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Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the s …

To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in …

The answer is no, but it's actually a deep question and leads to another metric on the Teichmüller space of surfaces. The minimum quasi-conformal constant of a map in a given homotopy class is the Tei …

To expand on Lee's answer, recall that Teichmüller space can be divided up into a "thin part" (where some geodesic is short, or equivalently where some conformal annulus has large modulus), and its co …

Here's another answer that may be more satisfying: Use convexity. Short summary: length of weighted simple closed multi-curves defines a convex function on the space of measured foliations with respec …