This paper exhibits closed manifolds \(M\) covered by \(S^{2n-1}\times \mathbb{R}^ k\) for all \(n\geq 2\) and for sufficiently large \(k\), with fundamental groups of infinite virtual cohomological dimension. These examples are based on results of \textit{M. S. Raghunathan} [Math. Ann. 266, 403-419 (1984; Zbl 0513.22008)] on lattices in covers of spin and symplectic groups and address a problem first raised by Wall.