Cubulable Kähler groups

Abstract

We prove that a Kähler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a $CAT\left(0\right)$ 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 Kähler 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 Kähler groups on irreducible, locally finite $CAT\left(0\right)$ cubical complexes, under the assumption that there is no fixed point in the visual boundary.