## Some hyperbolic three-manifolds that bound geometrically

24 Jun 2020  ·  Kolpakov Alexander, Martelli Bruno, Tschantz Steven T. ·

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. read more

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Geometric Topology