# Groups, Geometry, and Dynamics

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**Volume 11, Issue 4, 2017, pp. 1253–1279**

**DOI: 10.4171/GGD/428**

Published online: 2017-12-07

Fibered commensurability and arithmeticity of random mapping tori

Hidetoshi Masai^{[1]}(1) Tohoku University, Japan

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.

*Keywords: *Random walk, mapping class group, fibered commensurability, arithmetic 3-manifold

Masai Hidetoshi: Fibered commensurability and arithmeticity of random mapping tori. *Groups Geom. Dyn.* 11 (2017), 1253-1279. doi: 10.4171/GGD/428