On lower central series quotients of finitely generated algebras over Z

Let A be an associative unital algebra, Bk its successive quotients of lower central series and Nk the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of Bk's and Nk's have been of great interest since the pioneering work of [11]. In this paper, we will concentrate on the case where A is a noncommutative polynomial algebra over Z modulo a single homogeneous relation. Both the torsion part and the free part of Bk's and Nk's are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with Appendix A about the torsion subgroups of Nk(An(Z)) and some open problems.