Abstract
The problem of calculating volumes of non-Euclidean polytopes was already studied by Lobachevsky and Schlafli and is of actual interest, in particular, in connection with the structure of volume spectra of hyperbolic space forms. A brief summary of these results and an introduction to the volume problem for hyperbolic polytopes are given. The covolumes of some hyperbolic five-dimensional manifolds are computed; these are built upon hyperbolic crystals of simple combinatorial metrical type, and their volumes are all commensurable with Riemann’s zeta function at three.
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