[Paper Review] Fibered commensurability and arithmeticity of random mapping tori

Hidetoshi Masai|arXiv (Cornell University)|Aug 2, 2014

Geometric and Algebraic Topology30 references3 citations

TL;DR

This paper establishes that for random walks on the mapping class group of a finite-type surface, the probability of generating a non-minimal mapping class in its fibered commensurability class decays exponentially. As a consequence, the set of mapping classes yielding arithmetic or asymmetric hyperbolic 3-manifolds (via mapping tori) is exponentially small or large, respectively, under mild conditions on the random walk measure.

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.

Motivation & Objective

• To prove that the set of minimal elements in fibered commensurability classes is exponentially large under a random walk on the mapping class group.
• To show that for surfaces with punctures, the probability of generating an arithmetic mapping torus decays exponentially.
• To establish that for closed surfaces, the probability of generating an asymmetric mapping torus is exponentially high.
• To extend results on random pseudo-Anosov elements to fibered commensurability and arithmeticity via geometric group theory and random walk techniques.

Proposed method

• Use random walks on the mapping class group $\mathrm{Mod}(S)$ with a finite, non-elementary, and non-lifting-supporting measure $\mu$.
• Prove that the set of primitive elements is exponentially large using growth estimates on translation length in the curve complex.
• Establish that symmetric elements—those liftable with respect to any finite covering—are exponentially rare by analyzing fixed points in the projective measured foliation space.
• Apply the uniqueness of minimal elements in fibered commensurability classes to reduce arithmeticity questions to conjugacy classes of minimal elements.
• Leverage results from Bowditch-Maclachlan-Reid on finite commensurability classes of arithmetic manifolds to bound the number of minimal elements yielding arithmetic tori.
• Use Bachman-Schleimer's theorem on isometry groups of closed hyperbolic 3-manifolds to show asymmetric mapping tori are exponentially likely for closed surfaces.

Experimental results

Research questions

• RQ1 What is the asymptotic probability that a random mapping class is minimal in its fibered commensurability class?
• RQ2 How common are arithmetic mapping tori among random mapping classes when the surface has at least one puncture?
• RQ3 What is the probability that a random mapping torus of a closed surface is asymmetric?
• RQ4 Under what conditions on the random walk measure does the minimality of random mapping classes hold with exponentially high probability?
• RQ5 Can the fibered commensurability class of a random mapping class be characterized via geometric and dynamical properties of the associated pseudo-Anosov element?

Key findings

• The set of primitive elements in the mapping class group is exponentially large with respect to any measure $\mu$ satisfying Condition 1.2.
• The set of symmetric elements—those liftable with respect to any finite covering—is exponentially small under the same measure.
• The set of minimal elements in their fibered commensurability class is exponentially large, as a consequence of primitivity and non-symmetry.
• For surfaces with at least one puncture, the set of mapping classes yielding arithmetic mapping tori is exponentially small.
• For closed surfaces, the set of mapping classes yielding asymmetric mapping tori is exponentially large.
• The number of conjugacy classes of minimal elements giving rise to arithmetic mapping tori is finite, implying a uniform upper bound on translation length for such elements.