Research on the deformation space of hyperbolic structures on manifolds

本研究的主要目的是研究具有非空奇异集的三维双曲锥流形M的变形。Fujii和Ochiai首席研究员构造了一个求解Fuchsian型常微分方程组E的算法，该算法描述了M的变形。事实上，Fujii和Ochiai利用计算机成功地求解了常微分方程组，并实现了该算法。这一结果发表在《公共科学图书馆》杂志上。Res.Inst Math。SCI 43(2007)。Fujii具体地发现了曲面上双曲结构的退化与常微分方程组E的确定性奇点汇合之间的某种联系。这一结果发表在《九州数学》杂志上。61(2007)。这一结果在2006年6月在东京都大学举办的题为“双曲群在特定流形和相关主题上的行动”的研讨会上作了报告。在此基础上，Fujii构造了一个离散群G的测地自动机结构，并利用计算机程序计算了G的增长函数。这一结果于2008年1月在冈山大学举行的“Riemann曲面和离散群”会议上发表。UE研究了Seifert三维流形的Fukumoto-Furuta不变量和Neumann-Siebenmann不变量。UE还研究了它的自旋有理同调协边不变性。森下研究员研究了三维拓扑和算法之间的类比。特别是Morishita研究了SL Chem-Simons理论和Hida-Mazur理论之间的一些关系。研究人员川久美从微分几何的角度研究了Riemann模空间。实际上，Kawazumi发现了映射类群的某些化学形式与Johnson同态之间的某种联系。

The main purpose of this research was to study deformations of a 3-dimensional hyperbolic cone manifold M with non-empty singular set. The head investigator Fujii and the investigator Ochiai constructed an algorithm for solving ordinary differential equations E of Fuchsian type which describe deformations of M. In fact, Fujii and Ochiai succeeded in solving the ordinary differential equations by making use of computers with performing the algorithm. This result was reported in the journal, Publ. Res.Inst Math. Sci 43 (2007). Fujii found some relation between the degeneration of hyperbolic structures on a surface and the confluence of deterministic singular points of ordinary differential equations E concretely. This result was reported in the joumal, Kyushu J.Math. 61 (2007). This result was reported in the workshop, "Actions of hyperbolic groups on specific manifolds and related topics", at Tokyo Metropolitan University, in June 2006. Furthermore, Fujii constructed a geodesic automatic structure for some discrete group G and Computed the growth function of G by making use of a computer program. This result was reported at tie conference, "Riemann surfaces and discrete groups", at Okayama University in January, 2008.The investigator Ue researched the Fukumoto-Furuta invariant for Seifert 3-manifolds and the Neumann-Siebenmann invariant. Ue also studied its spin rational homology cobordism invariance. The investigator Morishita researched the analogy between 3-dimensional topology and Arithmetic. In particular, Morishita studied some relation between SL Chem-Simons theory and Hida-Mazur- theory. The investigator Kawazumi researched Riemann moduli space from the view points of differential geometry. In fact, Kawazumi found some relation between some Chem forms and Johnson homomorphisms for the mapping class groups.