Ends of covering 3-manifolds

覆盖 3 歧管的末端

• 批准号: 05640132
• 负责人: SOMA Teruhiko
• 金额: $ 0.96万
• 依托单位: Tokyo Denki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1993
• 资助国家: 日本
• 起止时间: 1993 至 1994
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-05640132/
• 关键词: hyperbolic 3-manifolds; ending lamination; bounded cohomology; topologically tame end; geometrically tame end; rigidity theorem; covering theorem; closed surface; hyperbolic 3-manifold; topologically tame end; 双曲線多様体; 3次元多様体; エンド不変量; 有界コホモロジー

Through the studies of ends of hyperbolic 3-manifolds, we obtained some results concering bounded cohomology.First, we showed, by using a certain hyperbolic 3-manifold, that the naturally defined pseudonorm on the third bounded cohomology H^3_(Z*Z ; R) is not a norm. As a corollary to this result, it is shown that, for any group G admitting a surjective homomorphism f : G*Z*Z,the pseudonorm on H^3_(G ; R) is not a norm.Next, we presented a rigidity theorem of certain hyperbolic 3-manifolds of infinite volume. Let SIGMA_g be a closed, connected, orientable surface of genus g>1. For any hyperbolic 3-manifold M homotopy-equivalent to SIGMA_g, the volume of M is infinite. Here, we consider the case where M has no geometrically finite ends, that is, M is doubly-degenerated. If the infimum inj(M) of injectivity radii at all points in M is positive, then by Minsky's Ending Lamination Theorem, the hyperbolic structure on M is determined only by its ending laminations. For any such M,M' with inj(M) >0, inj(M') > 0, we presented a condition equivalent to that M and M' have the same ending laminations in terms of the fundamental classes [omega_M], [omega_<M'>] defined as elements of H^3_(SIGMA_g ; R). Though [omega_M] = [omega_<M'>] is a sufficient condition for M isometric to M', we proved that a (formaly) weaker condition can be a necessary and sufficient condition for that.Furthermore, by using R.Canary's Covering Theorem, we showed that a topologically tame Kleinian group G is geometrically finite if and only if the funtametal class of G in H^3_(G ; R) is zero. As an application, we proved that, for any group G with a surjevtive homomorphism f : G*Z*Z,the dimension of H^3_(G ; R) is the cardinarity of continuum.

通过对双曲3-流形端点的研究，我们得到了一些关于有界上同调的结果：首先，我们利用某个双曲3-流形证明了第三有界上同调H ^3_（Z*Z ; R）上自然定义的端点不是范数;作为这一结果的推论，证明了对任何允许满同态f：G*Z*Z的群G，H^3_（G ; R）上的范数不是范数.设SIGMA_g是亏格g>1的闭连通可定向曲面.对于任意与SIGMA_g同伦等价的双曲3-流形M，M的体积是无穷大的.这里，我们考虑M没有几何有限端点的情况，即M是双重退化的。如果M上所有点的内射半径的下确界inj（M）都是正的，则根据Minsky的终结层定理，M上的双曲结构仅由它的终结层决定。对于任意这样的M，M'，其中inj（M）>0，inj（M'）> 0，我们给出了一个等价的条件，即M和M'具有相同的以H ^3_（SIGMA_g ; R）中元素定义的基本类[omega_M]，[omega_<M'>]表示的终止层。虽然[omega_M] = [omega_<M '>]是M等距于M'的充分条件，但我们证明了一个（形式上）较弱的条件也可以是M等距于M '的充分必要条件，并利用R.Canary的覆盖定理证明了拓扑驯服的Kleinian群G是几何有限的当且仅当G在H^3_（G ; R）中的基金属类为零.作为应用，我们证明了对任意满同态为f：G*Z*Z的群G，H^3_（G ; R）的维数是连续统的基数。

项目成果

Teruhiko Soma: "A rigidity theorem for Haken manifolds" Math. Proc. Cambridse Phil. Soc. (発表予定).

Teruhiko Soma：“Haken 流形的刚性定理”，Cambridse Phil。

Teruhiko Soma: "A rigidity theorem for Haken manifolds" Math. Proc. Cambridge Phil. Soc.(to appear).

Teruhiko Soma：“哈肯流形的刚性定理”数学。

Teruhiko Soma: "Covering 3‐manifolds with almost compact interior" Quart.J.Math.Oxford. 44. 345-353 (1993)

Teruhiko Soma：“用几乎紧凑的内部覆盖 3 个流形”Quart.J.Math.Oxford 44. 345-353 (1993)

Teruhiko Soma: "Equivariant almost homeomorphic maps between S^1 and S^2" Proc. Amer. Math. Soc.(発表予定).

Teruhiko Soma：“S^1 和 S^2 之间的等变几乎同胚映射”Proc。

Teruhiko Soma: "Rotation of spatial graphs" Topology Applications. (to appear).

Teruhiko Soma：“空间图的旋转”拓扑应用。