Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

TL;DR

This paper investigates the fundamental group of open manifolds with nonnegative Ricci curvature, showing under certain stability and volume growth conditions that the group is finitely generated and virtually abelian, with bounds depending on geometric parameters.

Contribution

It establishes new conditions under which the fundamental group is finitely generated and virtually abelian, linking tangent cone structures and volume growth to algebraic properties.

Findings

Fundamental group is finitely generated under tangent cone stability.

Presence of a normal abelian subgroup of finite index in the fundamental group.

Bounds on the index of the abelian subgroup depending on volume growth constant.

Abstract

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $\pi_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$ has Euclidean volume growth of constant at least $1-\epsilon(n)$, then $\pi_1(M)$ is finitely generated and $C(n)$-abelian.