Central limit theorems for mapping class groups and Out(FN)

Horbez, Camille

doi:10.2140/gt.2018.22.105

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Central limit theorems for mapping class groups and $\mathrm{Out}(F_N)$

Camille Horbez

Geometry & Topology 22 (2018) 105–156

DOI: 10.2140/gt.2018.22.105

Abstract

We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $Out (F_{N})$ , each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric on outer space). In the mapping class group case, this describes the spread of the hyperbolic length of a simple closed curve on the surface after applying a random product of mapping classes. In the case of $Out (F_{N})$ , this describes the spread of the length of primitive conjugacy classes in $F_{N}$ under random products of outer automorphisms. Both results are based on a general criterion for establishing a central limit theorem for the Busemann cocycle on the horoboundary of a metric space, applied to either the Teichmüller space of the surface or to the Culler–Vogtmann outer space.

Keywords

mapping class groups, Out(Fn), outer automorphism groups, random walks on groups, central limit theorem

Mathematical Subject Classification 2010

Primary: 20F65, 60B15

References

Publication

Received: 27 September 2015

Revised: 7 July 2016

Accepted: 6 January 2017

Published: 31 October 2017

Proposed: Martin Bridson

Seconded: Bruce Kleiner, Jean-Pierre Otal

Authors

Camille Horbez
Laboratoire de Mathématiques d’Orsay CNRS - Université Paris Sud Orsay France

Volume 22, issue 1 (2018)