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Steklov Mathematical Institute Seminar
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A cycle of work on the theory of regular decompositions of spaces S. S. Ryshkov, M. I. Shtogrin, N. P. Dolbilin |
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Abstract: A famous theorem of Voronoi states that for each primitive parallelohedron there is an affinely equivalent Dirichlet–Voronoi parallelohedron (existence theorem). Ryshkov proved that such a Dirichlet–Voronoi parallelohedron is unique up to similarity (uniqueness theorem). He also proved that for each type of It was proved that for a space of constant curvature and arbitrary dimension and for any discrete group of motions of it having a compact fundamental domain there are only finitely many combinatorial types of regular Dirichlet–Voronoi decompositions (Shtogrin). So-called polycycles, which have important applications, were investigated. A polycycle is defined to be a cellular decomposition of the disk that admits a continuous locally homeomorphic cellular map to a Platonic decomposition of the sphere, the Euclidean plane, or the Lobachevskii plane. A criterion was established for a given graph to be the edge skeleton of some polycycle (Shtogrin). It was shown that a simply connected It was shown that if a family of decompositions of a space (Euclidean or Lobachevskii) with a finite protoset and with a certain local rule is at most countable, then among these decompositions at least one is crystallographic. This theorem generalizes known results on uncountability of socalled aperiodic families (Dolbilin). |