The strong Green-Griffiths-Lang conjecture predicts that a complex quasi-projective variety X is of log general type if and only if there is a proper Zariski closed subset Z of X such that all the holomorphic maps from the punctured disks to X with essential singularity at the origin are all contained in Z. In this talk I will show that this conjecture holds if the fundamental group of X admits a big and reductive representation into the complex general linear group. The proof is based on non-abelian Hodge theories and Nevanlinna theory. This work is jointly with Benoit Cadorel and Katsutoshi Yamanoi.