Some hyperbolic three-manifolds that bound geometrically
Abstract
A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and $120$-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k\geqslant 1$, we build an orientable compact closed $3$-manifold tessellated by $16k$ right-angled dodecahedra that bounds a $4$-manifold tessellated by $32k$ right-angled $120$-cells. A notable feature of this family is that the ratio between the volumes of the $4$-manifolds and their boundary components is constant and, in particular, bounded.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2013
- DOI:
- 10.48550/arXiv.1311.2993
- arXiv:
- arXiv:1311.2993
- Bibcode:
- 2013arXiv1311.2993K
- Keywords:
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- Mathematics - Geometric Topology;
- 57N16;
- 52B11;
- 52C45
- E-Print:
- 10 pages, 1 figure