Divergence of projective structures and lengths of measured laminations

Abstract

Given a complex structure, we investigate diverging sequences of projective structures on the fixed complex structure in terms of Thurston's parametrization. In particular, we will give a geometric proof to the theorem by Kapovich stating that as the projective structures on a fixed complex structure diverge so do their monodromies. In course of arguments, we extend the concept of realization of laminations for PSL$(2,{\mathbf C})$-representations of surface groups.