Hyperbolicity and fundamental groups of complex quasi-projective varieties

It is natural to ask how fundamental groups $\pi_1(X)$ of complex quasi-projective varieties $X$ determine their hyperbolicity. In this paper we address this question when there is a linear representation $\varrho:\pi_1(X)\to GL_N(\mathbb{C})$. First, if $\varrho$ is big and the Zariski closure of $\varrho\big(\pi_1(X)\big)$ in $GL_N(\mathbb{C})$ is a semisimple algebraic group, then we prove that there is a proper Zariski closed subset $Z\subsetneqq X$ such that any closed irreducible subvariety $V$ of $X$ not contained in $Z$ is of log general type; any holomorphic map from the punctured disk $\Delta^*$ to $X$ with image not contained in $Z$ does not have essential singularity at the origin. In particular, $X$ is of log general type and all entire curves $f:\mathbb{C}\to X$ lie on $Z$. We provide examples to illustrate that the condition is sharp. Second, if $X$ is a special or hyperbolically special quasi-projective manifold in the sense of Campana, we prove that $\varrho\big(\pi_1(X)\big)$ is virtually nilpotent. We also construct examples to show that this result is optimal, thus disproving Campana's abelianity conjecture for quasi-projective manifolds. To prove the above theorems we develop new feature in non-abelian Hodge, geometric group and Nevanlinna theories. Along the way we prove: the strong Green-Griffiths-Lang conjecture for quasi-projective varieties $X$ admitting a morphism $a:X\to A$ with $\dim X=\dim a(X)$ where $A$ is a semi-abelian variety; a reduction theorem for Zariski dense representations $\pi_1(X)\to G(K)$ where $G$ is a reductive group over a non-Archimedean local field $K$.