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Topology/Geometry Seminar

The Heights theorem for integrable quadratic differentials on infinite Riemann surfaces

Dragomir Saric (CUNY)

Location:  zoom link: https://rutgers.zoom.us/j/96839448491?pwd=NHNWcVFKTWpkRDZWcVVhVm9mYTNGUT09
Date & time: Tuesday, 22 September 2020 at 3:50PM - 4:50PM

An integrable holomorphic quadratic differential on a Riemann surface induces a measured foliation of the surface by horizontal trajectories. A quadratic differential associates to each homotopy class of a closed curve  its height, i.e.-the infimum of the transverse measure over the homotopy class. Marden and Strebel proved that the space of quadratic differentials is in a one to one correspondence to the heights maps when the Riemann surface is of parabolic type.

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.

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.