Instability of the fundamental group for non-collapsed Ricci-limits

TL;DR

This paper constructs examples of 4-dimensional manifolds with non-negative Ricci curvature that have different fundamental groups but converge to the same limit space, challenging previous assumptions about stability.

Contribution

It provides explicit examples showing the fundamental group is not stable under Ricci limit convergence in non-collapsed settings.

Findings

Different fundamental groups can share the same Ricci limit space.

Counterexamples to fundamental group stability in Ricci limit theory.

Advances understanding of geometric limits in Riemannian geometry.

Abstract

We construct two sequences of closed $4$-dimensional manifolds with non-negative Ricci curvature, diameter bounded from above by $1$, and volume bounded from below by $v>0$, with different fundamental groups but with the same Gromov-Hausdorff limit. This provides a negative answer to the question posed in [J. Pan. Ricci Curvature and Fundamental Groups of Effective Regular Sets. Journal of Mathematical Study, 58(1):3--21, 2025].