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 π 1 ⁢ ( M ) π₁(M) is finitely generated. This result confirms Pan–Rong’s conjecture for dimension n = 4 n=4 . Additionally, we prove that there exists a universal constant C > 0 C>0 such that π 1 ⁢ ( M ) π₁(M) contains an Abelian subgroup of index at most 𝐶. More specifically, if π 1 ⁢ ( M ) π₁(M) is infinite, then π 1 ⁢ ( M ) π₁(M) is a crystallographic group of rank at most 3; if π 1 ⁢ ( M ) π₁(M) is finite, then π 1 ⁢ ( M ) π₁(M) is isomorphic to a quotient of the fundamental group of a spherical 3-manifold.