Prym Varieties and Teichmüller Curves

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This paper gives a uniform construction of infinitely many primitive Teichmuller curves V ⊂ Mg for g = 2, 3 and 4.

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Characterizations of lattice surfaces (2008, background): “Lattice surfaces have been classified in some restricted settings (see [4, 7, 9, 13, 16]), and many additional examples of lattice surfaces have been found (see [2, 8, 14 , 23, 24]).”
A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces (2013, background): “ing McMullen [36], that the translation surface belongs to the Prym locus or is simply Prym.”
The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3 (2012, background): “In M3, McMullen exhibited in [McM06a] infinitely many primitive Teichmüller curves with quadratic trace field.”
Cylinder deformations in orbit closures of translation surfaces (2013, methodology): “McMullen later generalized his approach to provide examples in genus 3, 4 and 5 [18], giving the Prym eigenform loci mentioned above.”
Braid groups and Hodge theory (2013, methodology): “By a construction of Thurston, this implies there is a holomorphic quadratic differential q on X such that the loops (Si) are represented by a cylinders of heights (hi) on (X, |q|), and the lifts of α and β are realized by affine automorphisms [Th1]; cf. [ Mc3 , §4].”