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 $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$ Then we study the fundamental groups of such manifolds with slow relative volume growth and sublinear diameter growth. We show that if $\mathrm{RV}(s)\ll s<sup>2$ as $s\to\infty$, then $π_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s<sup>{1+δ}$ 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.

• The paper establishes that slow relative volume growth (RV(s)=o(s^2)) under sublinear diameter conditions leads to a virtually abelian fundamental group.
• It leverages asymptotic cones and RCD space theory to connect precise volume asymptotics with geometric rigidity and topological constraints.
• Stronger conditions (RV(s)=o(s^(1+δ))) combined with pointwise Ricci positivity yield finiteness of the fundamental group, extending classical results.

Complete Manifolds with Nonnegative Ricci Curvature and Slow Relative Volume Growth

---

Introduction and Main Results

The paper investigates the interactions between the asymptotic geometry of complete noncompact Riemannian manifolds with nonnegative Ricci curvature ($\operatorname{Ric} \geq 0$) and the structure of their fundamental groups, with a focus on classes with "slow" relative volume growth at infinity. Classical results (Cheeger–Gromoll, Bonnet–Myers, Milnor–Gromov, Sormani, Wei, et al.) establish that for closed manifolds with $\operatorname{Ric} \geq 0$, the fundamental group is virtually abelian, while for noncompact manifolds, nonabelian and even infinitely generated fundamental groups are possible, though always virtually nilpotent.

The central concept introduced is the relative volume growth function $\mathrm{RV}(s)$:

$$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$

which, by Bishop–Gromov comparison, satisfies $\mathrm{RV}(s) \leq s^n$. The paper systematically replaces the requirement of linear (minimal) volume growth with conditions on the sub-polynomial asymptotics of $\mathrm{RV}(s)$.

The main results are:

• If $\mathrm{RV}(s) = o(s^2)$ as $s \to \infty$ and $M$ has sublinear diameter growth, then $\pi_1(M)$ is almost abelian: it contains a $\operatorname{Ric} \geq 0$0 subgroup of finite index, for $\operatorname{Ric} \geq 0$1.
• If $\operatorname{Ric} \geq 0$2 as $\operatorname{Ric} \geq 0$3 for some $\operatorname{Ric} \geq 0$4, $\operatorname{Ric} \geq 0$5 has sublinear diameter growth, and $\operatorname{Ric} \geq 0$6 at a point, then $\operatorname{Ric} \geq 0$7 is finite.

These extend previous results for manifolds with linear volume growth and significantly broaden the topological implications for classes with strictly subquadratic relative volume growth.

---

Technical Framework: Asymptotic Cones, RCD Spaces, and Volume Growth

A crucial technical innovation is the development and use of relative volume asymptotics and their invariance under basepoint (proved using standard comparison techniques). The paper's analysis leverages the structure theory of asymptotic cones of manifolds with $\operatorname{Ric} \geq 0$8 and the modern theory of $\operatorname{Ric} \geq 0$9 spaces (synthetic lower Ricci curvature bounds in the sense of Lott–Sturm–Villani and Ambrosio–Gigli–Savaré).

For a sequence $\mathrm{RV}(s)$0, the rescaled pointed manifolds $\mathrm{RV}(s)$1 converge (in a measured Gromov–Hausdorff sense) to an asymptotic cone $\mathrm{RV}(s)$2, which automatically satisfies the $\mathrm{RV}(s)$3 condition. Limit measures reflect rescaled normalized volume.

The function $\mathrm{RV}(s)$4 effectively controls the asymptotic “volume profile” of the extremal rays of $\mathrm{RV}(s)$5. Small $\mathrm{RV}(s)$6 (subquadratic or sublinear) imposes strong rigidity on the possible asymptotic cones and the topological structure at infinity.

---

Rigidity via Distributional Bakry–Émery Ricci Curvature

The central geometric rigidity arises from a detailed analysis of the asymptotic cones—and more precisely, their equivariant versions under the action of the covering group, leveraging the theory of $\mathrm{RV}(s)$7 spaces with group actions.

