Relatively geometric actions of Kahler groups on CAT(0) cube complexes. (with Corey Bregman and Kejia Zhu)

Abstract We prove that for n at least 2 a non-uniform lattice in PU(n,1) does not admit a relatively geometric action on a CAT(0) cube complex, in the sense of Einstein-Groves. If \Gamma is a torsion-free lattive in a non-comppact semisimple Lie group G without compact factors that admits a relatively geometric action on a CAT(0) cube complex then G is commensurable with SO(n,1). We also prove that if a Kahler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a CAT(0) cube complex, then it is virtually a surface group.