Compact Manifolds with Unbounded Nilpotent Fundamental Groups and Positive Ricci Curvature

TL;DR

The paper constructs a sequence of smooth $10$-manifolds $(M_k^{10},g_k)$ with ${\rm Ric}_{g_k}\ge 9$ whose fundamental groups are degree‑two extensions of the finite Heisenberg groups $H_3(\mathbb{Z}/k\mathbb{Z})$, showing these groups are not uniformly virtually abelian in $k$ and thus answering questions of Kapovitch–Wilking and Fukaya–Yamaguchi in the negative. The construction proceeds by first building simply connected $M_k^4$ carrying a smooth, effective action of $H_3(\mathbb{Z}/k\mathbb{Z})$ via a Gibbons–Hawking–type ansatz over $S^3$, then lifting the action to the (spin) frame bundle to obtain a free action in dimension $10$. A careful analysis yields uniform diameter bounds and Ricci curvature bounds on the base, followed by a small conformal perturbation to achieve ${\rm Ric}_{g_k}>0$ while preserving symmetry. Finally, passing to the oriented frame bundle and its spin double cover produces simply connected $M_k^{10}$ with a free action of a $\mathbb{Z}/2\mathbb{Z}$-extended Heisenberg group, yielding the desired fundamental-group behavior and demonstrating limits of conjectures relating nonnegative Ricci curvature to uniform abelianness.

Abstract

It follows from the work of Kapovitch and Wilking that a closed manifold with nonnegative Ricci curvature has an almost nilpotent fundamental group. Leftover questions and conjectures have asked if in this context the fundamental group is actually uniformly almost abelian. The main goal of this work is to construct examples $(M^{10}_k, g_k)$ with uniformly positive Ricci curvature ${\rm Ric}_{g_k}\geq 9$ whose fundamental groups cannot be uniformly virtually abelian.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 4.1
• proof
• Remark 4.2: Choice of $V$
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• Lemma 4.6