The Geometry of R{covered foliations
The Geometry of R{covered foliations
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R{覆盖叶面的几何形状
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发表时间：
1999
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通讯作者：
Danny Calegari
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作者：
Danny Calegari
我们从横向几何的角度研究了 3{流形的 R{覆盖叶状结构。对于环形 3{流形 M 中的 R{ 覆盖叶理，我们证明 f M 可以通过规范圆柱 S 1 R 部分压缩，其中 1(M ) 通过 Homeo(S 1 )Homeo(R) 的元素作用，其中 S 1 因子与 e F 每个叶子的无穷大处的圆规范地一致。我们构建了一对非常完整的真实叠片，它们彼此横向并横向于 F ，它们将 F 的每片叶子结合在一起。这对叠片可以被吹散以给出 F 的横向调节伪阿诺索夫流，类似于具有伪阿诺索夫单向性的圆上表面束的瑟斯顿结构定理。该结构存在的推论是，底层流形 M 是同伦刚性的，即与恒等式同伦的自同胚是恒等式的同位素。此外，无穷远的乘积结构在叶状结构 F 到 R{覆盖叶状结构的变形下是刚性的，从某种意义上说，1(M )i nHomeo((S 1 )t) 的表示对于由 t 参数化的族来说都是共轭的。另一个推论是环境流形具有字双曲基本群。最后，我们推测这些结果与证明紧叶 3{流形的几何化猜想的程序之间的联系。 AMS 分类编号 主要：57M50、57R30 次要：53C12
We study R{covered foliations of 3{manifolds from the point of view of their transverse geometry. For an R{covered foliation in an atoroidal 3{manifold M , we show that f M can be partially compactied by a canonical cylinder S 1 R on which 1(M ) acts by elements of Homeo(S 1 )Homeo(R), where the S 1 factor is canonically identied with the circle at innity of each leaf of e F . We construct a pair of very full genuine laminations transverse to each other and to F , which bind every leaf of F . This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov flow for F , analogous to Thurston’s structure theorem for surface bundles over a circle with pseudo-Anosov monodromy. A corollary of the existence of this structure is that the underlying manifold M is homotopy rigid in the sense that a self-homeomorphism homotopic to the identity is isotopic to the identity. Furthermore, the product structures at innity are rigid under deformations of the foliation F through R{covered foliations, in the sense that the representations of 1(M )i nHomeo((S 1 )t) are all conjugate for a family parameterized by t. Another corollary is that the ambient manifold has word-hyperbolic fundamental group. Finally we speculate on connections between these results and a program to prove the geometrization conjecture for tautly foliated 3{manifolds. AMS Classication numbers Primary: 57M50, 57R30 Secondary: 53C12