Polynomial identities in graded group rings, restricted lie algebras and p-adic analytic groups

Let G be any finitely generated group, and let K be a field of chacteristic p > 0. It is shown that the graded group ring gr(KG) satisfies a nontrivial polynomial identity if and only if the pro-p completion of G is p-adic analytic, i.e. Can be given the structure of a Lie group over the p-adic field ℚ_p. The proof applies theorems of Lazard, Quillen and Passman, as well as results on Engel Lie algebras and on dimension subgroups in positive characteristic.