Volume 22, issue 1 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 22
Issue 7, 3761–4380
Issue 6, 3145–3760
Issue 5, 2511–3144
Issue 4, 1893–2510
Issue 3, 1267–1891
Issue 2, 645–1266
Issue 1, 1–644

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Other MSP Journals
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(FN), 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(FN), this describes the spread of the length of primitive conjugacy classes in FN 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