Winning games for bounded geodesics in moduli spaces of quadratic differentials

TL;DR

This paper demonstrates that the set of bounded geodesics in Teichmuller space is a winning set for Schmidt's game, indicating a form of largeness that applies even to measure zero and meager sets.

Contribution

It establishes that bounded geodesics form a winning set in various settings within Teichmuller theory, extending the concept across different Riemann surfaces and quadratic differential strata.

Findings

Bounded geodesics form a winning set in Teichmuller space.

Results apply to all Riemann surfaces and quadratic differential strata.

The notion of largeness extends to measure zero and meager sets.

Abstract

We prove that the set of bounded geodesics in Teichmuller space are 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 disc and for intervals exchanges with any fixed irreducible permutation.