Nonnegative Ricci curvature and virtual abelianness in dimensions less than 12

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.