A Combination Theorem for Convex Hyperbolic Manifolds, with Applications to Surfaces in 3-manifolds
نویسنده
چکیده
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, and whose intersection is a separable subgroup, there are finite index subgroups which generate a subgroup that is an amalgamated free product. Constructions of infinite volume hyperbolic n-manifolds are described by gluing lower dimensional manifolds. It is shown that every slope on a cusp of a hyperbolic 3-manifold is a multiple immersed boundary slope. If the fundamental group a hyperbolic 3-manifold contains a maximal surface group not carried by an embedded surface then it contains a freely indecomposable group with second betti number at least 2.
منابع مشابه
J ul 2 00 5 A Combination Theorem For Convex Hyperbolic Manifolds , With Applications To Surfaces In 3 - Manifolds
We prove the convex combination theorem for hyperbolic n-manifolds. Many applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generated a subgroup that is an am...
متن کامل1 5 Ju l 2 00 5 A Combination Theorem For Convex Hyperbolic Manifolds , With Applications To Surfaces In 3 - Manifolds .
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic nspace, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamat...
متن کاملul 2 00 5 A Combination Theorem For Convex Hyperbolic Manifolds , With Applications To Surfaces In 3 - Manifolds
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic nspace, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamat...
متن کاملReferences for Geometrization Seminar References
[1] L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. Math. 72 (1960), pp. 413– 429 [2] F. Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. Math. 124 (1986), pp. 71–158 [3] D. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, in Analytical and Geometric Aspects of Hyperbolic Space, LMS 111 (198...
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Introduction In these lecture notes we will give a quick introduction to 3–manifolds, with a special emphasis on their fundamental groups. The lectures were held at the summer school 'groups and manifolds' held in Münster July 18 to 21 2011. In the first section we will show that given k ≥ 4 any finitely presented group is the fundamental group of a closed k–dimensional man-ifold. This is not t...
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