#### Volume 22, issue 1 (2018)

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

### Camille Horbez

Geometry & Topology 22 (2018) 105–156
##### 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 $Out\left({F}_{N}\right)$, 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 $Out\left({F}_{N}\right)$, 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
##### 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