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Let G be a finite group having a factorisation G = AB into subgroups A and B with B cyclic and A∩B = 1, and let b be a generator of B. The associated skew-morphism is the bijective mapping f W A → A well-defined by the equality baB = f .a/B, where a ∊ A. In this paper, we shall classify all skew-morphisms of cyclic p-groups, where p is an odd prime. [ABSTRACT FROM AUTHOR]

We prove the existence of infinitely many orientably-regular but chiral maps of every given hyperbolic type {m, k}, by constructing base examples from suitable permutation representations of the ordinary (2, k,m) triangle group, and then taking abelian p-covers. The base examples also help to prove that for every pair (k,m) of integers with 1/k + 1/m ⩽ 1/2, there exist infinitely many regular and infinitely many orientably-regular but chiral maps of type {m, k}, each with the property that both the map and its dual have simple underlying graph. [ABSTRACT FROM AUTHOR]

We use the term half-regular map to describe an orientable map with an orientation preserving automorphism group that is transitive on vertices and half-transitive on darts. We present a full classification of half-regular Cayley maps using the concept of skew-morphisms. We argue that half-regular Cayley maps come in two types: those that arise from two skew-morphism orbits of equal size that are both closed under inverses and those that arise from two equal-sized orbits that do not contain involutions or inverses but one contains the inverses of the other. In addition, half-regular Cayley maps of the first type are shown to be half-edge-transitive, while half-regular Cayley maps of the second type are shown to be necessarily edge-transitive. A connection between half-regular Cayley maps and regular hypermaps is also investigated. [ABSTRACT FROM AUTHOR]

Abstract: A map is a 2-cell decomposition of a closed surface. A map on an orientable surface is called regular if its group of orientation-preserving automorphisms acts transitively on the set of darts (edges endowed with an orientation). In this paper we investigate regular maps which are regular covers over platonic maps with a cyclic group of covering transformations. We describe all such maps in terms of parametrised group presentations. This generalises the work of Jones and Surowski [G.A. Jones, D.B. Surowski, Cyclic regular coverings of the Platonic maps, European J. Combin. 21 (2000) 333–345] classifying the cyclic regular coverings over platonic maps with branched points exclusively at vertices, or at face-centres. [Copyright &y& Elsevier]

Abstract: We study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps , where is a -group and is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2. [Copyright &y& Elsevier]

Abstract: In an earlier paper by A. Breda, R. Nedela and J. Širáň, a classification was given of all regular maps on surfaces of negative prime Euler characteristic. In this article we extend the classification to surfaces with Euler characteristic (equivalently, to non-orientable surfaces of genus ) for all odd primes p. [Copyright &y& Elsevier]

Abstract: In this paper we derive an enumeration formula for the number of hypermaps of a given genus and given number of darts in terms of the numbers of rooted hypermaps of genus with darts, where . Explicit expressions for the number of rooted hypermaps of genus with darts were derived by Walsh [T.R.S. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory B 18 (2) (1975) 155–163] for , and by Arquès [D. Arquès, Hypercartes pointées sur le tore: Décompositions et dénombrements, J. Combin. Theory B 43 (1987) 275–286] for . 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 are distinguished up to the action of orientation-preserving hypermap isomorphisms. The enumeration results can be expressed in terms of Fuchsian groups. [Copyright &y& Elsevier]

Abstract: In this paper we derive an enumeration formula for the number of hypermaps of given genus g and given number of darts n in terms of the numbers of rooted hypermaps of genus with m darts, where . Explicit expressions for the number of rooted hypermaps of genus g with n darts were derived by Walsh (1975) for , and by Arques (1987) for . 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. The enumeration results can be expressed in terms of Fuchsian groups. [Copyright &y& Elsevier]

A map is a 2-cell embedding of a connected graph into a closed surface. A map is orientable if the supporting surface is orientable. An orientable map is regular if its group of orientation-preserving automorphisms acts transitively on the darts. Using an equivalent algebraic description of regular maps and their coverings, we employ the theory of group extensions to classify the almost totally branched coverings of the platonic maps with non-abelian covering transformation groups, generalising the results of Hu, Nedela and Wang. [ABSTRACT FROM AUTHOR]

Abstract: We use the conceptual idea of “maps on orbifolds” and the theory of the non-Euclidean crystallographic groups (NEC groups) to enumerate rooted and unrooted maps (both sensed and unsensed) on surfaces regardless of genus. As a consequence we deduce a formula for the number of chiral pairs of maps. The enumeration principle used in this paper is due to Mednykh (2006) , it counts the number of conjugacy classes of subgroups in NEC groups which are in one-to-one correspondence with unrooted (sensed or unsensed) maps. [Copyright &y& Elsevier]

In the present paper we introduce a family of functors (called operations) of the category of hypermaps (dessins) preserving the underlying Riemann surface. The considered family of functors include as particular instances the operations considered by Magot and Zvonkin (2000), Singerman and Syddall (2003), and Girondo (2003). We identify a set of 10 operations in the above infinite family which produce vertex-transitive dessins out of regular ones. This set is complete in the following sense: if a vertex-transitive map arises from a regular dessin H applying an operation, then it can be obtained from a regular dessin on the same surface (possibly different from H ) applying one of the 10 operations. The statement includes the classical case when the underlying surface is the sphere. [ABSTRACT FROM AUTHOR]