Kaehler groups and duality

This is the topological part of two papers on the cohomology of Kaehler groups. In this paper we show that if a linear duality group of dimension larger than 6 is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number is non-zero. As a corollary a cocompact p-adic lattice of rank larger than 6 is never Kaehler.