Finite generation of fundamental groups for manifolds with nonnegative Ricci curvature whose universal cover is almost $k$-polar at infinity

Abstract: In this article, we prove that the fundamental group $\pi_1(M)$ of a complete open manifold $M$ with nonnegative Ricci curvature is finitely generated, under the condition that the Riemannian universal cover $\tilde M$ satisfies an "almost $k$-polar at infinity" condition. Additionally, such $\pi_1(M)$ is virtually abelian. Furthermore, we demonstrate that the base point of any tangent cone at infinity of such a manifold is nearly a pole. In the case where $\tilde M$ exhibits almost maximal Euclidean volume growth, we prove that $M$ deformation retracts to a closed submanifold $F$ which is diffeomorphic to a flat manifold, provided $M$ is not simply connected.