Complete manifolds with nonnegative Ricci curvature and slow relative volume growth

For any complete and noncompact manifold $M$ with $\mathrm{Ric}\ge 0$, we define a function $\mathrm{RV}(s)$ that describes the growth of relative volume asymptotically $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$ Then we study the fundamental groups of such manifolds with slow relative volume growth and sublinear diameter growth. We show that if $\mathrm{RV}(s)\ll s^2$ as $s\to\infty$, then $π_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s^{1+δ}$ for some $δ\in (0,1)$ and the Ricci curvature is positive at a point, then $π_1(M)$ is finite. These results generalize our previous work on complete manifolds with $\mathrm{Ric}\ge 0$ and linear (minimal) volume growth.