| The 3-manifold is the main subject in the research of the low-dimension topology, but it is complicated. Since the hyperbolic manifold is basal, we can obtain some properties of the complicated manifold by studying it. In this paper, we will consider the degenerating handle additions and other problems of the hyperbolic manifolds by combinatorial method.If we obtain a nonhyperbolic manifold by doing handle additions along the separating slopes which are on the boundary of a hyperbolic manifold, the handle additions are called degenerating handle additions, the slopes are called degenerating slopes. In this paper, we will estimate the geometrical intersection number of two degenerating slopes.In the case of torus boundary components, some accurate results have been given. In the case of non-torus boundary components, M.Scharlemann and Y-Q wu have proved the Theorem: suppose M is a hyperbolic 3-manifold with boundary, and α,β are two separating essential curves on a genus g > 1 boundary component of M, if M[α], M[β] are both nonhyperbolic, then Δ(α,β) ≤ 14. On base of this Theorem, in this paper, we'll give more fined results. That is: Suppose M is a hyperbolic 3-manifold with boundary, and α,β are two separating slopes on dM.(1) If M[α] is annular while M[β] is reducible, then Δ(α,β) ≤ 8;(2) If both M[α] and M[β] are annular, then Δ(α,β) ≤ 8;(3) if M[α] is annular while M[β] is toroidal, then Δ(α,β) ≤ 10. The structure of this paper are as follows:In the first chapter, we introduce the development and methods of the research of 3-manifolds theory. and also the background of the research and the results of this paper.In the second chapter, we expatiate the basic definitions and properties of curves and surfaces in 3-manifolds. we also introduce some theorems about the construction of 3-manifolds.In the third chapter, we expatiate the basic definitions and correlative conclusions of the graph theory.In the last chapter, we prove several lemmas and also give the proof of the central theorem.
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