On Finiteness of the Number of Boundary Slopes of Immersed Surfaces in 3-manifolds
نویسندگان
چکیده
For any hyperbolic 3-manifold M with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of ∂M or the volume of M is bounded above. When the volume is bounded above, then area of ∂M is bounded above and the length of closed geodesic on ∂M is bounded below. We say that a proper immersion of a surface F into M is an essential surface if it is incompressible and ∂-incompressible, meaning that the immersion induces an injection of the fundamental group and relative fundamental group. Let c be an essential simple loop on the boundary ∂M of a compact 3-manifold M . If there is a proper immersion of an essential surface F into M such that each component of ∂F is homotopic to a multiple of c, we call c a boundary slope of F . We are interested in the following two questions:
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