Nonnegative Ricci curvature and virtual abelianness in dimensions less than 12

Dimitri Navarro; Jiayin Pan; Xingyu Zhu

Nonnegative Ricci curvature and virtual abelianness in dimensions less than 12

Differential Geometry 2026-06-26 v1

Authors: Dimitri Navarro , Jiayin Pan , Xingyu Zhu

Abstract

For any complete Riemannian manifold $M^n$ with nonnegative Ricci curvature and sublinear diameter growth, we establish a dimensional constraint $n\ge 4s(s-1)+k+1$ if the fundamental group $\pi_1(M)$ contains a torsion-free nilpotent subgroup of rank $k$ and step $s\ge 2$ . As a consequence, if such a manifold $M$ has dimension $n<12$ , then $\pi_1(M)$ is almost abelian. The proof is based on a dimensional estimate for $\mathrm{RCD}(0,N)$ spaces admitting $\mathbb{R}$ -orbits of large Hausdorff dimension.

Keywords

Ricci curvature fundamental group virtual abelianness diameter growth RCD spaces

Cite

@article{arxiv.2606.27724,
  title  = {Nonnegative Ricci curvature and virtual abelianness in dimensions less than 12},
  author = {Dimitri Navarro and Jiayin Pan and Xingyu Zhu},
  journal= {arXiv preprint arXiv:2606.27724},
  year   = {2026}
}