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… 
View PDF on arXiv
11 Citations
Highly Influential Citations
1
Background Citations
3
Methods Citations
2
Results Citations
2

Figures from this paper

11 Citations

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
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
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
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
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
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
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
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 $\text{Out}(F_N)$
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…
Very small group actions on R-trees and dehn twist automorphisms
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…
...
...