Separation in the BNSR-invariants of the pure braid groups (Q2405867)

The BNSR invariants \(\Sigma^m(G)\) of a group \(G\), are defined using representations \(G \to \mathbb{R}\). They are a generalization, due to \textit{R. Bieri} and \textit{B. Renz} [Comment. Math. Helv. 63, No. 3, 464--497 (1988; Zbl 0654.20029)], of the BNS invariant, defined by \textit{R. Bieri} et al. [Invent. Math. 90, 451--477 (1987; Zbl 0642.57002)]. The definition is rather complicated, but knowing the BNSR invariants of a group sheds light on the finiteness properties of coabelian subgroups of \(G\), that is normal subgroups \(N\) for which \(G/N\) is abelian. In general, \(\Sigma^m(G)\) is an open subset of the ``character sphere'' \(S(G)\). The present paper investigates the BNSR-invariants of the pure braid groups \(P_n\). This is much more complicated than the invariants of the full braid groups \(B_n\), because the abelianization of \(B_n\) is simply \(\mathbb{Z}\), whereas the abelianization of \(P_n\) is \(\mathbb{Z}^{\binom{n}{2}}\). Therefore, the BNSR invariant of \(B_n\) lives in \(S^0\) whereas the invariant of \(P_n\) is in the sphere of dimension \(\binom{n}{2} - 1\). As the author notes, even the BNSR invariant \(\Sigma^2(P_4)\) is not fully known. Morse-theoretic methods, introduced by \textit{M. Bestvina} and \textit{N. Brady} [Invent. Math. 129, No. 3, 445--470 (1997; Zbl 0888.20021)], are used to glean information on \(\Sigma^m(P_n)\). Among the applications are that for \(n \geq 4\), certain subgroups of \(P_n\) are finitely-generated but not finitely presentable.