On the classification of rank two representations of quasiprojective fundamental groups

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 $ρ$ factors through a map $X\to Y$ with $Y$ either a DM-curve or a Shimura modular stack.

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