Hecguard Splittings and genetic structures of 3-manifolds

本课题取得的主要成果如下：1。Akiyoshi, Sakuma， Wada和Yamashita已经完成了一份预印本（256页），其中完整地阐述了Jorgensen关于准紫狐穿孔环面群的Ford域的理论，包括一个完整的证明。我们计划写一篇续文来解释我们将他的理论扩展到准紫红色穿孔环面空间之外，并解释2-桥结的桥结构与其互补的完全双曲结构之间的关系。Epstein-Penner通过Minkowski空间中的凸壳构造引入了有限体积尖头双曲流形的欧几里得分解。akiyoshii - sakuma将构造推广到（可能的）无限体积尖头双曲流形，并引入了这些流形的ef分解。此外，akiyoshii - sakuma - wada yamashita还研究了ep -分解与尖头超曲流形弯曲层合的关系。akiishi - miyachi - sakuma将Bowditch关于双曲被刺破环面束的McShane恒等式的变化推广到一般的双曲被刺破面束。

The main results obtained by this project are as follows.1.Akiyoshi, Sakuma, Wada and Yamashita have completed a preprint (256 pages) which gives a full exposition of Jorgensen's theory for the Ford domains of quasifuchsian punctured torus groups, including a full proof. We plan to write a sequel of the paper to explain our extension of his theory to the outside of the quasifuchsian punctured torus space and to explain the relationship between the bridge structure of a 2-bridge knot and the complete hyperbolic structure of its complement.2.Epstein-Penner has introduced the Euclidean decompositions of finite-volume cusped hyperbolic manifolds through a convex hull construction in the Minkowski space. Akiyoshi-Sakuma has generalized the construction to (possibly) infinite-volume cusped hyperbolic manifolds and introduced EPH-decompositions of these manifolds. Moreover, relation between the EPH-decompositions and the bending laminations of cusped hyper-bolic manifolds were studied by Akiyoshi-Sakuma-Wada Yamashita.3.Akiyoshi-Miyachi-Sakuma have generalized Bowditch's variation of McShane's identity for hyperbolic punctured torus bundles to general hyperbolic punctured surface bundles.