On the structure of the Jacobian group for circulant graphs

The Jacobian of a graph is defined as the maximal Abelian group generated by flows obeying two Kirchhoff’s laws. This notion, also known as the Picard group, sandpile group, or critical group, has been extensively studied by many authors in the past decade. This is an important algebraic invariant of a finite graph. At the same time, the structure of the Jacobian is known only in particular cases. The paper is devoted to the study of the structure of the Jacobian group for circulant graphs. For the simplest graphs in this family, the Jacobian group is explicitly described, and in the general case, and effective algorithm for calculating it is proposed.     

Enumeration of unrooted hypermaps of a given genus

In this paper we derive an enumeration formula for the number of hypermaps of a given genus g and given number of darts n in terms of the numbers of rooted hypermaps of genus γ≤g with m darts, where m|n. Explicit expressions for the number of rooted hypermaps of genus g with n darts were derived by Walsh [T.R.S. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory B 18 (2) (1975) 155-163] for g=0, and by Arquès [D. Arquès, Hypercartes pointées sur le tore: Décompositions et dénombrements, J. Combin. Theory B 43 (1987) 275-286] for g=1. We apply our general counting formula to derive explicit expressions for the number of unrooted spherical hypermaps and for the number of unrooted toroidal hypermaps with given number of darts. We note that in this paper isomorphism classes of hypermaps of genus g≥0 are distinguished up to the action of orientation-preserving hypermap isomorphisms. The enumeration results can be expressed in terms of Fuchsian groups.     

On the enumeration of circular maps with given number of edges

A map is a closed Riemann surface S with an embedded graph G such that S G is homeomorphic to a disjoint union of open disks. Tutte began a systematic study of maps in the 1960s, and contemporary authors are actively developing it. We introduce the concept of circular map and establish its equivalence to the concept of map admitting a coloring of the faces in two colors. The main result is a formula for the number of circular maps with given number of edges.     

On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus

We obtain an upper bound for the number of holomorphic mappings of a genus 3 Riemann surface onto a genus 2 Riemann surface in a series of cases. In particular, we establish that the number of holomorphic mappings of an arbitrary genus 3 Riemann surface onto an arbitrary genus 2 Riemann surface is at most 48. We show that this estimate is sharp and find pairs of Riemann surfaces for which it is attained.     

On the volume of a spherical octahedron with symmetries

The group Aff(ℚ) of affine transformations with rational coefficients acts naturally not only on the real line ℝ, but also on the p-adic fields ℚ_p. The aim of this note is to show that all these actions are necessary and sufficient to represent bounded μ-harmonic functions for a probability measure μ on Aff(ℚ) that is supported by a finitely generated subgroup, that is, to describe the Poisson boundary. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

On the Volume of a Symmetric Tetrahedron in Hyperbolic and Spherical Spaces

We obtain some elementary formulas for the volume of a symmetric tetrahedron in hyperbolic and spherical spaces.     

Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces

The aim of this paper is to count subgroups of a given index in the fundamental group of an orientable S ¹-bundle over a compact surface. The number of subgroups of index n turns out to be independent of the orientability of the base surface 𝔣, closed or bordered, and is expressed as a linear combination of the numbers of surface subgroups of indices m = n/l l∣n. For a closed base surface of characteristic χ the respective coefficients are equal to l ^–χm+2 or vanish depending on l,n and the Euler number of the S ¹ -bundle.