Behavior of geodesics and tilings of hyperbolic space

The presented work is a research in the field of the geometry of two-dimensional hyperbolic (equipped with a metric of constant negative curvature) manifolds and studies tilings in hyperbolic n-space of arbitrary dimension by polytopes. In part one we introduce a new method (method of colour multilaterals) to describe the global behavior of geodesics on a arbitrary hyperbolic manifolds of dimension two. The best behaved tilings are the face-to-face tilings by convex polytopes. Of a special interest are tilings in hyperbolic n-space. In part two the main results of this publication are obtained for tilings (isohedral non-isohedral face-to-face non- face-to-face) in the hyperbolic n-space of arbitrary dimension for any () by compact and non-compact polytopes and we describe their discrete isometry groups and properties. Torsion free groups are especially important.