Symmetric automorphisms of free groups, BNSR-invariants, and finiteness properties

The BNSR-invariants of a group $G$ are a sequence $\Sigma^1(G)\supseteq \Sigma^2(G) \supseteq \cdots$ 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 $\Sigma^{n-2}(P\Sigma Aut_n)\setminus \Sigma^{n-1}(P\Sigma Aut_n)$. We use this to prove that for $n\ge 2$, $\Sigma^{n-2}(\Sigma Aut_n)$ equals the full character sphere $S^0$ of $\Sigma Aut_n$ but $\Sigma^{n-1}(\Sigma Aut_n)$ is empty, so in particular the commutator subgroup $\Sigma Aut_n'$ 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.