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The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine and their followers at the beginning of the century. This theory considers finite graphs as analogs of Riemann surfaces and branched coverings of graphs as holomorphic mappings. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of half-edges of the graph. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus g > 1 does not exceed 84(g — 1). Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a discrete analog of the E. Bujalance and G. Gromadzki theorem improving one of Arakawa's results is obtained. [ABSTRACT FROM AUTHOR]

The isomorphism classes of hypermaps of a given genus g ≤ 6 and a given number d of darts are enumerated. The hypermaps of a given genus g are distinguished up to orientation preserving isomorphisms. The obtained results depend on recent progress in counting rooted hypermaps, in particular, by P. Zograf, M. Kazarian, A. Giorgetti, and T. Walsh. These results can be interpreted as an enumeration of conjugacy classes of subgroups in a free Fuchsian group of rank two with a genus restriction. Bibliography: 36 titles. [ABSTRACT FROM AUTHOR]

Several versions of the Riemann-Hurwitz theorem for branched coverings of graphs are presented. A larger class of graphs which may have not only multiple edges but also loops and darts is considered. This makes it possible to render the class of graphs closed with respect to morphisms arising as quotient maps for actions of finite groups. [ABSTRACT FROM AUTHOR]

The article discusses the Hurwitz' Theorem or the upper bound for groups acting on a Riemann surface. Topics include role of holomorphic mappings and Riemann surfaces, finite multigraphs and their automorphism groups, and rank of finite multigraph's fundamental groups to define the genus of graphs.

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser Ge of every edge e is dihedral of order 4 and the stabiliser Gv of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that Hv is the cyclic subgroup of index 2 in Gv. An analogous result is proved for orientably-regular embeddings. [ABSTRACT FROM AUTHOR]

We generalize the idea of reflexibility of a map on a surface by introducing certain integers as its 'exponents'. An exponent is any integer $e$ with the property that changing the cyclic permutation of edges around each vertex induced by the map to its $e$-th power gives rise to an isomorphic map. The exponents reduced modulo the least common multiple of the vertex valencies form an Abelian group, the exponent group $\mbox{Ex}\,(M)$ of the map $M$. Along with the automorphism group, the group in fact provides an additional measure of symmetry of ~$M$. The paper is devoted to developing the fundamentals of the theory of exponent groups of maps. Motivation comes from the problem of classification of regular maps with a given underlying graph. To this end, we prove that the number of non-isomorphic regular maps (if any) with a given underlying graph and the same map automorphism group is $|{\Bbb Z}_n^*:\mbox{Ex}\,(M)|$, $n$ being the valency of $M$. We calculate the exponent groups for some interesting families of regular maps including complete maps. Special attention is paid to the problem of how the antipodality of a map is reflected by its exponent group. In the final section we discuss several open problems.1991 Mathematics Subject Classification: 05C10, 05C25, 20F32. [ABSTRACT FROM PUBLISHER]