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

Alessandro Sisto; Samuel J. Taylor

doi:10.4310/MRL.2019.v26.n1.a14

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

Alessandro Sisto, Samuel J. Taylor

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

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 → ∞. Our setup is general and also applies to (relatively) hyperbolic groups and to 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_wn has length on the order of 1/log²(n).

Original language	English
Pages (from-to)	293-321
Number of pages	29
Journal	Mathematical Research Letters
Volume	26
Issue number	1
Early online date	7 Jun 2019
DOIs	https://doi.org/10.4310/MRL.2019.v26.n1.a14
Publication status	Published - 2019

ASJC Scopus subject areas

General Mathematics

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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 → ∞. Our setup is general and also applies to (relatively) hyperbolic groups and to Out(Fn). We then use this result to prove Rivin{\textquoteright}s conjecture that for a random walk (wn) on the mapping class group, the shortest geodesic in the hyperbolic mapping torus Mwn has length on the order of 1/log2(n).",

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AB - 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 → ∞. Our setup is general and also applies to (relatively) hyperbolic groups and to Out(Fn). We then use this result to prove Rivin’s conjecture that for a random walk (wn) on the mapping class group, the shortest geodesic in the hyperbolic mapping torus Mwn has length on the order of 1/log2(n).

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

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