Dense and empty BNSR-invariants of the McCool groups

An automorphism of the free group Fₙ is called pure symmetric if it sends each generator to a conjugate of itself. The group PSAutₙ of all pure symmetric automorphisms, and its quotient PSOutₙ by the group of inner automorphisms, are called the McCool groups. In this paper we prove that every BNSR-invariant Σᵐ of a McCool group is either dense or empty in the character sphere, and we characterize precisely when each situation occurs. Our techniques involve understanding higher generation properties of abelian subgroups of McCool groups, coming from the McCullough-Miller space. We also investigate further properties of the second invariant Σ² for McCool groups, using a general criterion due to Meinert for a character to lie in Σ².