Statistical convex-cocompactness for mapping class groups of non-orientable surfaces

Abstract

We show that a finite volume deformation retract $\mathcal{T}_{\varepsilon_t}^{-}(\mathcal{N}_g)/\mathrm{MCG}(\mathcal{N}_g)$ of the moduli space $\mathcal{M}(\mathcal{N}_g)$ of non-orientable surfaces $\mathcal{N}_g$ behaves like the convex core of $\mathcal{M}(\mathcal{N}_g)$, despite not even being quasi-convex. We then show that geodesics in the convex core leave compact regions with exponentially low probabilities, showing that the action of $\mathrm{MCG}(\mathcal{N}_g)$ on $\mathcal{T}_{\varepsilon_t}^{-}(\mathcal{N}_g)$ is statistically convex-cocompact. Combined with results of Coulon and Yang, this shows that the growth rate of orbit points under the mapping class group action is purely exponential, pseudo-Anosov elements in mapping class groups of non-orientable surfaces are exponentially generic, and the action of mapping class group on the limit set in the horofunction boundary is ergodic with respect to the Patterson-Sullivan measure. A key step of our proof relies on complexity length, developed by Dowdall and Masur, which is an alternative notion of distance on Teichm\"uller space that accounts for geodesics that spend a considerable fraction of their time in the thin part.