The BNSR-invariants of a group are a sequence of geometric invariants that reveal important information about finiteness properties of certain subgroups of . We consider the symmetric automorphism group and pure symmetric automorphism group of the free group and inspect their BNSR-invariants. We prove that for , all the “positive” and “negative” character classes of lie in . We use this to prove that for , equals the full character sphere of but is empty, so in particular the commutator subgroup is of type but not . Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.
"Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties." Michigan Math. J. 67 (1) 133 - 158, March 2018. https://doi.org/10.1307/mmj/1516330971