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On the topology of manifolds with nonnegative Ricci curvature and linear volume growth

Dimitri Navarro, Jiayin Pan, Xingyu Zhu

Abstract

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

On the topology of manifolds with nonnegative Ricci curvature and linear volume growth

Abstract

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) -manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup of finite index, where . Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

Paper Structure

This paper contains 19 sections, 52 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ricci limits and RCD spaces
  4. Equivariant Gromov--Hausdorff convergence
  5. Manifolds with linear volume growth
  6. Escape rate
  7. Equivariant asymptotic geometry of successive covering spaces
  8. Polar asymptotic cones and non-maximal escape rate
  9. Consequences of asymptotic $\mathbb{R}$-orbits
  10. Convergence of appropriate symmetric subsets to cylinders
  11. Statement of the Induction Theorem and an outline
  12. Convergence of appropriate domains to cubes
  13. Proof of the Induction Theorem
  14. Structure of fundamental groups
  15. Plane/halfplane rigidity for RCD$(0,N)$ spaces
  16. ...and 4 more sections

Key Result

Theorem A

If $M^n$ is an open manifold with $\mathrm{Ric}\ge 0$ and linear volume growth, then $\pi_1(M)$ contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$.

Theorems & Definitions (132)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Remark 1.5
  • Theorem 1.6: Induction Theorem
  • Theorem 1.7: Plane/Halfplane Rigidity
  • Definition 1.8
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 122 more