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| Format: | Preprint |
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2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1809.10220 |
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| _version_ | 1866916826928119808 |
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| author | Pan, Jiayin |
| author_facet | Pan, Jiayin |
| contents | We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $π_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$ has Euclidean volume growth of constant at least $1-ε(n)$, then $π_1(M)$ is finitely generated and $C(n)$-abelian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1809_10220 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups Pan, Jiayin Differential Geometry We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $π_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$ has Euclidean volume growth of constant at least $1-ε(n)$, then $π_1(M)$ is finitely generated and $C(n)$-abelian. |
| title | Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1809.10220 |