Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent

Lazard asked which $N$-series of a group arise from multiplicative filtrations of the integral group ring. Given an $N$-series $H_{\bullet }$ of a group $G$, let $A_{\bullet }$ be the filtration of the augmentation ideal spanned by products of elements $x-1$, $x\in H_a$, weighted by $a$. Every realizing filtration contains $A_{\bullet }$, so $H_{\bullet }$ is realizable exactly when the maps $H_n\mathrm{/}{H}_{n+1}\to A_n\mathrm{/}{A}_{n+1}$, $h\mapsto h-1$, are injective. We show that each kernel is the cokernel of an explicit map of normalized bar groups; for a finite group this makes realizability decidable by integer linear algebra. For a profinite group with a cofinal tower of finite quotients, an element $h\in H_n$ with $h-1\in A_{n+1}$ in every finite quotient satisfies $h-1\in A_{n+1}$ in the discrete group ring exactly when the widths of its Losey expressions are bounded. This descent can fail. Using Tahara’s class-three $2$-groups and the Hartl-Mikhailov-Passi description of the fourth dimension subgroup, we construct a countable product $P$ of finite $2$-groups and an element $h\in {\gamma }_3(P)$ that lies in the fourth dimension subgroup of every finite subproduct of $P$ but not in $D_4(P)$.