Pointed Spherical Tilings and Hyperbolic Virtual Polytopes

The paper presents an introduction to the theory of hyperbolic virtual polytopes from the viewpoint of combinatorial rigidity theory. Namely, we give a shortcut for a reader who is acquainted with the notions of Laman graphs, 3D liftings and pointed tilings. From this viewpoint, a hyperbolic virtual polytope is a stressed pointed graph embedded in the sphere S. The advantage of such a presentation is that it gives an alternative and the most convincing proof of existence of hyperbolic polytopes (and therefore, counterexamples to A.D. Alexandrov’s conjecture).