Let $G$ be a group of finite cohomological dimension $n$ (e.g., of type FP). If $i\leq n$, does there exist a subgroup $H$ of $G$ of cohomological dimension $i$?
Do you know any example where such a phenomenon does not hold?
Example. Let $G$ be a RAAG (right-angled Artin group) on a finite graph $\Gamma$. Then, the cohomological dimension of $G$ equals the clique number of $\Gamma$, i.e., the maximal number of pairwise adjacent vertices in $\Gamma$. This amounts to saying that a RAAG $G$ has cohomological dimension $n$ if the maximal abelian subgroup of $G$ has rank $n$, i.e., it is isomorphic with $\mathbb Z^n$. In turn, this implies that $G$ contains $\mathbb Z^i$ for all $i\leq n$, so that the above question has affirmative answer for the class of RAAGs. Notice that for a group $G$ it is not necessary to contain abelian subgroups of rank equal to the cohomological dimension.
Example. If $G$ has cohomological dimension $2$, then $G$ is torsion free and hence contains an infinite cyclic subgroup, which clearly has cohomological dimension $1$.