Complete and noncompact manifolds with nonnegative Ricci curvature are known to have at least linear volume growth by the work of Calabi and Yau. In joint work with Dimitri Navarro and Xingyu Zhu, we prove two results on the fundamental groups of these manifolds with linear volume growth. First, the fundamental group contains a subgroup \mathbb{Z}^k of finite index. Second, if the Ricci curvature is positive, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive \mathbb{Z}-folded covering spaces and a plane/halfplane rigidity result for RCD spaces.