Geodesics Currents and Counting Problems

Abstract

For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\overline{\Sigma }$, and for every compactly supported filling current $\alpha$, we compute as $L\to \infty$, the number of mapping classes $\varphi$ so that $f(\varphi (\alpha ))\le L$. As an application, when the surface in question is closed, we prove a lattice counting theorem for Teichmüller space equipped with the Thurston metric.

17 pages. In the new version, the main theorem is stated for all compact surfaces. To appear in GAFA

References in corpus (3)

• Convexity of length functions and Thurston's shear coordinates
• Counting curves, and the stable length of currents
• What's wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces

Cited by in corpus (3)

• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves
• From curves to currents
• Valuations on the character variety: Newton polytopes and Residual Poisson Bracket