Vol 226, No 5 (2017)

Primitive monodromy groups of rational functions P/Q, where Q is a polynomial with no multiple roots and deg P > deg Q+1 are classified. There are 17 families of such functions which are not Belyi functions. Only one family from the list contains functions that have five critical values. All the remaining families consist of functions with at most four critical values and constitute one-dimensional strata in the Hurwitz space. The action of the braid group on generators of their monodromy groups is computed and the corresponding megamaps are drawn.

The result extends the classification of primitive edge rotation groups of weighted trees obtained by the author and Zvonkin and is also a generalization of the classification of primitive monodromy groups of polynomials obtained by P. Müller.

In general, the Hurwitz numbers count the branched covers of the Riemann sphere with prescribed ramification data or, equivalently, the factorizations of a permutation with prescribed cycle structure data. In the present paper, the study of monotone orbifold Hurwitz numbers is initiated. These numbers are both variations of the orbifold case and generalizations of the monotone case. These two cases have previously been studied in the literature. We derive a cut-and-join recursion for monotone orbifold Hurwitz numbers, determine a quantum curve governing their wave function, and state an explicit conjecture relating them to topological recursion. Bibliography: 27 titles.

The regular maps and the arc-transitive maps on surfaces with nonempty boundary are classified. It is shown that it is unrealistic to expect a similar classification of edge-transitive maps on such surfaces.

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.

The paper contains a survey of the current state of a constructive part in dessin d’enfants theory. Namely, it is devoted to actual establishing the correspondence between the Belyi pairs and their combinatorial-topological representations. This correspondence is established in terms of equivalence of categories, and all relevant categories are introduced. Several connections with arithmetic are discussed. One of the sections presents a possible generalization of the theory, in which three branch points, allowed for the Belyi functions, are replaced by four. Several directions for further research are presented. Bibliography: 80 titles.