Spectral theorems for random walks on mapping class groups and $\text{Out}(F_N)$

Abstract

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 mapping class group case, we first observe that the drift of the random walk in the curve complex is also equal to the linear growth rate of the translation lengths in this complex. By using a contraction property of typical Teichmüller geodesics, we then lift the above fact to the realization of the random walk on the Teichmüller space. For the case of $\text{Out}(F_N)$, we follow the same procedure with the free factor complex in place of the curve complex, and the outer space in place of the Teichmüller space. A general criterion is given for making the lifting argument possible.

45 pages, 3 figures. arXiv admin note: text overlap with arXiv:1506.07244

References in corpus (7)

• The boundary of the complex of free factors
• Sublinear deviation between geodesics and sample paths
• The boundary of the free splitting graph and the free factor graph
• Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings
• Two extensions of Thurston's spectral theorem for surface diffeomorphisms
• Central limit theorem for linear groups
• Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space

Cited by in corpus (2)

• Degree growth for tame automorphisms of an affine quadric threefold
• Random trees in the boundary of Outer space