Work

Dense and empty BNSR-invariants of the McCool groups

Year: 2026

Type: article

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 $\mathrm {PSAut}_n$ of all pure symmetric automorphisms and its quotient $\mathrm {PSOut}_n$ by the group of inner automorphisms are called the McCool groups . In this article, 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 $\Sigma ^2$ .

Source: Canadian Journal of Mathematics

DOI: 10.4153/s0008414x26102089

Authors: Mikhail Ershov, Matthew C. B. Zaremsky

Institutions: University of Virginia, University at Albany, State University of New York

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

OpenAlex ID: W4414581704