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Award ID contains: 2304698

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  1. (, Advances in Mathematics)
    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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    Free, publicly-accessible full text available July 1, 2026
  2. (, Journal of Mathematical Study)
    For a Gromov-Hausdorff convergent sequence of closed manifolds $$M_i^n\ghto X$$ with $$\Ric\ge-(n-1)$$, $$ \mathrm{diam}(M_i)\le D$$, and $$\mathrm{vol}(M_i)\ge v>0,$$ we study the relation between $$\pi_1(M_i)$$ and $$X$$. It was known before that there is a surjective homomorphism $$\phi_i:\pi_1(M_i)\to \pi_1(X)$$ by the work of Pan--Wei. In this paper, we construct a surjective homomorphism $$\psi_i$$ from the interior of the effective regular set in $$X$$ back to $$M_i$$. These surjective homomorphisms $$\phi_i$$ and $$\psi_i$$ are natural in the sense that their composition $$\phi_i \circ \psi_i$$ is exactly the homomorphism induced by the inclusion map from the effective regular set to $$X$$. 
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  3. ; (, Advances in Mathematics)
    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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