  We continue the study of the lower central series and its associated graded
components for a free associative algebra with n generators, as initiated by B.
Feigin and B. Shoikhet. We establish a linear bound on the degree of tensor
field modules appearing in the Jordan-Hoelder series of each graded component,
which is conjecturally tight. We also bound the leading coefficient of the
Hilbert polynomial of each graded component. As applications, we confirm
conjectures of P. Etingof and B. Shoikhet concerning the structure of the third
graded component.