The main rigidity theorems are established via distributional Bakry–Émery Ricci curvature lower bounds in the sense of [Mondino–Rybarz], applied to general warped product spaces with regularity $\mathrm{RV}(s)$8. In particular:

• If an $\mathrm{RV}(s)$9 space $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$0 admits a free $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$1-action whose quotient is a ray, and if the relative measure of certain regions ("strips") grows sub-quadratically at infinity, then $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$2 is isometric to a Euclidean halfplane.
• For a space with an isometric $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$3-action fixing a point and quotient a ray, if the measure grows super-quadratically near the basepoint, then $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$4 must be isometric to a ray, with the $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$5-action trivial.

These rigidity results utilize fine measure-theoretic regularity and the behavior of warping functions under synthetic curvature bounds, culminating in splitting theorems for certain asymptotic behaviors of the relative volume.

---

Inductive Structure and Fundamental Group Implications

A systematic induction is developed for towers of $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$6 (or, more generally, torsion-free nilpotent) covering subgroups. At each stage, the possible asymptotic cones of the covering spaces are restricted by the rigidity theorems and volume conditions, resulting ultimately in spaces isometric to products of Euclidean halflines.

This control of the asymptotic geometry and covering space topology implies:

• Under the slow relative volume growth condition $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$7, any finitely generated virtually nilpotent covering group must be (virtually) abelian, a higher-dimensional extension of the classical Milnor–Gromov–Cheeger–Gromoll finiteness/abelianity dichotomy for closed/open cases.
• Under even slower relative volume growth ($$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$8, $$\mathrm{RV}(s) = \limsup_{r \to \infty} \frac{\operatorname{vol} B_{rs}(p)}{\operatorname{vol} B_r(p)}, \quad s \geq 1,$$9), positivity of Ricci at a point implies finiteness of the fundamental group, extending the Bonnet–Myers–Milnor results to broad open manifold classes.

A key technical point is that stronger conditions on relative volume growth (not just absolute volume growth) are essential for these conclusions; the authors pose open questions regarding the sharpness of these results and possible exceptions.

---

Implications, Context, and Future Directions

This work substantially generalizes rigidity and finiteness theorems for the topological structure of noncompact manifolds with nonnegative Ricci curvature, based solely on relative volume growth rates at infinity, rather than classical absolute growth. It relates the asymptotic geometry of the universal cover and its asymptotic cones (an analytic/geometric concept) to the algebraic structure of the fundamental group.

Practically, this highlights that for manifolds with sufficiently slow volume escape—even in the infinite, non-splitting, noncompact case—strong constraints are imposed on large-scale topology. The precise dichotomy at quadratic relative volume asymptotics is established as sharp; examples are constructed to show the necessity and limits of these threshold exponents.

Theoretically, the approach introduces a flexible measure-theoretic toolkit for analyzing equivariant Gromov–Hausdorff limits, well-adapted to the study of manifolds with synthetic Ricci lower bounds and group actions. The distributional Bakry–Émery method and developments in non-smooth differential geometry (RCD spaces) play a central role.

Future directions include a deeper classification of manifolds with discrete volume growth rates, exploration of possible non-abelian structures just above the quadratic threshold, applications to Alexandrov geometry, and extension of measure-theoretic rigidity techniques to settings with lower regularity and variable (not just nonnegative) Ricci bounds.

---

Conclusion

The paper rigorously elucidates the intricate relationship between the analytic property of slow relative volume growth and the algebraic structure of the fundamental group for open manifolds with $\mathrm{RV}(s) \leq s^n$0. By introducing the relative volume growth function $\mathrm{RV}(s) \leq s^n$1 and leveraging $\mathrm{RV}(s) \leq s^n$2 structure theory and distributional curvature bounds, the authors establish sharp new virtual abelianness and finiteness results for the fundamental group, generalizing and extending many classical theorems in Riemannian geometry. The analytic methods and inductive topological strategy set the stage for further advances in global differential geometry and metric measure rigidity.