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Hyperbolic manifolds given by non-face-to-face incidences.
by
Florin Damian
State University of Moldova
One of well-known hyperbolic manifolds is obtained as the complement of the Whitehead link to the three-dimensional sphere S3. This manifold has as its geodesic submanifold the sphere wits 3 cusps, which is two-sided embedded. The same manifold can be constructed from the non-compact regular octahedron of finite volume. The sphere with 3 cusps corresponds to some hyperfaces of this octahedron. Some neconstructions of the manifold along the mentioned submanifold yield manifolds for which the octahedron has non-face-to-face incidences. In the communications we will give analogous examples of hyperbolic n-manifolds (n=4, 5) obtained by non-face-to-face incidences of hyperfaces of some fundamental polytopes [2, 3]. We would like to remark that the rich symmetry group of submanifold is essential in this construction. For the three-dimensional Euclidean torus all non-face-to-face incidences of the parallelohedron were classified from the combinatorial point of view by M.I. Shtogrin. For the hyperbolic spaces Hn (n=3, 4, 5) this method permits to construct some complicated non-face-to-face tile-transitive tilings [1].
[1] Balcan V., Damian F. Despre descompuneri"not face-to-face" pentru E3 in stereoedre a formelor spatiale euclidiene diferite de tor. Simpozionul stiintific al Academiei de Studii Economice din Moldova 1997, Chisinau, p. 446-447.
[2] Damian F. Symmetry and complete hyperbolic manifolds of finite volume. Satel. Conf. of ECM'96 Symmetry and antisymmetry in Mathematics, Formal Languages and Computer Science. Brasov, 1996, p.39-40.
[3] Damian F.L. Hyperbolic 5-manifolds with cusps over non-toric euclidian spatial form. International Conference Dedicated to the 90th Anniversary of L.S.Pontryagin. Algebra, Geometry and Topology, Moscow, 1998, pp. 114-116.(in russian)
Date received: March 1, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caet-09.