Dehn surgery on Anosov flows
Dehn surgery on Anosov flows
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DOI：
10.1007/bfb0061421
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发表时间：
1983
影响因子：
0.9
通讯作者：
S. Goodman
中科院分区：
文献类型：
--
作者：
S. Goodman
最近发现了三维流形上Anosov流的几个新的有趣的例子。在文献[2]中，Handel和瑟斯顿给出了非代数Anosov流的第一个例子。在它们的构造中，测地线流沿着不可压缩的环面被分割，并用Dehn扭曲来调节，保持了Anosov结构，但得到的流形既不是Sl-丛，也不是Sl上的环丛。它是一个图流形，并且流是体积保持的。Franks和Williams[1]最近给出了一个不保体积的Anosov流的例子，因此也不是代数的。流形是沿边界环面标识的图形8结的补码的两个副本。每个部分都有一个双曲结构，但不是整个流形。在这篇文章中，我们给出了更多支持Anosov流的封闭3-流形的新例子。这是通过在任何Anosov流的周期轨道附近进行保留经度的Dehn手术来实现的。这些例子中的许多也是非代数的。此外，所产生的一些流形是双曲线的，因此是阿托尔型的，回答了文献[2]中提出的一个问题。
Several new and interesting examples of Anosov flows on 3-manifolds have been found recently. In [2], Handel and Thurston give the first examples of non-algebraic Anosov flows. In their construction a geodesic flow is cut apart along an incompressible torus and reglued with a Dehn twist, preserving an Anosov structure, but the resulting manifold is neither an Sl-bundle or a torus-bundle over Sl. It is a graph manifold, and the flow is volume-preserving. Franks and Williams [1] have recently given an example of an Anosov flow which is not volume-preserving; hence, also not algebraic. The manifold is two copies of the complement of the figure eight knot, identified along the boundary torus. Each piece has a hyperbolic structure, but not the entire manifold.In this paper, we produce more new examples of closed 3-manifolds which support Anosov flows. This is accomplished by doing a Dehn surgery, preserving longitudes, on a neighborhood of a periodic orbit of any Anosov flow. Many of these examples are also non-algebraic. Further, some of the manifolds produced are hyperbolic, and hence atoroidal, answering a question posed in[2].