On Wiman’s theorem for graphs

The aim of the paper is to find discrete versions of the Wiman theorem which states that the maximum possible order of an automorphism of a Riemann surface of genus g≥2g≥2 is 4g+24g+2. The role of a Riemann surface in this paper is played by a finite connected graph. The genus of a graph is defined as the rank of its homology group. Let ZNZN be a cyclic group acting freely on the set of directed edges of a graph XX of genus g≥2g≥2. We prove that N≤2g+2N≤2g+2. The upper bound N=2g+2N=2g+2 is attained for any even gg. In this case, the signature of the orbifold X/ZNX/ZN is (0;2,g+1)(0;2,g+1), that is X/ZNX/ZN is a tree with two branch points of order 22 and g+1g+1 respectively. Moreover, if N<2g+2N<2g+2, then N≤2gN≤2g. The upper bound N=2gN=2g is attained for any g≥2g≥2. The latter takes a place when the signature of the orbifold X/ZNX/ZN is (0;2,2g)(0;2,2g).