On hyperbolic 5-manifolds of finite volume
by
Florin Damian
Inst. of Math. & Comp. Sci., Acad. of Sci. and Moldova State University

Possibly the first hyperbolic 5-manifolds of finite volume which share a fundamental polyhedron is constructed.

Using the variation of a metric parameter, a non-compact polytope of finite volume tessellating the hyperbolic spaces H⁵ is constructed. A very simple identification of its 48 hyper-faces is given, which yields a non-compact orientable hyperbolic manifold M⁵ with cusps over the most symmetric 4-dimensional torus [2]. The stabilizer of any point of the torus is isomorphic to F₄ - group of symmetry of regular 24-cells. Two-sided embedded totally geodesic 4-submanifolds M⁴ are indicated. Different metric reconstructions [4] of M⁵ along M⁴ permit to obtain new 5-manifolds, in particular, non-orientable, with cusps over atoroidal euclidean spaces forms, etc.

For clarity the analogues of this construction in dimensions 3 and 4 are shown. The same constructions n=3, 4 see also in [1, 3, 4].

1. Davis M.W., A hyperbolic 4-manifold. Proceeding of the American Mathematical Sosiety, 93, no. 2, 1985, pp. 325-328

2. Damian F.L., Hyperbolic 5-manifolds with cusps over non-toric euclidian spatial fofm, International Conference Dedicated to the 90th Anniversary of L.S.Pontryagin. Algebra, Geometry and Topology, Moscow, 1998, pp. 114-116.(in russian)

3. Dubravko Ivansi\'c, Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron. Preprint 1997-052, Mat. Sci. Res. Inst., Berkeley, 1997.

4. Wielemberg N.J., Hyperbolic 3-manifolds which share a fundamental polyhedron, Proceedings of the 1978 Stony Brook Conference (Ann. Math. Stud. V. 97) Princeton Univ. Press, 1981. P. 505- 513.

Date received: June 17, 1999

Copyright © 1999 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 # cacy-42.