# 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} }

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…

## 11 Citations

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

- Mathematics
- 2015

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

- Mathematics
- 2016

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

- Mathematics
- 2016

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

- MathematicsErgodic 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…

Degree growth for tame automorphisms of an affine quadric threefold

- Mathematics
- 2018

In this paper, we consider the degree sequences of the tame automorphisms preserving an affine quadric threefold. Using some valuatives estimates derived from the work of Shestakov-Umirbaev and the…

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…

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

- Mathematics
- 2017

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

- MathematicsProceedings 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

- MathematicsGeometry & 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…

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…

## References

SHOWING 1-10 OF 57 REFERENCES

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

- Mathematics
- 2015

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

- MathematicsErgodic 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…

Contracting elements and random walks

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2018

We define a new notion
of contracting element of a group and we show that contracting elements coincide with
hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class…

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…

Random walks on weakly hyperbolic groups

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2018

Let G be a countable
group which acts by isometries on a separable,
but not necessarily proper, Gromov hyperbolic space X. We say the
action of G is weakly hyperbolic if G contains two independent…

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

- Mathematics
- 2014

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…