Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open 4-Manifolds

2025/02/05 by Hongzhi Huang, Huang, Hongzhi, Xian-Tao Huang +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2502.03259

openalex publication_date 2025/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of M has Euclidean volume growth, then the fundamental group π₁(M) is finitely generated. This result confirms Pan-Rong's conjecture \citePR18 for dimension n = 4. Additionally, we prove that there exists a universal constant C>0 such that π₁(M) contains an abelian subgroup of index ≤ C. More specifically, if π₁(M) is infinite, then π₁(M) is a crystallographic group of rank ≤ 3. If π₁(M) is finite, then π₁(M) is isomorphic to a quotient of the fundamental group of a spherical 3-manifold.

Cited by

• Fundamental groups of open manifolds with nonnegative Ricci curvature and universal cover Euclidean volume growth

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