Complete manifolds with nonnegative Ricci curvature and slow relative volume growth

Abstract

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 $$\mathrm{RV}(s)=\underset{r\to \infty }{\mathrm{limsup}}\frac{\mathrm{vol}{B}_{rs}(p)}{\mathrm{vol}{B}_r(p)},s\ge 1.$$ 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 ${\pi }_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s^{1+\delta }$ for some $\delta \in (0,1)$ and the Ricci curvature is positive at a point, then ${\pi }_1(M)$ is finite. These results generalize our previous work on complete manifolds with $\mathrm{Ric}\ge 0$ and linear (minimal) volume growth.