Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910815245828096 |
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| author | Huang, Hongzhi Huang, Xian-Tao |
| author_facet | Huang, Hongzhi Huang, Xian-Tao |
| contents | 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 $π_1(M)$ is finitely generated. This result confirms Pan-Rong's conjecture \cite{PR18} for dimension $n = 4$. Additionally, we prove that there exists a universal constant $C>0$ such that $π_1(M)$ contains an abelian subgroup of index $\le C$. More specifically, if $π_1(M)$ is infinite, then $π_1(M)$ is a crystallographic group of rank $\le 3$. If $π_1(M)$ is finite, then $π_1(M)$ is isomorphic to a quotient of the fundamental group of a spherical 3-manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03259 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds Huang, Hongzhi Huang, Xian-Tao Differential Geometry 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 $π_1(M)$ is finitely generated. This result confirms Pan-Rong's conjecture \cite{PR18} for dimension $n = 4$. Additionally, we prove that there exists a universal constant $C>0$ such that $π_1(M)$ contains an abelian subgroup of index $\le C$. More specifically, if $π_1(M)$ is infinite, then $π_1(M)$ is a crystallographic group of rank $\le 3$. If $π_1(M)$ is finite, then $π_1(M)$ is isomorphic to a quotient of the fundamental group of a spherical 3-manifold. |
| title | Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2502.03259 |