Fill Radius and the Fundamental Group

TL;DR

This paper establishes a connection between the fill radius of a manifold's universal cover and the algebraic structure of its fundamental group, showing bounded fill radius implies the group is virtually free.

Contribution

It proves that a closed Riemannian manifold with universal cover of bounded fill radius has a virtually free fundamental group, linking geometric and algebraic properties.

Findings

Universal cover with bounded fill radius implies virtually free fundamental group.

Provides insights into conjectures on positive isotropic curvature.

Connects geometric fill radius to algebraic group properties.

Abstract

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.