Research on the deformation space of hyperbolic structures on manifolds

本文主要研究了具有非空奇异集的三维双曲锥流形M的变形问题。首席研究员Fujii和研究员Ochiai构建了一个算法，用于求解描述M.事实上，藤井和落合成功地解决了常微分方程，利用计算机与执行的算法。这一结果发表在Publ. Res.Inst Math.Sci43（2007）期刊上。Fujii具体地发现了曲面上双曲结构的退化与常微分方程E的确定性奇点的汇合之间的某种联系。这一结果报道在Kyushu J.Math.61（2007）的期刊中。这一结果在2006年6月东京都立大学的“双曲群在特定流形上的作用及相关主题”研讨会上报告。此外，藤井还为某个离散群G构造了测地线自动结构，并利用计算机程序计算了G的增长函数。这一结果在2008年1月的冈山大学“Riemann曲面与离散群”会议上发表。还研究了它的自旋有理同调配边不变性。研究员森下研究了三维拓扑和算术之间的类比。特别地，Morishita研究了SL Chem-Simons理论和Hida-Mazur理论之间的一些关系。研究者川澄从微分几何的角度研究了黎曼模空间。实际上，Kawazumi发现了映射类群的某些Chem形式与约翰逊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.