  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.
