Regular polyhedra related to projective linear groups

AbstractFor each odd prime p, there is a regular polyhedron Πp of type {3,p} with 12(p2−1) vertices whose rotation group is PSL(2,p); its complete group is PSL(2,p) × Z2 or PGL(2,p) as p ≡ 1 or 3 (mod 4). If p ≡ 1 (mod 4), then the group of Πp contains a central involution, and identification of antipodal vertices under this involution yields another regular polyhedron Πp2 of type {3,p} with 14(p2)−1) vertices and group PSL(2,p). Realizations of the polyhedra in euclidean spaces are briefly described

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

The Geometry of T-Varieties

This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.Comment: 42 pages, 17 figures. v2: minor changes following the referee's suggestion

Computing symmetry groups of polyhedra

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used for instance in integer linear programming.Comment: 20 pages, 1 figure; containing a corrected and improved revisio