Mathematics – Geometric Topology
Scientific paper
2008-09-29
Mathematics
Geometric Topology
31 pages, 5 figures, revised version
Scientific paper
A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses 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 on the curve complex of a random walk grows linearly in the length of the walk, and similarly, that distance in the curve complex between the disc sets of a random Heegaard splitting grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability one.
Maher Joseph
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