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… 

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References

SHOWING 1-10 OF 57 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
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
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
  • A. Sisto
  • Mathematics
    Journal 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
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
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
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
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
...
...