Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds

Let $M$ be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of $M$ has Euclidean volume growth, then the fundamental group ${\pi }_1(M)$ is finitely generated. This result confirms Pan-Rong’s conjecture \cite{PR18} for dimension $n=4$. Additionally, we prove that there exists a universal constant $C>0$ such that ${\pi }_1(M)$ contains an abelian subgroup of index $\le C$. More specifically, if ${\pi }_1(M)$ is infinite, then ${\pi }_1(M)$ is a crystallographic group of rank $\le 3$. If ${\pi }_1(M)$ is finite, then ${\pi }_1(M)$ is isomorphic to a quotient of the fundamental group of a spherical 3-manifold.