Splitting and Slow Volume Growth for Open Manifolds with Nonnegative Ricci Curvature

In \cite{NPZ24}, Navarro-Pan-Zhu proved that the fundamental group of an open manifold with nonnegative Ricci curvature and linear volume growth contains a subgroup isomorphic to $\mathbb{Z}^k$ with finite index. They further asked whether the existence of a torsion-free element in the fundamental group forces the universal cover to split off an isometric $\mathbb{R}$-factor (Question 1.3 of \cite{NPZ24}). In this article, we provide an affirmative answer to this question. Specifically, we prove that if an open manifold with nonnegative Ricci curvature has linear volume growth, then its universal cover is isometric to a metric product $\mathbb{R}^k \times N$, where $N$ is an open manifold with linear volume growth and $k$ is the integer such that $\pi_1(M)$ contains a $\mathbb{Z}^k$-subgroup of finite index. As a direct consequence, if the Ricci curvature is positive at some point, then the fundamental group is finite. We also establish that for an open manifold $M$ with nonnegative Ricci curvature, if the infimum of its volume growth order is strictly less than $3$ and $\tilde{M}$ has Euclidean volume growth, then the universal cover $\tilde{M}$ splits off an $\mathbb{R}^{n-2}$-factor. As an application, if $M$ has first Betti number $b_1 = n-2$ and $\tilde{M}$ has Euclidean volume growth, then its universal cover admits such a splitting. This result provides a partial answer to \cite[Question 1.6]{PY24}.