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

Abstract

The BNSR-invariants of a group $G$ are a sequence ${\Sigma }^1\left(G\right)\supseteq {\Sigma }^2\left(G\right)\supseteq \cdots$ of geometric invariants that reveal important information about finiteness properties of certain subgroups of $G$. We consider the symmetric automorphism group ${\Sigma Aut}_n$ and pure symmetric automorphism group ${P\Sigma Aut}_n$ of the free group $F_n$ and inspect their BNSR-invariants. We prove that for $n\ge 2$, all the “positive” and “negative” character classes of ${P\Sigma Aut}_n$ lie in ${\Sigma }^{n-2}\left({P\Sigma Aut}_n\right)\setminus {\Sigma }^{n-1}\left({P\Sigma Aut}_n\right)$. We use this to prove that for $n\ge 2$, ${\Sigma }^{n-2}\left({\Sigma Aut}_n\right)$ equals the full character sphere $S^0$ of ${\Sigma Aut}_n$ but ${\Sigma }^{n-1}\left({\Sigma Aut}_n\right)$ is empty, so in particular the commutator subgroup ${\Sigma Aut}_n\text{'}$ is of type $F_{n-2}$ but not $F_{n-1}$. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.