Teichmueller dynamics

seminar: Multi-geodesic tesselations of surfaces and uniformisation

speaker: Samuel Lelievre (University of Warwick)

abstract: Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special cases the surface is decomposed into polygons geodesic for both the flat and the hyperbolic metric. This is the case for certain surfaces which are translation and half-turn tiled by an Euclidean rectangle. We will explain how exploring their multi-geodesic tessellation provides a mechanical way to reconstruct their uniformising Fuchsian group and allows to describe their Teichmueller disk in terms of Fenchel-Nielsen coordinates. In many cases the tiling by rectangles allows to recover an equation for the corresponding algebraic curve, providing a bridge between the algebraic equation and the hyperbolic structure deduced from the multi-geodesic tessellation; in other words solving the uniformization problem for such curves.

timetable:
Fri 25 May, 11:15 - 12:15, Sala Conferenze Centro De Giorgi
<< Go back