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The first part of this paper is a survey on Teichmüller curves and Veech groups, with emphasis on the special case of origamis where much stronger tools for the investigation are available than in the general case. In the second part we study a particular configuration of origami curves in genus 3: A “base” curve is intersected by infinitely many “transversal” curves. We determine their Veech groups and the closure of their locus in M 3.

A comb of origami curves in the moduli space M 3 with three dimensional closure