Depth of Pleated Surfaces in Toroidal Cusps of Hyperbolic 3-manifolds
نویسنده
چکیده
Let F be a closed essential surface in a hyperbolic 3-manifold M with a toroidal cusp N . The depth of F in N is the maximal distance from points of F in N to the boundary of N . It will be shown that if F is an essential pleated surface which is not coannular to the boundary torus of N then the depth of F in N is bounded above by a constant depending only on the genus of F . The result is used to show that an immersed closed essential surface in M which is not coannular to the torus boundary components of M will remain essential in the Dehn filling manifold M(γ) after excluding Cg curves from each torus boundary component of M , where Cg is a constant depending only on the genus g of the surface.
منابع مشابه
Surfaces in Three-manifolds with Hyperbolic Fundamental Group
We show that if a closed irreducible three-manifold with hyperbolic fundamental group contains a surface subgroup satisfying a certain geometric regularity assumption, then the surface is either quasiconvex or a virtual fiber. In the latter case, the manifold is hyperbolic. The regularity condition ensures that we may find algebraic bounds on the surface group which are analogous to the diamete...
متن کاملA Survey of Length Series Identities for Surfaces, 3-manifolds and Representation Varieties
We survey some of our recent results on length series identities for hyperbolic (cone) surfaces, possibly with cusps and/or boundary geodesics; classical Schottky groups; representations/characters of the one-holed torus group to SL(2,C); and hyperbolic 3 manifolds obtained by hyperbolic Dehn surgery on punctured torus bundles over the circle. These can be regarded as generalizations and variat...
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...
متن کاملMinimal Surfaces in Geometric 3-manifolds
In these notes, we study the existence and topology of closed minimal surfaces in 3-manifolds with geometric structures. In some cases, it is convenient to consider wider classes of metrics, as similar results hold for such classes. Also we briefly diverge to consider embedded minimal 3-manifolds in 4-manifolds with positive Ricci curvature, extending an argument of Lawson to this case. In the ...
متن کامل