Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity

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 $\mathrm{\Gamma }\tilde{p}$, where $\mathrm{\Gamma }={\pi }_1(M,p)$ acts on the universal cover $(\tilde{M},\tilde{p})$. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of $\mathrm{\Gamma }\tilde{p}$. We also give the first example of a manifold $M$ of $\mathrm{R}\mathrm{i}\mathrm{c}>0$ and ${\pi }_1(M)=\mathbb{Z}$ but with a varying orbit growth order.