Asymptotic structures of non-compact hyperbolic 3-manifoIds and differential geometry

非紧双曲3-流形的渐近结构和微分几何

• 批准号: 12640063
• 负责人: OHSHIKA Ken'ichi
• 金额: $ 2.11万
• 依托单位: Osaka University (2001-2003)The University of Tokyo (2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640063/
• 关键词: Hyperbolic 3-manifold; Kleinian group; Deformation space; ending lamination; tameness; Klein群; end invariant; bounded cohomology; 双曲多様体; 関数群; R-樹; Schottky群; 凸芯; 幾何的極限

We have been studying the topological properties of the deformation spaces of hyperbolic structures on 3-manifolds making use of differential geometry and low-dimensional manifold theory. In particular, we investigated the behavior of hyperbolic structures on the boundaries of quasi-conformal deformation spaces, which is known to coincide with the boundaries of the entire deformation spaces. As a result of this line of research, Ohshika proved that Bers-Thurston conjecture, which states that every finitely generated Kleinian group would be an algebraic limit of quasi-conformal deformations of a geometrically finite group, follows once Marden's tameness conjecture is proved to be true. This should be an important progress towards the solution of the ultimate problem of classifying the hyper-bolic structures on a given 3-manifold with finitely generated fundamental group.

利用微分几何和低维流形理论研究了三维流形上双曲结构形变空间的拓扑性质。特别地，我们研究了双曲结构在拟共形变形空间边界上的行为，这是已知与整个变形空间的边界重合的。作为这一研究的结果，Ohshika证明了，一旦Marden的温和性猜想被证明为真，Bers-瑟斯顿猜想，即每个有限生成的Klein群将是几何有限群的拟共形变形的代数极限，就会推出。这应该是朝着解决基本群有限生成的给定三维流形上的双曲结构分类的最终问题的一个重要进展。

项目成果

原靖浩, 南範彦: "Borsuk-Ulam type theorems for compact Lie group actions"Proc.AMS. 132. 903-909 (2004)

Yasuhiro Hara、Norihiko Minami：“紧李群作用的 Borsuk-Ulam 型定理”Proc.AMS. 132. 903-909 (2004)

足立正, 遠藤久顕: "リーマン面の退化族の諸相"数学. 56. 49-72 (2004)

Tadashi Adachi、Hisaaki Endo：“黎曼曲面上简并群的方面”，数学 56. 49-72 (2004)。

I.Nagasaki: "Linearity of dimension functions for semi-linear G-spheres"Proc.AMS. 130. 1843-1850 (2002)

I.Nagasaki：“半线性 G 球体尺寸函数的线性”Proc.AMS。

Hara, Minami: "Borsuk-Ulam type theorems for compact Lie group actions"Proc.AMS. 132. 903-909 (2004)

Hara, Minami：“紧致李群作用的 Borsuk-Ulam 型定理”Proc.AMS。

Ohshika, Ken'ichi: "Density of geometrically finite Kleinian groups. Hyperbolic spaces and discrete groups, II (Kyoto, 2001)"Surikaisekikenkyusho Kokyuroku. No.1270. 93-100 (2002)

Ohshika，Kenichi：“几何有限 Kleinian 群的密度。双曲空间和离散群，II（京都，2001 年）”Surikaisekikenkyusho Kokyuroku。