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 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, where =₁ (M, p) acts on the universal cover (M, p). Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of. We also give the first example of a manifold M of Ric>0 and ₁ (M) =Z but with a varying orbit growth order.