Prüfer $2$-group and Milnor's Conjecture on Fundamental Groups

Abstract

We construct complete, one-ended Riemannian manifolds in dimensions four and five having strictly positive Ricci curvature, with fundamental group isomorphic to the Prüfer $2$-group $C_{2^{\infty }}$ and any subgroup $\Gamma \le \mathbb{Q}/\mathbb{Z}$, respectively. This gives counterexamples to Milnor's conjecture in the two remaining dimensions.

69 pages; improved exposition, add full construction for all subgroups of $\mathbb{Q}/\mathbb{Z}$ in 5 dimension, and the complete description of volume growth