Dense and empty BNSR-invariants of the McCool groups

TL;DR

This paper studies the BNSR-invariants of McCool groups, showing they are either dense or empty in the character sphere and providing precise characterizations for these cases.

Contribution

It characterizes the density and emptiness of BNSR-invariants in McCool groups and analyzes the second invariant using a new criterion.

Findings

BNSR-invariants are either dense or empty in the character sphere.

Precise conditions for when each case occurs are established.

Further properties of the second invariant  are investigated.

Abstract

An automorphism of the free group $F_n$ is called pure symmetric if it sends each generator to a conjugate of itself. The group $\mathrm{PSA}_n$ of all pure symmetric automorphisms and its quotient $\mathrm{PSO}_n$ by the group of inner automorphisms are called the McCool groups. In this paper we prove that every BNSR-invariant $\Sigma^m$ 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 $\Sigma^2$ for McCool groups using a general criterion due to Meinert for a character to lie in $\Sigma^2$.