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

Abstract

We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on 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 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.

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

“…Using techniques of , Dahmani and Horbez [, Proposition 1.9] proved. Proposition Let be a separable geodesic Gromov hyperbolic metric space, with hyperbolicity constant .…”

“…For a pseudo‐Anosov element , let be its translation length in . The following result is [, Theorem 3.1]. Theorem Let be a nonelementary probability measure on with finite support.…”

“…We begin with reviewing some background on random walks on groups. This is a vast subject, see, for example, [7,15,18] for more details.…”

The stratum of random mapping classes

“…In this section, we establish that along almost every sample path ω, for sufficiently large n, the invariant Teichmüller geodesic for the pseudo-Anosov element w n , has a subsegment, whose length grows linearly in n, which fellow travels the Teichmüller geodesic sublinearly tracked by ω. This uses a result of Dahmani and Horbez [DH15] and the fellow travelling result, Theorem 1.4. We fix a basepoint X ∈ T (S).…”

“…Next, we show that if g lies in the semigroup generated by the support of µ, there is a positive probability that the geodesic γ ω tracked by a sample path ω, fellow travels the invariant geodesic γ g for distance at least D. Ergodicity of the shift map on Mod(S) Z then implies that a positive proportion of subsegments of γ ω of length D fellow travel some translate of γ g . We then use work of Dahmani and Horbez [DH15] which shows that for almost all sample paths ω, for sufficiently large n, all elements w n are pseudo-Anosov, with invariant geodesics γ w n which fellow travel γ ω for a distance which grows linearly in n. In particular, this implies that γ w n fellow travels a sufficiently long subsegment of a translate of γ g , and so lies in the principal stratum.…”