Separation in the BNSR-invariants of the pure braid groups

Abstract

We inspect the BNSR-invariants Σm(Pn) of the pure braid groups Pn, using Morse theory. The BNS-invariants Σ1 (Pn) were previously computed by Koban, McCammond, and Meier. We prove that for any 3≤m≤n, the inclusion Σm−2 (Pn) ⊆ Σm−3 (Pn) is proper, but Σ∞(Pn) = Σn−2 (Pn). We write down explicit character classes in each relevant Σm−3 (Pn)\Σm−2 (Pn). In particular we get examples of normalsubgroups N ≤ Pn with Pn/N ∼= Z such that N is of type Fm−3 but not Fm−2, for all 3 ≤ m ≤ n.