Alexander invariants and cohomology jump loci in group extensions (Q6609501)

This paper studies three different versions (integral/rational/mod-\(p\)) of Alexander invariants of a group \(G\), defined as follows:\N\begin{itemize}\N\item The \textbf{(integral) Alexander invariant} \(B(G)\) of the group \(G\) is defined as \(B(G):=G'/G''\), where \(G':=[G,G]\) and \(G''=[G',G']\). \(B(G)\) has the structure of a \(\mathbb Z[G_{\mathrm{ab}}]\)-module induced by conjugation in the maximal metabelian quotient \(G/G''\) of \(G\), where \(G_{\mathrm{ab}}:=G/G'\) is the abelianization of \(G\).\N\item The \textbf{rational Alexander invariant} \(B_{\mathbb Q}(G)\) of the group \(G\) is defined as \(B_{\mathbb Q}(G):=G'_{\mathbb Q}/G''_{\mathbb Q}\), where \(G'_{\mathbb Q}=\sqrt{G'}:=\langle g\in G\mid g^m\in G'\text{ for some }m\in\mathbb N\rangle\), and \(G''_{\mathbb Q}:=\sqrt{[G'_{\mathbb Q},G'_{\mathbb Q}]}\). \(B_{\mathbb Q}(G)\) has the structure of a \(\mathbb Z[G_{\mathrm{abf}}]\)-module induced by conjugation in the maximal torsion-free metabelian quotient \(G/G''_{\mathbb Q}\) of \(G\), where \(G_{\mathrm{abf}}:=G/G'_{\mathbb Q}\) is the maximal torsion-free abelian quotient of \(G\).\N\item For every prime \(p\), the \textbf{mod-\(p\) Alexander invariant} \(B_p(G)\) of the group \(G\) is defined as \(B_p(G):=G'_p/G''_p\), where \(G'_p:=\langle G^p, [G,G]\rangle\) is the subgroup generated by \(G^p:=\langle g^p\mid g\in G\rangle\) and \(G'\), and \(G''_p:=\langle \left(G'_p\right)^p,\left[G'_p,G'_p\right]\rangle\). \(B_p(G)\) has the structure of a \(\mathbb Z_p[H_1(G;\mathbb Z_p)]\)-module induced by conjugation in the maximal metabelian \(p\)-quotient \(G/G''_{p}\) of \(G\).\N\end{itemize}\N\NThe three versions (integral, rational and mod-\(p\)) of the Alexander invariant have a topological interpretation as the first homology of certain covering spaces associated to the group \(G\). More concretely, let \(X\) be a connected topological space such that \(\pi_1(X)\cong G\). Then,\N\begin{itemize}\N\item \(B(G)\cong H_1(X^{\mathrm{ab}},\mathbb Z)\), where \(X^{\mathrm{ab}}\) is the universal abelian cover of \(X\).\N\item \(B_{\mathbb Q}(G)\cong H_1(X^{\mathrm{abf}},\mathbb Z)\), where \(X^{\mathrm{abf}}\) is the universal torsion-free abelian cover of \(X\).\N\item \(B_p(G)\cong H_1(X^{(p)},\mathbb Z_p)\), where \(X^{(p)}\) is the \(p\)-congruence cover of \(X\), namely the one induced by the composition \N\[\N\pi_1(X)\twoheadrightarrow G_{\mathrm{ab}}\cong H_1(X;\mathbb Z)\twoheadrightarrow G/G'_p\cong H_1(X;\mathbb Z_p),\N\]\Nwhere the first epimorphism is the abelianization and the second one is induced by the projection \(\mathbb Z\twoheadrightarrow\mathbb Z_p\).\N\end{itemize}\N\NThis paper provides an extension of known results for the classical (integral) invariants to the rational and mod-\(p\) versions, and it establishes connections between them. For example, the relation of the integral Alexander invariants of \(G\) with its lower central series given by Massey's correspondence is extended to the rational and mod-\(p\) Alexander invariants and versions of the lower central series. The relationship of Alexander invariants with other (co)homological invariants such as characteristic and resonance varieties is made explicit.\N\NAs suggested by the title, the paper studies the behavior of the above mentioned (co)homological invariants with respect certain group extensions, which for the most part are of the form \N\[\N1\to K\to G\to Q\to 1, \N\]\Nwhere \(Q\) is an Abelian group (resp. torsion-free abelian group/elementary abelian \(p\)-group) and the sequence remains exact when passing to the abelianizations (resp. maximal torsion-free abelian quotients/first homology with \(\mathbb Z_p\) coefficients). The goal of this is to relate the different invariants of \(G\) to those of \(K\). The language used is algebraic, but the motivation to study such short exact sequences in the integral and rational cases comes from geometry, specifically from the setting of central complex hyperplane arrangement complements \(M\) with Milnor fiber \(F\) having trivial algebraic monodromy: in that case \(G=\pi_1(M)\), \(K=\pi_1(F)\), and \(Q=\mathbb Z\). The exposition is detailed and mostly self-contained, making it a very good resource for people wanting to learn more about the subject.