The Heights Theorem for infinite Riemann surfaces

TL;DR

This paper extends the Heights Theorem to all surfaces with a first-kind fundamental group, establishing the injectivity of the horizontal map for integrable holomorphic quadratic differentials on hyperbolic Riemann surfaces.

Contribution

It generalizes the Heights Theorem beyond parabolic surfaces to all surfaces with a first-kind fundamental group and analyzes the boundedness of measured laminations.

Findings

Horizontal map is injective for all surfaces with a hyperbolic metric.

Bounded measured laminations correspond to surfaces with bounded geodesic pants decompositions.

Unbounded laminations occur when surfaces have shrinking closed geodesics.

Abstract

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric. This was established by Strebel in the case of the unit disk. When a hyperbolic surface has a bounded geodesic pants decomposition, the horizontal map assigns a bounded measured lamination to each integrable holomorphic quadratic differential. When surface has a sequence of closed geodesics whose lengths go to zero, then there exists an integrable holomorphic quadratic differential whose horizontal measured lamination is not bounded. We also give a sufficient condition for the non-integrable holomorphic quadratic differential to give rise to bounded measured laminations.