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

TL;DR

This paper investigates the fundamental groups of open manifolds with nonnegative Ricci curvature and linear volume growth, revealing their algebraic structure and conditions for finiteness, using advanced geometric analysis techniques.

Contribution

It establishes that such manifolds have fundamental groups containing a finite index subgroup isomorphic to al^k, and that positive Ricci curvature implies a finite fundamental group.

Findings

Fundamental group contains a al^k subgroup of finite index.

Positive Ricci curvature implies a finite fundamental group.

Uses equivariant asymptotic geometry and rigidity results for RCD spaces.

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.