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A Teichmüller curve V ⊂ M g V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems.
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Abstract This paper introduces a space of nonabelian modular symbols 𝒮 ( V ) {{\mathcal{S}}(V)} attached to any hyperbolic Riemann surface V ,and applies it to obtain new results on polygonal billiards and holomorphic 1-forms.In particular, it shows the scarring behavior of periodic trajectories for billiardsin a regular polygon is governed by a countable set of measureshomeomorphic to ω ω + 1 {\omega^{\omega}+1} .more » « less
