The fundamental group of compact K{\"a}hler threefolds

Let $X$ be a compact K{\"a}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $\pi\_1(X)\simeq \pi\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact K{\"a}hler manifolds of algebraic dimension $\dim(X)-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact K{\"a}hler threefolds.