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

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 $\text{Out}(F_N)$, each time under a finite second moment condition on the measure (either with respect to the Teichm\"uller 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 $\text{Out}(F_N)$, 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 Teichm\"uller space of the surface, or to Culler--Vogtmann's outer space.
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Article dans une revue
Geometry and Topology, Mathematical Sciences Publishers, 2018, 22 (1), pp.105-156

https://hal.archives-ouvertes.fr/hal-01187723
Contributeur : Marie-Annick Guillemer <>
Soumis le : jeudi 27 août 2015 - 14:20:36
Dernière modification le : jeudi 15 novembre 2018 - 11:56:37

### Identifiants

• HAL Id : hal-01187723, version 1
• ARXIV : 1506.07244

### Citation

Camille Horbez. Central limit theorems for mapping class groups and $\text{Out}(F_N)$. Geometry and Topology, Mathematical Sciences Publishers, 2018, 22 (1), pp.105-156. 〈hal-01187723〉

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