Filtered ends, proper holomorphic mappings of Kahler manifolds to   Riemann surfaces, and Kahler groups
  The main goal of this paper is to prove that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. This also provides a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler 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. The main results are also used to show that any properly ascending HNN extension with finitely generated base group, as well as Thompson's groups V, T, and F, are not Kahler. This version contains details which will not appear in a version submitted for publication. 
Terrence Napier, Mohan Ramachandran