Suppose $M$ is a (closed, connected, oriented, smooth) manifold.

If $M$ is aspherical, i.e., if the inversal covering $\tilde{M}$ is contractible, $M$ is a $B\pi_1(M)$. This is often enforced by geometry, for instance it holds if $M$ admits a metric of non-positive sectional curvature (Cartan--Hadamard).

We deduce that the inequality $$\text{vcd}(\pi_1(M)) \leq \text{dim}(M)$$ holds (it is actually an equality in this case).

I learned recently that for a simply-connected Lie group $G$ and a lattice $\Gamma < G$, we can pick a maximal compact $K < G$, then $K \backslash G$ is contractible, hence $K \backslash G / \Gamma \sim_{\mathbf Q} B\Gamma$. Thus for $M = G/\Gamma$, the above inequality is also satisfied.

Furthermore, and going in the same direction, Mostow proved that fundamental groups of arbitrary homogeneous spaces, if they are solvable, have rank at most the dimension of the space. The above inequality thus also holds in this case.

However, it is (of course) easy to construct manifolds for which the above inequality fails, as any finitely presented group is the fundamental group of some $d$-manifold for all $d \geq 4$.

Is there a common generalization of being aspherical or a homogeneous space of the type described that ensures that the above inequality is satisfied? What are other conditions on $M$ that ensure that it holds?

assumessolvability of the fundamental group. $\endgroup$