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 $\mathrm{\Sigma }Au{t}_n$ and pure symmetric automorphism group $P\mathrm{\Sigma }Au{t}_n$ 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 $P\mathrm{\Sigma }Au{t}_n$ lie in $\Sigma^{n-2}(P\Sigma Aut_n)\setminus \Sigma^{n-1}(P\Sigma Aut_n)$\mathrm{.}Weusethistoprovethatfor$n\ge 2$,$\Sigma^{n-2}(\Sigma Aut_n)$equalsthefullcharactersphere$S^0$of$\Sigma Aut_n$but$\Sigma^{n-1}(\Sigma Aut_n)$ is empty, so in particular the commutator subgroup $\mathrm{\Sigma }Au{t}_n^{\mathrm{\prime }}$ 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.