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Creators/Authors contains: "Ye, Zhu"

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  1. ; (, 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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