Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties

Abstract

The BNSR-invariants of a group G are a sequence Σ1(G)⊇Σ2(G)⊇⋯ of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAutn and pure symmetric automorphism group PΣAutn of the free group Fn and inspect their BNSR-invariants. We prove that for n≥2, all the “positive” and “negative” character classes of PΣAutn lie in Σn−2(PΣAutn)∖Σn−1(PΣAutn). We use this to prove that for n≥2, Σn−2(ΣAutn) equals the full character sphere S0 of ΣAutn but Σn−1(ΣAutn) is empty, so in particular the commutator subgroup ΣAutn' is of type Fn−2 but not Fn−1. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.