We describe geometric methods that allow to build and investigate hyperbolic manifolds of dimensions 3, 4 and 5, with certain predefined properties, such as cusps geometry, the geometry of a totally geodesic submanifold, etc. Some result of these constructions look like as a generalization of 2-dimensional pants. Also we use these examples and methods of metric reconstruction to obtain non-face-to-face incidence schemes for fundamental polyhedra. As a result, new manifolds and some exotic tilings on universal coverage are obtained. The talk will be focused on the transfer of methods of discrete geometry to topology and vice versa.