Fill Radius and the Fundamental Group

In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: ''Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free.'' We explain the relevance of this theorem to some conjectures on positive isotropic curvature and 2-positive Ricci curvature.