Nonnegative Ricci curvature, stability at infinity, and finite generation of fundamental groups

TL;DR

This paper investigates the fundamental group of open manifolds with nonnegative Ricci curvature, proving finite generation under certain tangent cone conditions, thus confirming a special case of Milnor's conjecture.

Contribution

It establishes finite generation of the fundamental group for manifolds with a unique tangent cone at infinity and Euclidean volume growth, under specific geometric conditions.

Findings

Fundamental group is finitely generated under tangent cone conditions.

Confirms Milnor conjecture for manifolds with unique tangent cone and Euclidean volume growth.

Shows stability at infinity implies finite generation of the fundamental group.

Abstract

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $\pi_1(M)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and the unique tangent cone at infinity.