Kähler manifolds and fundamental groups of negatively $δ$-pinched manifolds

Abstract

The fundamental group of a Riemannian manifold with $δ$-pinched negative curvature, $δ>1/4$, cannot be the fundamental group of a quasicompact Kähler manifold. The proof also implies that a non-uniform lattice in $F_{4(-20)}$ cannot be the fundamental group of a quasicompact Kähler manifold. We also construct examples in the spirit of Gromov-Thurston to show that our result is a non-trivial extension of the previously known result that a non-uniform lattice in real hyperbolic space in dimension at least 3 cannot be the fundamental group of a quasicompact Kähler manifold.