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We study the rigidity problems for open (complete and noncompact) $$n$$-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of $$M$$ properly contains a Euclidean $$\mathbb{R}^{k-1}$$, then the first Betti number of $$M$$ is at most $n-k$; moreover, if equality holds, then $$M$$ is flat. Next, we study the geometry of the orbit $$\Gamma\tilde{p}$$, where $$\Gamma=\pi_1(M,p)$$ acts on the universal cover $$(\widetilde{M},\tilde{p})$$. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of $$\Gamma\tilde{p}$$. We also give the first example of a manifold $$M$$ of $$\mathrm{Ric}>0$$ and $$\pi_1(M)=\mathbb{Z}$$ but with a varying orbit growth order.
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This content will become publicly available on July 1, 2026
Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups
We study the fundamental group of an open $$n$$-manifold $$M$$ of nonnegative Ricci curvature with additional stability conditions on $$\widetilde{M}$$, the Riemannian universal cover of $$M$$. We prove that if every asymptotic cone of $$\widetilde{M}$$ is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric cone, then $$\pi_1(M)$$ 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 by a constant $C(n,L)$. In particular, our result implies that if $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$1-\epsilon(n)$$, then $$\pi_1(M)$$ is finitely generated and $C(n)$-abelian.
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- Award ID(s):
- 2304698
- PAR ID:
- 10623650
- Publisher / Repository:
- Elsevier Inc.
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 474
- Issue:
- C
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 110310
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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