Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3–manifolds
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چکیده
A closed, and say orientable, Riemannian 3–manifold (M, ρ) is hyperbolic if the metric ρ has constant sectional curvature κρ = −1. Equivalently, there is a discrete and torsion free group Γ of isometries of hyperbolic 3–space H3 such that the manifolds (M, ρ) and H3/Γ are isometric. It is well-known that the fundamental group π1(M) of every closed 3–manifold which admits a hyperbolic metric is a non-elementary Gromov hyperbolic group and hence that it is is infinite and does not contain free abelian subgroups of rank 2. A 3–manifold M whose fundamental group does not have subgroups isomorphic to Z2 is said to be atoroidal. Another well-known property of those 3–manifolds which admit a hyperbolic metric is that they are irreducible, ie every embedded sphere bounds a ball. Surprisingly, these conditions suffice to ensure that a closed 3–manifold M admits a hyperbolic metric.
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تاریخ انتشار 2007