Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds

TL;DR

The paper resolves Pan-Rong's conjecture in dimension $4$ by proving that an open $4$-manifold with nonnegative Ricci curvature has a finitely generated fundamental group whenever its universal cover exhibits Euclidean volume growth. It develops a framework based on equivariant Gromov-Hausdorff limits and topological regularity of almost-splitting level sets (via Bruè–Pigati–Semola) to show finite generation and, moreover, virtual abelianness with a universal bound on the index of an abelian subgroup. In the infinite case, the fundamental group is crystallographic of rank at most $3$, containing a free abelian normal subgroup of index $C$, while in the finite case it is a quotient of a spherical $3$-manifold group; a corollary yields pole-type properties for asymptotic cones. Together, these results illuminate the structure of $\\pi_1(M)$ under nonnegative Ricci curvature with Euclidean-volume growth and connect to stability phenomena in noncollapsed Ricci limit spaces, with broad implications for the geometry and topology of open manifolds.

Abstract

Let $M$ be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of $M$ has Euclidean volume growth, then the fundamental group $π_1(M)$ is finitely generated. This result confirms Pan-Rong's conjecture \cite{PR18} for dimension $n = 4$. Additionally, we prove that there exists a universal constant $C>0$ such that $π_1(M)$ contains an abelian subgroup of index $\le C$. More specifically, if $π_1(M)$ is infinite, then $π_1(M)$ is a crystallographic group of rank $\le 3$. If $π_1(M)$ is finite, then $π_1(M)$ is isomorphic to a quotient of the fundamental group of a spherical 3-manifold.

Theorems & Definitions (60)

• Theorem A
• Theorem 1.1
• Theorem B
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• Lemma 3.2