Random Heegaard splittings

Consider a random walk on the mapping class group, and let w n be the location of the random walk at time n. A random Heegaard splitting M(w n ) is a 3-manifold obtained by using w n as the gluing map between two handlebodies. We show that the joint distribution of (w n , w n −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 w n acting on the curve complex, and the distance between the disk sets of M(w n ) in the curve complex, grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability 1.

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