Winning games for bounded geodesics in moduli spaces of quadratic differentials

Abstract

We prove that the set of bounded geodesics in Teichmuller space is a winning set for Schmidt's game. This is a notion of largeness in a metric space that can apply to measure $0$ and meager sets. We prove analogous closely related results on any Riemann surface, in any stratum of quadratic differentials, on any Teichmuller disk and for intervals exchanges with any fixed irreducible permutation.