Complete manifolds with nonnegative Ricci curvature and slow relative volume growth

TL;DR

This paper investigates the fundamental groups of complete, noncompact manifolds with nonnegative Ricci curvature, showing that slow relative volume growth constrains the groups to be almost abelian or finite.

Contribution

It introduces a new volume growth function and establishes conditions under which the fundamental group is almost abelian or finite, generalizing previous results.

Findings

If RV(s) << s^2, then the fundamental group is almost abelian.

If RV(s) << s^{1+δ} with δ in (0,1) and Ricci positive somewhere, the fundamental group is finite.

The results extend prior work on manifolds with linear 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 $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$ 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 $\pi_1(M)$ 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.