An extension theorem in the theory of regular tilings and its applications

The extension theorem gives necessary and sufficient conditions under which a convex polytope admits a regular (tile-transitive) tiling of the d-dimensional space of constant curvature. The extension theorem was first stated in [N. P. Dolbilin, Dokl. Akad. Nauk SSSR {bf 230} (1976), no.~3, 516--519; [msn] MR0420414 (54 #8428) [/msn]]. In the paper under review a detailed proof of the extension theorem is given. Also, from this theorem the known theorems on fundamental polytopes (by Venkov and by Coxeter) are derived.