We show that if Teichm\"uller geodesics spend enough time in the thick part of moduli space, they display CAT(-1)-type properties. In particular, they exponentially contract along strongly stable leaves. As an application we prove two closing lemmas.
Orthospectra of Geodesic Laminations and Dilogarithm Identities on Moduli Space
Coordinates for the moduli space of flat PSL(2,R)-connections
The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary
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