Homology of iterated semidirect products of free groups
Let \(P_n\) denote the pure braid group on \(n\) strands. There is a split epimorphism \(P_{n+1}\to P_n\) whose kernel is a free group \(F_n\) on \(n\) generators. The action of \(P_n\) on \(F_n\) is given by the (restriction of the) well-known Magnus embedding of the full braid group into \(\Aut(F_n)\). In this way \(P_{n+1}\) has the structure of an iterated semidirect product of free groups. This is the situation analyzed in the paper under review. Other groups admitting such a structure include fundamental groups of complements of fiber-type arrangements, including the discriminantal arrangements of \textit{V. V. Shekhtman} and \textit{A. N. Varchenko} [Invent. Math. 106, No. 1, 139-194 (1991; Zbl 0754.17024)], and of total spaces of certain nonlinear fibrations with punctured-surface fibers. Suppose \(G\) has a finite filtration \(1=G_0\leq G_1\leq\ldots\leq G_\ell=G\) such that \(G_k\) is a split extension of \(G_{k-1}\) by a finitely-generated free group. An Eilenberg-MacLane space \(K(G,1)\) can be constructed using fibered products of bouquets of circles, with monodromy determined by the semidirect product structure. Using the bundle structure and methods of combinatorial group theory, specifically the Fox calculus, the authors construct a finite, free resolution \(C_\bullet(G)\) of \(\mathbb{Z}\) as a module over the group ring \(\mathbb{Z}[G]\). This construction yields an algorithm for computing the cohomology of \(G\), with coefficients in an arbitrary \(G\)-module. This cohomology, in case \(G=P_{n,k}:=\ker(P_n\to P_k)\), has applications to solutions of the Knizhnik-Zamolodchikov equations of conformal field theory. A cohomology vanishing theorem is deduced which strengthens previously known results. The algorithm is also used to compute the Betti numbers and algebraic monodromy of Milnor fibers associated with Coxeter arrangements (mainly of type \(A_\ell\)), and with classical discriminants, producing some previously unseen results. If \(\Gamma\) is a group which acts on \(G\) compatibly with the filtration, there is an induced action on the resolution \(C_\bullet(G)\). This is used to construct new representations of the full braid group, the pure braid group, and the subgroup \(P_{n,k}\), which generalize the Burau and Gassner representations. The representations of the full braid group do not factor through the Hecke algebra, and are sensitive enough to detect braids in the kernel of the Burau representation.
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