Separation in the BNSR-invariants of the pure braid groups

2015/07/30 by Matthew C. B. Zaremsky, Zaremsky, Matthew C. B. · 1 citation
Mathematics · #20F36 #20F63 #57M07 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F36 #msc:20F63 #msc:57M07

paper · pdf · doi:10.48550/arxiv.1507.08597

21 pages, 3 figures

arxiv created 2015/07/30 · arxiv updated 2015/07/31

Abstract

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

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