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

2016/11/22 by Alessandro Sisto, Sisto, Alessandro, Samuel J. Taylor +1 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1611.07545

openalex publication_date 2016/11/22 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28

Abstract

We show that the largest subsurface projection distance between a marking and\nits image under the nth step of a random walk grows logarithmically in n, with\nprobability approaching 1 as n tends to infinity. Our setup is general and also\napplies to (relatively) hyperbolic groups and to \Out(F_n). We then\nuse this result to prove Rivin's conjecture that for a random walk (w_n) on\nthe mapping class group, the shortest geodesic in the hyperbolic mapping torus\nM_wn has length on the order of 1/ \log²(n).\n

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