Mixing Chiral Polytopes

An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A "mixing" construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral

Bi-rotary Maps of Negative Prime Characteristic

Bi-orientable maps (also called pseudo-orientable maps) were introduced by Wilson in the seventies 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 local-orientation-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

Two infinite families of Archimedean maps of higher genera

The well known infinite families of prisms and antiprisms on the sphere were, for long time, not considered as Archimedean solids for reasons not fully understood. In this paper we describe the first two infinite families of Archimedean maps on higher genera which we call “generalized” prisms and “generalized” antiprisms

Census of quadrangle groups inclusions

In a classical result of 1972 Singerman classifies the inclusions between triangle groups. We extend the classification to a broader family of triangle and quadrangle groups forming a particular subfamily of Fuchsian groups. With two exceptions, each inclusion determines a finite bipartite map (hypermap) on a 2-dimensional spherical orbifold that encodes the complete information and gives a graphical visualisation of the inclusion. A complete description of all the inclusions is contained in the attached tables

Quadrangle groups inclusions

In this paper we generalise Singerman’s results on triangle group inclusions to the broader class of generalised quadrangle groups, that is, Fuchsian groups with signature of genus 0 and generated by three or four elliptic generators. For any possible inclusion P<Q P<Q we also give the number of non-conjugate subgroups of Q isomorphic to P

Hypermap operations of finite order

Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups and of the groups ?=C2*C2*C2 and ?+=F2. We will consider the elements of finite order in these two groups, and the operations they induce.<br/

Constructions of chiral polytopes of small rank

An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4 and 5

Classification of thin regular map representations of hypermaps

There are two well known maps representations of hypermaps, namely the Walsh and the Vince map representations, being dual of each other. They correspond to normal sub groups of index two of a free product Γ = (C2 × C2) ∗ C2 which decompose as “elemen tary” free product C2 ∗ C2 ∗ C2. However Γ has three normal subgroups that decompose as “elementary” free product C2 ∗ C2 ∗ C2, the third of these sbgroups giving the less known petrie-path map representation. By relaxing the “elementary” free product condition to free product of rank 3, and under the extra condition “words of smaller length” on the genera tors, we prove that the number of map representations of hypermaps increases to 15 (up to a restrictedly dual), all of which described in this paper.publishe