Heegaand spliffings and hyperbolic structures of 3-manifolds

3-流形的heegard分裂的研究一直是3-流形理论的重要课题之一，我们对“非双曲”3-流形的heegard分裂已经有了深入的认识。然而，我们对双曲流形的理解还远远不能令人满意。特别是，据我们所知，双曲结构和heegard分裂之间没有任何关系。在本课题中，我们证明了双桥结补的双曲结构与其桥结构密切相关，双曲补是一种高度分裂。实际上，我们利用双桥结构给出了双桥结补的双曲结构的具体构造。更精确地说，我们在2桥结补上构造了一个连续的双曲锥流形结构族，该结构族沿上下隧道具有奇点，其中锥角从0到2π变化。锥角为0的锥流形结构对应于拟fuchsian一次刺破环面空间的有理边界群，锥角为2π的锥流形结构对应于2桥结补的双曲结构。为了证明这一结果，我们给出了Jorgensen关于拟fuchsian一次穿刺环面群的显式表述和部分充分证明，并将该理论推广到拟fuchsian一次穿刺空间外的群。Masaaki Wada为这个项目开发的计算机程序“OPTI”将Jorgensen的理论及其推广可视化，它不仅是这个项目不可或缺的工具，也是研究Teichmuller空间不可或缺的工具。我们希望我们在这个项目中得到的结果是双曲结构与3流形heegard分裂之间关系研究的开始。

The study of Heegaard splittings of 3-manifolds has been one of the most important themes in 3-manifold theory, and we already have deep understanding of the Heegaard splittings of "non-hyperbolike" 3-manifolds. However, our understanding of those of hyperbolic manifolds is far from satisfaction. In particular, as far as we know, no relationship between the hyperbolic structures and the Heegaard splittigs had been known.In this project we have proved that the hyperbolic structure of a 2-bridge knot complement is intimately related with its bridge structure, which is a kind of Heegaard splitting. In fact, we have given a concrete construction of the hyperbolic structure of a 2-bridge knot complement by using the 2-bridge structure. To be more precise, we have constructed a continuous family of hyperbolic cone-manifold structures on a 2-bridge knot complement which have singularities along the upper and lower tunnels, where the cone angle varies from 0 to 2π. The cone-manifold structure with cone angle 0 corresponds to a rational boundary group of the quasi-Fuchsian once-punctured torus space and that with cone angle 2π gives the hyperbolic structure of the 2-bridge knot complement. To establish this result, we have given an explicit formulation and a full proof to (a part of) the theory announced by Jorgensen on the quasi-Fuchsian once-punctured torus groups, and generalized the theory to that for the groups outside the quasi-Fuchsian once-punctured space. The computer program "OPTI" developed by Masaaki Wada for this project visualizes Jorgensen's theory and its generalization, and it has been an indispensable tool not only for this project but also for the study of Teichmuller spaces. We hope the result we have obtained in this project is the beginning of the study of the relationship between the hyperbolic structures and the Heegaard splittings of 3-manifolds.