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

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 $C_{2^{\mathrm{\infty }}}$. This gives counterexamples to Milnor’s conjecture in the two remaining dimensions.