Multi-geodesic Tessellations, Fractional Dehn Twists and Uniformization of Algebraic Curves

To illustrate what this paper is about we first consider a classical example of a translation surface. Consider two copies of a regular Euclidean pentagon as on the left of figure 1. Identifying opposite parallel sides by translations one obtains a Riemann surface with a natural locally flat metric with one cone type point of total angle 6π. Since this is a compact Riemann surface it is also an algebraic curve and in this case it is well known that this is the curve defined by y = x − 1. Since the surface is of genus 2 it is also a hyperbolic surface. The hyperbolic metric can easily be recovered by taking two copies of a regular hyperbolic pentagon (with interior angle π/5) in the unit disk, attached as on the right of figure 1 and with opposite sides identified by appropriate hyperbolic transformations (this construction is adapted from the one given in [Ku-Nä]).