On Wiman’s theorem for graphs

Abstract

a b s t r a c tThe 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 ≥ 2 is 4g +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 Z N be a cyclic group acting freely on the set of directed edges of a graph X of genus g ≥ 2. We prove that N ≤ 2g + 2. The upper bound N = 2g + 2 is attained for any even g. In this case, the signature of the orbifold X /Z N is (0; 2, g + 1), that is X /Z N is a tree with two branch points of order 2 and g + 1 respectively. Moreover, if N < 2g + 2, then N ≤ 2g. The upper bound N = 2g is attained for any g ≥ 2. The latter takes a place when the signature of the orbifold X /Z N is (0; 2, 2g). IntroductionKlein's quartic curve, x 3 y + y 3 z + z 3 x = 0, admits the group PSL 2 (7) as its full group of conformal automorphisms. It is characterized as the curve of smallest genus realizing the upper bound 84(g − 1) on the order of a group of conformal automorphisms of a curve of genus g > 1, given by Hurwitz [9] in 1893. Around the same time, Wiman [15] characterized the curves w 2 = z 2g+1 − 1 and w 2 = z(z 2g − 1), g > 1, as the unique curves of genus g admitting cyclic automorphism groups of the largest and the second largest possible order (4g + 2 and 4g, respectively). The modern proof of these and similar results is contained in the paper by K. Nakagawa [13].Over the last decade, counterparts of many theorems from the classical theory of Riemann surfaces were derived in the discrete case [1,3,5,12]. In these theorems, the finite connected graphs play the role of algebraic curves and the conformal automorphisms are replaced by harmonic ones. We say that a finite group acts harmonically on a graph if it acts freely on the set of directed edges. Following [1] we define the genus of a graph as the rank of its homology group. Then the upper bound on the order of a group acting harmonically on a graph of genus g > 1 is 6(g − 1). This result was obtained by S.Corry [5].The aim of the present paper is to find a discrete version of the Wiman theorem.Let Z N be a cyclic group acting freely on the set of directed edges of a graph X of genus g ≥ 2. We prove that N ≤ 2g + 2. The upper bound N = 2g + 2 is attained for any even g. Moreover, if N < 2g + 2, then N ≤ 2g. The upper bound N = 2g is attained for any g ≥ 2. We describe also the signature of the quotient graphs X /Z N arising in these cases. See Theorems 3 and 4 for explicit statements of the results. *

Fixed points of cyclic groups acting purely harmonically on a graph

“…Let Z n be a cyclic group acting harmonically a graph X of genus g ≥ 2. Recall the following results proved in [6]. First of all, we get n ≤ 2g + 2.…”

“…Proof. By Theorem 3 from [6], we know that if T is an automorphism of order n = 2g + 2 then the signature of the orbifold X/Z n is (0; 2, g + 1). Note that by Proposition 1 if the signature of orbifold X/Z n is equal to (γ; m 1 , .…”

“…An automorphism of a graph X is said to be harmonic if it generates a cyclic group acting harmonically on X. 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].…”

Accola theorem on hyperelliptic graphs

“…Taking into consideration the fact that a trivial group is assigned to any edge, the definition of a covering of graphs of groups can be formulated as follows. This definition was introduced in [12]. To illustrate the notion of a covering in the category of graphs of groups, we provide a basic example.…”

“…Also we employ the theory of graphs of groups (or the Bass-Serre theory) to uniformize the coverings of a graph just as it works for Riemann surfaces. This approach was proposed by A. Mednykh and I. Mednykh [12].…”