On fundamental groups of elliptically connected surfaces

Abstract

A compact complex manifold $X$ is called elliptically connected if any pair of points in $X$ can be connected by a chain of elliptic or rational curves. We prove that the fundamental group of an elliptically connected compact complex surface is almost abelian. This confirms a conjecture which states that the fundamental group of an elliptically connected K\"ahler manifold must be almost abelian.