Profinite completions and cohomology jump loci

Abstract

Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$, under two hypotheses treated separately: that the loci are finite unions of torsion-translated subtori, which holds for smooth quasi-projective varieties, and that $\widehat{G}$ determines the Betti numbers of the finite cyclic covers of $X$ in degrees $\le q$, which holds unconditionally for $q=1$ and in all degrees when $X$ is aspherical and $G$ is good in the sense of Serre. We show also that $\widehat G$ determines the graded abelian groups $\mathrm{gr}_r(G/W(G))$, torsion included, for every verbal subgroup $W(G)$; the cases $W(G)=1$ and $W(G)=G''$ give the lower central series quotients and the Chen groups. For hyperplane arrangements $\mathcal{A}$, it follows that no arithmetic Zariski pair is distinguished by any of these invariants, while two known lattice-isomorphic pairs show that the profinite completion of the arrangement group $G(\mathcal{A})$ is not combinatorially determined, and does not determine $G(\mathcal{A})$.