Closed manifolds with positive Ricci curvature and large nilpotent fundamental groups

In 2011, V. Kapovitch and B. Wilking proved that in any given dimension $n$ there is a constant $C(n)$ such that the fundamental group of a closed $n$-dimensional Riemannian manifold with nonnegative Ricci curvature has a nilpotent subgroup with index less than $C(n)$. A natural question left in their work is whether one can strengthen this result by replacing "nilpotent" with "abelian". I will discuss joint work with E. Bruè and A. Naber where we answer Kapovitch and Wilking’s question in the negative.