The stratum of random mapping classes

Abstract

We consider random walks on the mapping class group whose support generates a nonelementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff [KP15].We will refer to condition (3) above as the principal stratum assumption.

Random Subgroups of Acylindrically Hyperbolic Groups and Hyperbolic Embeddings

“…Finally, a version of Proposition 10.4 is shown for the mapping class group acting on Teichmüller space, for µ with finite support, by Gadre and Maher [GM16], and independently by Baik, Gekhtman and Hamenstädt [BGH16]. A significantly simpler version of these arguments works in the setting of acylindrically hyperbolic groups, but we present the details below for the convenience of the reader.…”

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

“…Different but related recurrence properties for axes of random pseudo‐Anosov elements have been obtained independently at the same time by Gadre and Maher in .…”

Stable Strata of Geodesics in Outer Space

“…In [GM16], Gadre and Maher shed light on these questions. They proved that if the support of a random walk on M CG(S) is "sufficiently large" and contains a principal pseudo-Anosov g, then for every X ∈ T (X) and for ν X -a.e.…”