TY - JOUR

T1 - Fibered commensurability and arithmeticity of random mapping tori

AU - Masai, Hidetoshi

N1 - Funding Information:
Acknowledgments. The author would like to thank Ingrid Irmer, Joseph Maher, Makoto Sakuma and Giulio Tiozzo for helpful conversations. He would especially like to thank Joseph Maher for suggesting to use the work [6] to prove Theorem 1.1. He started this work when he was in ICERM, Brown University. Thanks also goes to ICERM and JSPS for supporting the visit. He would also like to thank Brian Bowditch for bringing the paper [5] in his attention. An earlier version of this paper contained a gap in the proof of Theorem 1.3. The author would like to thank referee(s) for pointing out the gap and careful reading. This work was partially supported by JSPS Research Fellowship for Young Scientists.
Publisher Copyright:
© 2017 European Mathematical Society.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017

Y1 - 2017

N2 - 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 theminimality,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.

AB - 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 theminimality,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.

KW - Arithmetic 3-manifold

KW - Fibered commensurability

KW - Mapping class group

KW - Random walk

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U2 - 10.4171/GGD/428

DO - 10.4171/GGD/428

M3 - Article

AN - SCOPUS:85038873130

VL - 11

SP - 1253

EP - 1279

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

SN - 1661-7207

IS - 4

ER -