Asia Ivić Weiss

Chirality and projective linear groups

In recent years the term ‘chiral’ 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[ω]) are used to construct regular polytopes.

Monodromy Groups and Self-Invariance

For every polytope P there is the universal regular polytope of the same rank as P corre- sponding to the Coxeter group C = (∞, . . . , ∞). For a given automorphism d of C, using mon- odromy groups, we construct a combinatorial structure Pd. When Pd is a polytope isomorphic to P we say that P is self-invariant with respect to d, or d-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a d-automorphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of suchpolyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (du- ality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

Self-duality of chiral polytopes

We prove that self-dual chiral polytopes of odd rank possess a polarity, that is, an involutory duality, and give an example showing this is not true in even ranks. Properties of the extended groups, that is of the groups of automorphisms and dualities, of self-dual chiral polytopes are discussed in detail.

Incidence-polytopes of type {6, 3, 3}

From the hyperbolic honeycomb {6, 3, 3} we derive a class of abstract polytopes whose cells are isomorphic to the toroidal maps {6, 3}b,c and vertex figures to tetrahedra. We give a criterion on the finiteness of these incidence-polytopes in terms of the group PSL± (2, ℤ[ω]), leading, among other things, to the explicit recognition of the groups in some interesting special cases.

Regular 4-polytopes related to general orthogonal groups

For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p }. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.

Semisymmetric graphs from polytopes

Every 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.

Quaternionic modular groups

Abstract Matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n -space or hyperbolic ( n +1)-space H n +1 . In particular, groups operating in the hyperbolic plane or hyperbolic 3-space may be represented by 2×2 matrices whose entries are rational integers or real or imaginary quadratic integers. The theory is extended here to groups operating in H 4 or H 5 and matrices over one of the three basic systems of quaternionic integers. Quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of H 4 and H 5 . For four-dimensional groups the division ring of quaternions is treated as a Clifford algebra. Results in hyperbolic 5-space derive from the homeomorphism of inversive 4-space and the quaternionic projective line.

Free extensions of chiral polytopes

Abstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If ^C is a chiral polytope with regular facets J-, then among all chiral polytopes with facets %^ there is a universal such polytope P, whose group is a certain amalgamated product of the groups of ^C and J. Finite extensions are also discussed.

Incidence-polytopes with toroidal cells

A type of partially ordered structures called incidence-polytopes generalizes the notion of polyhedra in a combinatorial sense. The concept includes all regular polytopes as well as many well-known configurations. We use hyperbolic geometry to derive certain types of incidence-polytopes whose cells are isomorphic to maps of type {4, 4}, {6, 3}, or {3, 6} on a torus. For these structures we give a criterion on the finiteness in terms of groups of 2 × 2 matrices, leading among other things to the explicit recognition of the groups in some interesting special cases.

Problems on Polytopes, Their Groups, and Realizations

SummaryThe paper gives a collection of open problems on abstract polytopes that were either presented at the \emph{Polytopes Day in Calgary} or motivated by discussions at the preceding \emph{Workshop on Convex and Abstract Polytopes\/} at the Banff International Research Station in May~2005.