course: Infinite Teichmüller spaces via hyperbolic geometry
speaker: Dragomir Saric (Queen's College London)abstract: Let X be an infinite area hyperbolic surface. The Teichmuller space T(X) of the surface X has been extensively studied using the quasiconformal deformations of X by various authors.
We study T(X) via the deformations of hyperbolic metrics on X. Our first tool in deforming the hyperbolic metrics are earthquakes. We characterize which earthquakes deform the base hyperbolic metric on X in a bounded fashion such that the new metric is also in T(X). Furthermore, we establish a homeomorphism between the Teichmüller space T(X) and the space of earthquake measures (joint with Miyachi). Using the Liouville currents, we introduce Thurston's bordification of T(X) and prove that it equals to the space $PML{bdd}(X)$ of projective bounded measured laminations on X (analogous to Bonahon's work for closed surfaces). The universal Teichmüller space $T(\mathbb{H})$ is the Teichmüller space of the hyperbolic plane $\mathbb{H}$. We give a parametrization of the universal Teichmüller space $T(\mathbb{H})$ in terms of shear coordinates on the Farey tesselation of the hyperbolic plane $\mathbb{H}$. Furthermore, we describe the tangent space and the complex structure of $T(\mathbb{H})$ in terms of shear coordinates.
We also study the hyperbolic metrics on X via its length spectrum (as started by Shiga, and further developed by Alessandrini, Liu, Papadopuolos and Su, Basmajian, Kinjo, Kim etc). The Thurston's boundary to T(X) using the length spectrum is introduced and compared to the Thurston's boundary using the Liouville currents.
timetable:
Tue 27 May, 14:30 - 16:00, Aula Magna Dipartimento di Matematica
Wed 28 May, 9:30 - 11:00, Aula Magna Dipartimento di Matematica
Thu 29 May, 11:30 - 12:30, Aula Magna Dipartimento di Matematica
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