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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.00182 |
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| _version_ | 1866909212484829184 |
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| author | Huang, Hongzhi |
| author_facet | Huang, Hongzhi |
| contents | In this article, we prove that the fundamental group $π_1(M)$ of a complete open manifold $M$ with nonnegative Ricci curvature is finitely generated, under the condition that the Riemannian universal cover $\tilde M$ satisfies an "almost $k$-polar at infinity" condition. Additionally, such $π_1(M)$ is virtually abelian. Furthermore, we demonstrate that the base point of any tangent cone at infinity of such a manifold is nearly a pole. In the case where $\tilde M$ exhibits almost maximal Euclidean volume growth, we prove that $M$ deformation retracts to a closed submanifold $F$ which is diffeomorphic to a flat manifold, provided $M$ is not simply connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00182 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite generation of fundamental groups for manifolds with nonnegative Ricci curvature whose universal cover is almost $k$-polar at infinity Huang, Hongzhi Differential Geometry In this article, we prove that the fundamental group $π_1(M)$ of a complete open manifold $M$ with nonnegative Ricci curvature is finitely generated, under the condition that the Riemannian universal cover $\tilde M$ satisfies an "almost $k$-polar at infinity" condition. Additionally, such $π_1(M)$ is virtually abelian. Furthermore, we demonstrate that the base point of any tangent cone at infinity of such a manifold is nearly a pole. In the case where $\tilde M$ exhibits almost maximal Euclidean volume growth, we prove that $M$ deformation retracts to a closed submanifold $F$ which is diffeomorphic to a flat manifold, provided $M$ is not simply connected. |
| title | Finite generation of fundamental groups for manifolds with nonnegative Ricci curvature whose universal cover is almost $k$-polar at infinity |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2312.00182 |