Classification of regular maps of Euler characteristic −3p

Journal of Combinatorial Theory, Series B, 2007

Möbius regular maps are surface embeddings of graphs with doubled edges such that (i) the automorphism group of the embedding acts regularly on flags and (ii) each doubled edge is a centre of a Möbius band on the surface. In the first part of the paper we give an abstract characterisation of Möbius regular maps with a given automorphism group in terms of two dihedral subgroups intersecting in a special way. As an application we exhibit an interesting correspondence between Möbius regular maps of valence 6 and 3-arc-transitive cubic graphs. The second part of the paper deals with constructions of Möbius regular maps on certain classes of simple groups. The main result here is an exact enumeration of all such maps on PSL(2, q) groups. A number of other results related to the two main topics are presented.

2020

If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types then the map is said to be a Semi-equivelar map. In this article we classify all semi-equivelar and vertex transitive maps on the surface of Euler genus 3, i.e., on the surface of Euler characteristic -1.

Indian Journal of Pure and Applied Mathematics

Semi Equivelar maps are generalizations of Archimedean solids to surfaces other than 2-sphere. Semi Equivelar Maps were introduced by Upadhyay et. al. in 2014. They also studied semi equivelar maps on the surface of Euler characteristics v ¼ À1. In this article we classify all the semi equivelar maps on this surface with upto 12 vertices. We show that there are exactly four such maps. We also prove that there are at least 10 semi equivelar maps on this surface. We compute their Automorphism groups and show that none of these maps are vertex transitive. Keywords Vertex transitive maps Á Archimedean solids Á Semi-Equivelar Maps Á Automorphism group AMS classification 52B70 Á 52C20 Á 57M20 Á 57N05 1 Introduction Semi equivelar maps are generalizations of the maps on surfaces of Archemedian solids to the maps on surfaces other than the 2-sphere. It is observed that not much study has been made of maps on this surface except for the study of weakly neighbourly polyhedral maps (see [9]) and some semi equivelar maps (see [1] and [2]). In this article we attempt to give a classification of semi equivelar maps on this surface with at most 12 vertices and study their automorphism group. Recall that a surface is a closed 2-dimensional manifold without boundary. A Polyhedral complex is a finite collection X of convex polytopes, in Euclidean space E n such that the following two conditions are satisfied for X : first, if r 2 X and s is a face of r, then s 2 X and second, if r 1 and r 2 2 X then r 1 \ r 2 is a face of r 1 and r 2. The dimension d of X is the maximum of the dimensions of the elements in X. We also call X a polyhedral d-complex. In this article the objects of our study are polyhedral 2-complexes. For a 2dimensional polyhedral complex X, the elements of dimension 0, 1 an 2 are called vertex, edges and faces. Let v be a vertex of X. Then star of v, denoted by st(v), is a polyhedral complex fr 2 X : fvg rg and the link of a vertex v in X, denoted by lk(v) is the polyhedral complex fr 2 ClðstðvÞÞ : r \ fvg ¼ /g, where Cl(A) denote closure of A. If the link of each vertex in X is topologically a circle then X is also called a combinatorial 2-manifold. The set Aut(X) for a polyhedral complex X is a collection of all the automorphism

European Journal of Combinatorics, 2001

A map is a cell decomposition of a closed surface; it is regular if its automorphism group acts transitively on the flags, mutually incident vertex-edge-face triples. The main purpose of this paper is to establish, by elementary methods, the following result: for each positive integer w and for each pair of integers p ≥ 3 and q ≥ 3 satisfying 1/ p + 1/q ≤ 1/2, there is an orientable regular map with face-size p and valency q such that every non-contractible simple closed curve on the surface meets the 1-skeleton of the map in at least w points. This result has several interesting consequences concerning maps on surfaces, graphs and related concepts. For example, MacBeath's theorem about the existence of infinitely many Hurwitz groups, or Vince's theorem about regular maps of given type ( p, q), or residual finiteness of triangle groups, all follow from our result.

Discrete Mathematics, 2015

In this paper we classify the reflexible and chiral regular oriented maps with p faces of valency n, and then we compute the asymptotic behaviour of the reflexible to chiral ratio of the regular oriented maps with p faces. The limit depends on p and for certain primes p we show that the limit can be 1, greater than 1 and less than 1. In contrast, the reflexible to chiral ratio of regular polyhedra (which are regular maps) with Suzuki automorphism groups, computed by Hubard and Leemans [13], has produced a nill asymptotic ratio.

Annals of Combinatorics, 2019

Bi-orientable maps (also called pseudo-orientable maps) were introduced by Wilson in the 1970s to describe non-orientable maps with the property that opposite orientations can consistently be assigned to adjacent vertices. In contrast to orientability, which is both a combinatorial and topological property, bi-orientability is only a combinatorial property. In this paper we classify the bi-orientable maps whose localorientation-preserving automorphism groups act regularly on arcs, called here bi-rotary maps, of negative prime Euler characteristic. Unlike other classification results for highly symmetric maps on such surfaces, we do not use the Gorenstein-Walter result on the structure of groups with dihedral Sylow 2-subgroups.

Advances in Geometry, 2008

(co-author Gian Pietro Pirola) We study dominant rational maps from a general surface in P^3 to surfaces of general type. We prove restrictions on the target surfaces, and special properties of these rational maps. We show that for small degree the general surface has no such map. Moreover a slight improvement of a result of Catanese, on the number of moduli of a surface of general type, is also obtained.

arXiv: Combinatorics, 2015

We prove that infinite regular and chiral maps take place on surfaces with at most one end. Moreover, we prove that an infinite regular or chiral map on an orientable surface with genus can only be realized on the Loch Ness monster, that is, the topological surface of infinite genus with one end.

Topology and its Applications, 1984

A generic map of a smooth surface M to an oriented smooth surface N is an immersion except on a compact family of curves where it may have fold or cusp singularities. If the domain is oriented, the map is homotopic to one having no cusps while, if nonorientable, it is homotopic to a map having at most one or no cusps depending upon whether the genus of M is odd or even. Conditions under which a generic map is the composition of an immersion of M into N x R followed by projection to N are given. Finally, an immersion of P* into R-', whose projection to R2 has a fold locus consisting of a single curve containing a single cusp, is described. The purpose of this paper is to describe a slight improvement upon a theorem of Levine concerning the elimination of cusps of a generic map from one compact oriented surface to another and to elaborate upon results of HaeAiger concerning the factorization of maps of surfaces into the plane as the composition of an immersion followed by a projection.

Transactions of the American …, 1998

The present paper is devoted to the classification of irregular surfaces of general type with pg ≥ 3 and nonbirational bicanonical map. Our main result is that, if S is such a surface and if S is minimal with no pencil of curves of genus 2, then S is the symmetric product of a curve of genus 3, and therefore pg = q = 3 and K 2 = 6.