# Spectral Theorems for Random Walks on Mapping Class Groups and Out $(\boldsymbol{F}_{\boldsymbol{N}})$

@article{Dahmani2015SpectralTF,
title={Spectral Theorems for Random Walks on Mapping Class Groups and Out \$(\boldsymbol\{F\}\_\{\boldsymbol\{N\}\})\$},
author={François Dahmani and Camille Horbez},
journal={International Mathematics Research Notices},
year={2015},
volume={2018},
pages={2693-2744}
}
• Published 22 June 2015
• Mathematics
• International Mathematics Research Notices
We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time $n$ of the random walk to the Lyapunov exponent of the walk, which gives the typical growth rate of the length of a curve -- or of a conjugacy class in $F_N$ -- under a random product of diffeomorphisms/automorphisms. In the…

## Figures from this paper

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

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

### Topological Entropy of Random Walks on Mapping Class Groups

For any pseudo-Anosov diffeomorphism on a closed orientable surface $S$ of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the

### Some dynamics of random walks on the mapping class groups

For any pseudo-Anosov diffeomorphism on a closed orientable surface $S$ of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the

### The stratum of random mapping classes

• Mathematics
Ergodic Theory and Dynamical Systems
• 2017
We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map

### The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

• Mathematics
Proceedings of the London Mathematical Society
• 2019
We study the smallest positive eigenvalue λ1(M) of the Laplace–Beltrami operator on a closed hyperbolic 3‐manifold M which fibers over the circle, with fiber a closed surface of genus g⩾2 . We show

### Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings

• Mathematics
• 2017
Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a

### Random outer automorphisms of free groups: Attracting trees and their singularity structures

• Mathematics
• 2018
We prove that a "random" free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its

### Equidistribution of closed geodesics along random walk trajectories with respect to the harmonic invariant measure

We prove that for suitable random walks on isometry groups of $CAT(-1)$ spaces, typical sample paths eventually land on loxodromic elements which equidistribute with respect to a flow invariant

### Random walks, WPD actions, and the Cremona group

• Mathematics
Proceedings of the London Mathematical Society
• 2021
We study random walks on the Cremona group. We show that almost surely the dynamical degree of a sequence of random Cremona transformations grows exponentially fast, and a random walk produces

### Random trees in the boundary of outer space

• Mathematics
Geometry &amp; Topology
• 2022
We prove that for the harmonic measure associated to a random walk on Out$(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This

## References

SHOWING 1-10 OF 50 REFERENCES

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

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

### Statistics and compression of scl

• Mathematics
Ergodic Theory and Dynamical Systems
• 2014
Abstract We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting non-degenerately on

### The Poisson boundary of the mapping class group

• Mathematics
• 1994
Abstract. A theory of random walks on the mapping class group and its non-elementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the

### Counting Growth Types of Automorphisms of Free Groups

Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy

### Hyperbolic extensions of free groups

• Mathematics
• 2014
Given a finitely generated subgroup $\Gamma \le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension

### Random extensions of free groups and surface groups are hyperbolic

• Mathematics
• 2015
In this note, we prove that a random extension of either the free group $F_N$ of rank $N\ge3$ or of the fundamental group of a closed, orientable surface $S_g$ of genus $g\ge2$ is a hyperbolic group.

### Moduli of graphs and automorphisms of free groups

• Mathematics
• 1986
This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study

### Ergodic decompositions for folding and unfolding paths in Outer space

• Mathematics
• 2014
We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique

### The horoboundary of outer space, and growth under random automorphisms

We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions