn of order n ≥ 2g − 1. We show that in the most cases, the automorphism acts xed point free, while for groups of order 2g and 2g − 1 it can have one or two xed points.

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Section: Introductionmentioning

confidence: 99%