By Theorem 5 of "The Algebraic Characterization of Geometric 4-Manifolds" by Hillman, if $G$ is an elementary amenable group of finite cohomological dimension, then $h(G)\leq cd(G)$.
My question is that is there a bigger class of finitely presented groups of finite cohomological dimension such that $h(G)\leq cd(G)$?
By Corollary 2.5 of "Cohomology of rings" by Brown, if $cd(G)<\infty$, then $G$ is torsion free.