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 condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $\pi_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$ has Euclidean volume growth of constant at least $1-\epsilon(n)$, then $\pi_1(M)$ is finitely generated and $C(n)$-abelian.