Of a special interest are tilings in hyperbolic n-space . It is natural to extend the study of tiling problems to the hyperbolic plane as well as hyperbolic spaces of higher dimension. In this work we consider Karoly Böröczky tilings in hyperbolic space in arbitrary dimension, study some properties and some useful consequences of this Böröczky's construction. In the given work it will be shown, that Böröczky tiling has one more remarkable property using them it is simple to make examples of not face-to-face tilings of the hyperbolic n-dimensional space composed of congruent (equal),...

The main purpose of this work is the given a new constructive method for solving the problem of the behavior of geodesic on hyperbolic surfaces of genus g, k punctures and with n geodesic boundary components. At first:1) we obtain a complete classification of all possible geodesic curves on the simplest hyperbolic 2-manifolds (hyperbolic horn; hyperbolic cylinder; parabolic horn (cusp), hyperbolic pants); 2) on surface of genus 2; Finally: 3) on compact closed hyperbolic surface without boundarie (general case); 4) on hyperbolic surface of genus g and with n geodesic boundary components; 5)...