Nonnegative Ricci curvature, metric cones, and virtual abelianness

TL;DR

This paper proves that open manifolds with nonnegative Ricci curvature and certain geometric conditions have virtually abelian fundamental groups, linking their asymptotic geometry to algebraic properties of their fundamental groups.

Contribution

It establishes conditions under which the fundamental group of such manifolds is virtually abelian, especially when the universal cover is conic at infinity and has specific volume growth.

Findings

Fundamental group contains an abelian subgroup of finite index under given conditions.

Bound on the index of the abelian subgroup depending on dimension and volume growth.

Connection between asymptotic cone geometry and algebraic structure of the fundamental group.

Abstract

Let $M$ be an open $n$-manifold with nonnegative Ricci curvature. We prove that if its escape rate is not $1/2$ and its Riemannian universal cover is conic at infinity, that is, every asymptotic cone $(Y,y)$ of the universal cover is a metric cone with vertex $y$, then $\pi_1(M)$ contains an abelian subgroup of finite index. If in addition the universal cover has Euclidean volume growth of constant at least $L$, we can further bound the index by a constant $C(n,L)$.