Cubulable K\"ahler groups
  We prove that a K\"ahler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free Abelian factor. Similarly, we prove that a closed aspherical K\"ahler manifold with a cubulable fundamental group has a finite cover which is biholomorphic to a topologically trivial principal torus bundle over a product of Riemann surfaces. Along the way, we prove a factorization result for essential actions of K\"ahler groups on irreducible, locally finite CAT(0) cubical complexes, under the assumption that there is no fixed point in the visual boundary. 
Thomas Delzant, Pierre Py
math.GT math.AG math.GR