Fibered commensurability and arithmeticity of random mapping tori

Abstract

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.

Earlier versions contain a gap in Section 4 (Lemma 4.8 of v1 and v2 is false). 25 pages (10 pages longer)

References in corpus (2)

• Exponential decay in the mapping class group
• On commensurability of fibrations on a hyperbolic 3-manifold

Cited by in corpus (1)

• On commensurability of quadratic differentials