Random Heegaard splittings
Random Heegaard splittings
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DOI：
10.1112/jtopol/jtq031
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发表时间：
2008-09
影响因子：
1.1
通讯作者：
Joseph Maher
中科院分区：
文献类型：
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作者：
Joseph Maher
考虑映射类组上的随机游走，并令wn为时间n处随机游走的位置。随机Heegaard分裂M（wn）是通过使用wn作为两个非线性体之间的胶合映射而获得的3-流形。我们证明了（wn，wn-1）的联合分布是渐近独立的，并且收敛于由随机游动定义的调和测度和反射调和测度的乘积。我们利用这一点来证明wn作用于曲线复形的平移长度，以及曲线复形中M（wn）的圆盘集之间的距离，在n中线性增长。特别是，这意味着随机Heegaard分裂是双曲的渐近概率为1。
Consider a random walk on the mapping class group, and let wn be the location of the random walk at time n. A random Heegaard splitting M(wn) is a 3‐manifold obtained by using wn as the gluing map between two handlebodies. We show that the joint distribution of (wn, wn−1) is asymptotically independent, and converges to the product of the harmonic and reflected harmonic measures defined by the random walk. We use this to show that the translation length of wn acting on the curve complex, and the distance between the disk sets of M(wn) in the curve complex, grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability 1.