A random tunnel number one 3–manifold does not fiber over the circle
A random tunnel number one 3–manifold does not fiber over the circle
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随机隧道 1 号 3 – 歧管不在圆上形成光纤
DOI：
10.2140/gt.2006.10.2431
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发表时间：
2005
影响因子：
2
通讯作者：
D. Thurston
中科院分区：
文献类型：
--
作者：
N. Dunfield;D. Thurston
我们解决这个问题：一个三维流形在圆上纤维化有多常见？考虑这一点的一个动机是深入了解相当难以理解的虚纤维化猜想。对于一类特殊的隧道数为1的三维流形，我们提供了令人信服的理论和实验证据，证明了重复是一个非常罕见的性质。事实上，在各种精确的意义上，它发生的概率为0。我们的主要定理是，这对于随机隧道1号三维流形的测量层模型是正确的。第一个要素是K Brown的算法，该算法可以决定给定的隧道1号3流形纤维是否在圆上。继阿戈尔，哈斯和W瑟斯顿的领导下，我们实现布朗的算法非常有效地工作的背景下，火车轨道/间隔交流。为了分析所得到的算法，我们推广Kerckhoff的工作，以了解完整亏格2区间交换的分裂序列的动态。结合所有这些与一个“魔术分裂序列”和工作的Mirzakhani证明了主要定理。3流形的情况与随机2生成1关联群形成了鲜明的对比;特别是，我们证明了这样的群“纤维”的概率严格在0和1之间。
We address the question: how common is it for a 3‐manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3‐manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3‐manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3‐manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown’s algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a “magic splitting sequence” and work of Mirzakhani proves the main theorem. The 3‐manifold situation contrasts markedly with random 2‐generator 1‐relator groups; in particular, we show that such groups “fiber” with probability strictly between 0 and 1.