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Separation in the BNSR-invariants of the pure braid groups. (English) Zbl 1397.20057

The BNSR invariants \(\Sigma^m(G)\) of a group \(G\), are defined using representations \(G \to \mathbb{R}\). They are a generalization, due to R. Bieri and B. Renz [Comment. Math. Helv. 63, No. 3, 464–497 (1988; Zbl 0654.20029)], of the BNS invariant, defined by 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 M. Bestvina and 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.

MSC:

20F65 Geometric group theory
57M07 Topological methods in group theory
20F36 Braid groups; Artin groups
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