Filtered ends, proper holomorphic mappings of Kähler manifolds to riemann surfaces, and kähler groups

The main result of this paper is that a connected bounded geometry complete Kähler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson's groups V, T, and F, are not Kähler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact Kähler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2.