By Weiping Li, Loretta Bartolini, Jesse Johnson, Feng Luo, Robert Myers
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Example text
Let U be a component of M(0,μ] . The set of points removed from U is: {x ∈ U |d(x, M \ U ) > d}. When U is a solid torus neighborhood of a closed short geodesic γ, the set of points removed is {x ∈ U |d(x, γ) ≤ d(X, γ) − d}, and is either empty or an open solid torus neighborhood of γ. In the first case U ⊂ X and in the second case we remove a neighborhood of γ. When U is a cusp, (M \ X) ∩ U is isotopic to U . This establishes (2). Let γ ⊂ M be a geodesic of length less than 2R. Then for every p ∈ γ, injM (p) < R.
Note that a convex polyhedron is not required to be of bounded diameter or finite sided (that is, the intersection of finitely many half spaces). 5. Vi is a convex polyhedron that projects onto Vi Proof. It is immediate that Vi is a convex polyhedron. xi , p˜] is the shortest geodesic from p˜ to any preimage of Given any p˜ ∈ Vi , [˜ xi , p˜] is the shortest geodesic from the projection {x1 , . . , xN }. The projection of [˜ p) ∈ Vi . As p˜ was an arbitrary point of p˜ to {x1 , . . , xN }. It follows easily that π(˜ of Vi , we see that Vi projects into Vi .
For another geometric study of the triangulation of the thick part see Breslin’s [2]. 1 (Jørgensen, Thurston). Let μ > 0 be a Margulis constant for H3 . Then for any d > 0 there exists a constant K > 0, depending on μ and d, so that for any complete finite volume hyperbolic 3-manifold M , Nd (M[μ,∞) ) can be triangulated using at most KVol(M ) tetrahedra. The manifold Nd (M[μ,∞) ) is the closed d-neighborhood of the μ-thick part of M and is denoted by X throughout this paper. By the Margulis Lemma, M \ X consists of disjoint cusps and open solid tori, and each of these solid tori is a regular neighborhood of an embedded closed geodesic.