Some hyperbolic three-manifolds that bound geometrically

Kolpakov, Alexander; Martelli, Bruno; Tschantz, Steven

doi:10.1090/proc/12520

Some hyperbolic three-manifolds that bound geometrically
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by Alexander Kolpakov, Bruno Martelli and Steven Tschantz PDF

Proc. Amer. Math. Soc. 143 (2015), 4103-4111 Request permission

Erratum: Proc. Amer. Math. Soc. 144 (2016), 3647-3648.

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.

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