Some hyperbolic three-manifolds that bound geometrically

@inproceedings{Kolpakov2015SomeHT,
  title={Some hyperbolic three-manifolds that bound geometrically},
  author={Alexander Kolpakov and Bruno Martelli and Steven T. Tschantz},
  year={2015}
}
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… Expand

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References

SHOWING 1-10 OF 29 REFERENCES
Constructing hyperbolic manifolds which bound geometrically
Let H denote hyperbolic n-space, that is the unique connected simply connected Riemannian manifold of constant curvature −1. By a hyperbolic n-orbifold we shall mean a quotient H/Γ where Γ is aExpand
All flat three-manifolds appear as cusps of hyperbolic four-manifolds
Abstract There are ten diffeomorphism classes of compact, flat 3-manifolds. It has been conjectured that each of these occurs as the boundary of a 4-manifold whose interior admits a complete,Expand
Three-Dimensional Manifolds Defined by Coloring a Simple Polytope
In the present paper we introduce and study a class of three-dimensional manifolds endowed with the action of the group ℤ23 whose orbit space is a simple convex polytope. These manifolds originateExpand
Hyperbolic four-manifolds with one cusp
We introduce an algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologicallyExpand
On right-angled reflection groups in hyperbolic spaces
We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space $\Bbb H^n$ may only exist if $n\leq 14.$ We also provide a family of such polyhedra of dimensionsExpand
ALMOST FLAT MANIFOLDS
1.1. We denote by V a connected ^-dimensional complete Riemannian manifold, by d = d(V) the diameter of V, and by c = c(V) and c~ = c~(V), respectively, the upper and lower bounds of the sectionalExpand
On the number of ends of rank one locally symmetric spaces
Let Y be a noncompact rank one locally symmetric space of finite volume. Then Y has a finite number e.Y/ > 0 of topological ends. In this paper, we show that for any n2 N , the Y with e.Y/ n that areExpand
Gravitational instantons of constant curvature
In this paper, we classify all closed flat 4-manifolds that have a reflective symmetry along a separating totally geodesic hypersurface. We also give examples of small-volume hyperbolic 4-manifoldsExpand
The orientability of small covers and coloring simple polytopes
Small Cover is an -dimensional manifold endowed with a Z2 action whose orbit space is a simple convex polytope . It is known that a small cover over is characterized by a coloring of which satisfiesExpand
Convex polytopes, Coxeter orbifolds and torus actions
0. Introduction. An n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n. In this paper we investigate certain group actions on manifolds, whichExpand
...
1
2
3
...