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

TL;DR

This paper proves that subsurface projection distances grow logarithmically in random walks and uses this to confirm Rivin's conjecture that shortest geodesic lengths in random hyperbolic mapping tori decrease on the order of 1/ log^2(n).

Contribution

It establishes the growth rate of subsurface projection distances for random walks and confirms Rivin's conjecture on geodesic lengths in random mapping tori.

Findings

Subsurface projection distances grow logarithmically with high probability.

Shortest geodesic lengths in random mapping tori are on the order of 1/ log^2(n).

Results apply to hyperbolic groups and Out(F_n).

Abstract

We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. 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 $1/ \log^2(n)$.