Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer $q>1$ and every topological group $G$, with realizations $B(q,G)$ that filter the classifying space $BG$. In particular for $q=2$ this yields a single space $B(2,G)$ assembled from all the n-tuples of commuting elements in $G$. Homotopy properties of the $B(q,G)$ will be described for finite groups, and cohomology calculations provided for compact Lie groups. Recent results on understanding both the number and stable homotopy type of the components of related spaces of representations will also be discussed.