On the fundamental group of a compact Kähler manifold (Q1210416)

We prove that the fundamental group of a compact Kähler manifold cannot be an extension of a group with infinitely many ends by a finitely generated group. This implies that the fundamental group of such a manifold is indecomposable in a very strong sense. For example, it cannot be expressed as a nontrivial free product (or HNN extension) amalgamated over a finite group. The special case of free products had been established earlier by \textit{M. Gromov} in [C. R. Acad. Sci., Paris, Sér. I 308, No. 3, 67-70 (1989; Zbl 0661.53049)]. Our theorem is proved by a slight refinement of his methods. For the reader's convenience we have included careful proofs of some of Gromov's key results.