22 research outputs found
Regular Incidence Complexes, Polytopes, and C-Groups
Regular incidence complexes are combinatorial incidence structures
generalizing regular convex polytopes, regular complex polytopes, various types
of incidence geometries, and many other highly symmetric objects. The special
case of abstract regular polytopes has been well-studied. The paper describes
the combinatorial structure of a regular incidence complex in terms of a system
of distinguished generating subgroups of its automorphism group or a
flag-transitive subgroup. Then the groups admitting a flag-transitive action on
an incidence complex are characterized as generalized string C-groups. Further,
extensions of regular incidence complexes are studied, and certain incidence
complexes particularly close to abstract polytopes, called abstract polytope
complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder,
A. Deza, and A. Ivic Weiss (eds), Springe
Chirality and projective linear groups
AbstractIn recent years the term âchiralâ has been used for geometric and combinatorial figures which are symmetrical by rotation but not by reflection. The correspondence of groups and polytopes is used to construct infinite series of chiral and regular polytopes whose facets or vertex-figures are chiral or regular toroidal maps. In particular, the groups PSL2(Zm) are used to construct chiral polytopes, while PSL2(Zm[i]) and PSL2(Zm[Ï]) are used to construct regular polytopes
Semisymmetric graphs from polytopes
AbstractEvery finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
Hexagonal extensions of toroidal maps and hypermaps
The rank 3 concept of a hypermap has recently been generalized to a higher rank structure in which hypermaps can be seen as âhyperfacesâ but very few examples can be found in literature. We study finite rank 4 structures obtained by hexagonal extensions of toroidal hypermaps. Many new examples are produced that are regular or chiral, even when the extensions are polytopal. We also construct a new infinite family of finite nonlinear hexagonal extensions of the tetrahedron.The authors would like to thank two anonymous referees for their numerous helpful and insightful comments. This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, by NSERC and by the Portuguese Foundation for Science and Technology (FCT-Fundação para a CiĂȘncia e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.publishe
Constructing highly regular expanders from hyperbolic Coxeter groups
A graph is defined inductively to be -regular if
is -regular and for every vertex of , the sphere of radius
around is an -regular graph. Such a graph is said
to be highly regular (HR) of level if . Chapman, Linial and
Peled studied HR-graphs of level 2 and provided several methods to construct
families of graphs which are expanders "globally and locally". They ask whether
such HR-graphs of level 3 exist.
In this paper we show how the theory of Coxeter groups, and abstract regular
polytopes and their generalisations, can lead to such graphs. Given a Coxeter
system and a subset of , we construct highly regular quotients
of the 1-skeleton of the associated Wythoffian polytope ,
which form an infinite family of expander graphs when is indefinite and
has finite vertex links. The regularity of the graphs in
this family can be deduced from the Coxeter diagram of . The expansion
stems from applying superapproximation to the congruence subgroups of the
linear group .
This machinery gives a rich collection of families of HR-graphs, with various
interesting properties, and in particular answers affirmatively the question
asked by Chapman, Linial and Peled.Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest
Vinber
Highly symmetric hypertopes
We study incidence geometries that are thin and residually
connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytopes theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case
and C+-groups in the chiral case