Mathematical Research Letters

Volume 26 (2019)

Number 1

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

Pages: 293 – 321



Alessandro Sisto (Department of Mathematics, Eidgenössische Technische Hochschule, Zürich, Switzerland)

Samuel J. Taylor (Department of Mathematics, Temple University, Philadelphia, Pennsylvania, U.S.A.)


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 $1 / \log^2 (n)$.

Received 2 December 2016

Accepted 11 June 2017

Published 7 June 2019