Let N g (f ) denote the number of rooted maps of genus g having f edges. An exact formula for N g (f ) is known for g = 0 (Tutte, 1963), g = 1 (Arques, 1987), g = 2, 3 (Bender and Canfield, 1991). In the present paper we derive an enumeration formula for the number Θ γ (e) of unrooted maps on an orientable surface S γ of a given genus γ and with a given number of edges e. It has a form of a linear combination i,j c i,j N g j (f i ) of numbers of rooted maps N g j (f i ) for some g j γ and f i e. The coefficients c i,j are functions of γ and e. We consider the quotient S γ /Z of S γ by a cyclic group of automorphisms Z as a two-dimensional orbifold O. The task of determining c i,j requires solving the following two subproblems:(a) to compute the number Epi o (Γ, Z ) of order-preserving epimorphisms from the fundamental group Γ of the orbifold O = S γ /Z onto Z ; (b) to calculate the number of rooted maps on the orbifold O which lifts along the branched covering S γ → S γ /Z to maps on S γ with the given number e of edges.The number Epi o (Γ, Z ) is expressed in terms of classical number-theoretical functions. The other problem is reduced to the standard enumeration problem of determining the numbers N g (f ) for some g γ and f e. It follows that Θ γ (e) can be calculated whenever the numbers N g (f ) are known for g γ and f e. In the end of the paper the above approach is applied to derive the functions Θ γ (e) explicitly for γ 3. We note that the function Θ γ (e) was known only for γ = 0 (Liskovets, 1981). Tables containing the numbers of isomorphism classes of maps with up to 30 edges for genus γ = 1, 2, 3 are presented.

No abstract

No abstract

Abstract. In this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

No abstract