Largest projections for random walks and shortest curves in random mapping tori

We show that the largest subsurface projection distance between a marking and its image under the $n \textrm{th}$ step of a random walk grows logarithmically in $n$, with probability approaching $1$ as $n \to \infty$. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$.

We then use this result to prove Rivin’s conjecture that for a random walk ($w_n$) on the mapping class group, the shortest geodesic in the hyperbolic mapping torus $M_{w_n}$ has length on the order of $1 / \log^2 (n)$.

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Received 2 December 2016

Accepted 11 June 2017

Published 7 June 2019