On the classification of rank two representations of quasiprojective fundamental groups

Abstract

Suppose $X$ is a smooth quasiprojective variety over $\cc$ and $ρ: π_1(X,x) \to SL(2,\cc)$ is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then $\rho$ factors through a map $X\to Y$ with $Y$ either a DM-curve or a Shimura modular stack.

minor changes in exposition, citations

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• Rank 2 local systems and abelian varieties
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• A family of flat connections on the projective space having dihedral monodromy and algebraic Garnier solutions
• Rank 2 local systems and abelian varieties II
• Orbifold groups, quasi-projectivity and covers
• Absolute sets of rigid local systems
• Algebraic isomonodromic deformations of the five punctured sphere arising from quintic plane curves
• Les groupes de Burger-Mozes ne sont pas kählériens
• Pluriharmonic maps into buildings and symmetric differentials
• A variety that cannot be dominated by one that lifts
• Dominant classes of projective varieties
• Algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere and finite braid group orbits on character varieties