Random Heegaard splittings

By Joseph Maher. 2008

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Consider a random walk on the mapping class group, and let wn be the location of the random walk at time n. A random Heegaard splitting M(wn) is a 3‐manifold obtained by using wn as the gluing map between two handlebodies. We show that the joint distribution of (wn, wn−1) is asymptotically independent, and converges to the product of the harmonic and reflected harmonic measures defined by the random walk. We use this to show that the translation length of wn acting on the curve complex, and the distance between the disk sets of M(wn) in the curve complex, grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability 1.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4531f1ab57456b5404e983aa17326208fbaba48f