Let $G$ be a finitely presented group on and let $ma(G) = n$ be the minimum of $\left| K \right|$ ($K \subseteq G$) over all group presentations $G \cong F\{A\} / H$ such that $G / {H \vee [K, K]}$ is commutative.
If $ma(G) = 0$ then $G$ is abelian and if $G = F\{A\}$ then $ma(G) = \left|A\right|$.
Thus $ma$ measures the the numbers of 'steps' $G$ is away from being commutative.
Is this construction useful? Does it have a name?