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On the topology of manifolds with nonnegative Ricci curvature and linear volume growth
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abstract
Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.
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Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds
Open 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth of the universal cover have finitely generated, virtually abelian fundamental groups.
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