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LINEAR GROWTH OF TRANSLATION LENGTHS OF RANDOM ISOMETRIES ON GROMOV HYPERBOLIC SPACES AND TEICHMÜLLER SPACES

Part of: Low-dimensional topology Stochastic processes Riemann surfaces Special aspects of infinite or finite groups

Published online by Cambridge University Press:  06 November 2023

Hyungryul Baik Open the ORCID record for Hyungryul Baik [Opens in a new window] ,
Inhyeok Choi  and
Dongryul M. Kim Open the ORCID record for Dongryul M. Kim [Opens in a new window]
Hyungryul Baik*
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon, 34141, South Korea
Inhyeok Choi
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon, 34141, South Korea (inhyeokchoi48@gmail.com)
Dongryul M. Kim
Affiliation:
Department of Mathematics, Yale University, 219 Prospect Street, New Haven, CT 06511, USA (dongryul.kim@yale.edu)
*
hrbaik@kaist.ac.kr
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Abstract

We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov, and almost every random walk on $\mathrm {Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk.

We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmüller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.

Keywords

Random walkGromov hyperbolic spacespectral theoremTeichmüller spaceweakly hyperbolic group

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups 30F60: Teichmüller theory
Secondary: 57M60: Group actions in low dimensions 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 45
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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