Fundamental groups of open manifolds with nonnegative Ricci curvature and universal cover Euclidean volume growth

In this note, we will give an positive answer to Pan-Rong’s conjecture that for an open manifold with nonnegative Ricci curvature, if its universal cover has Euclidean volume growth, then its fundamental group is finitely generated. Moreover the fundamental group is virtually abelian. The same result has been given by H.Huang-X.Huang for dimension 4. In fact, we will show the fundamental group finitely generated in a more general condition.