Fundamental groups of open manifolds with nonnegative Ricci curvature and universal cover Euclidean volume growth

TL;DR

This paper proves that open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated, virtually abelian fundamental groups, extending previous results to more general conditions.

Contribution

It confirms Pan-Rong's conjecture and generalizes the finite generation of fundamental groups under broader conditions.

Findings

Fundamental groups are finitely generated.

Fundamental groups are virtually abelian.

Results extend previous theorems to more general cases.

Abstract

In this note, we will give an positive answer to Pan-Rong's conjecture that for an open manifold with nonnegative Ricci curvature, if its universal cover has Euclidean volume growth, then its fundamental group is finitely generated. Moreover the fundamental group is virtually abelian. The same result has been given by H.Huang-X.Huang for dimension 4. In fact, we will show the fundamental group finitely generated in a more general condition.