Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

We study the fundamental group of an open n-manifold M of nonnegative Ricci curvature with additional stability conditions on $\tilde{M}$, the Riemannian universal cover of M. We prove that if every asymptotic cone of $\tilde{M}$ is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric cone, then ${\pi }_1\left(M\right)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\tilde{M}$ 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 $\tilde{M}$ has Euclidean volume growth of constant at least $1-ϵ\left(n\right)$, then ${\pi }_1\left(M\right)$ is finitely generated and $C\left(n\right)$-abelian.