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 ≥ 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.
Some hyperbolic three-manifolds that bound geometrically
MARTELLI, BRUNO;
2015-01-01
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 ≥ 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.File | Dimensione | Formato | |
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