随机映射环面的纤维可积性与算术性
Fibered commensurability and arithmeticity of random mapping tori
摘要：
本文考虑有限型曲面的映射类群上的随机游动。我们假设随机游动是由一个概率测度决定的，它的支撑是有限的，并生成一个非初等子群$H$。我们进一步假设$H$不是只由关于任何一个覆盖的提升构成。然后证明了这样的随机游动在其纤维化的可映射类中给出非极小映射类的概率是指数衰减的。作为极小性的一个应用，我们证明了对于曲面至少有一个穿刺的情况，一个随机游动给出具有算术映射环面的映射类的概率是指数衰减的。我们还证明了一个随机游动产生的非对称映射环面指数高概率封闭的情况下。
英文摘要：
We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup $H$. We further assume that $H$ is not consisting only of lifts with respect to any one covering. Then we prove that the probability that such a random walk gives a non-minimal mapping class in its fibered commensurability class decays exponentially. As an application of the minimality, we prove that for the case where a surface has at least one puncture, the probability that a random walk gives mapping classes with arithmetic mapping tori decays exponentially. We also prove that a random walk gives rise to asymmetric mapping tori with exponentially high probability for closed case.
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