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Dense and empty BNSR-invariants of the McCool groups

Mikhail Ershov, Matthew C. B. Zaremsky

TL;DR

The paper determines that the higher BNSR-invariants Σ^m for McCool groups PSAut_n and PSOut_n are either dense or empty on their character spheres, with precise thresholds: Σ^{n-2}(PSAut_n) is dense while Σ^{n-1}(PSAut_n) is empty, and for PSOut_n, Σ^{n-3} is dense while Σ^{n-2} is empty (for n ≥ appropriate bounds). The authors apply the Meier–Meinert–Van Wyk density criterion together with the Whitehead poset and McCullough–Miller space to establish density via ∞-generation by abelian subgroups. They also develop tools for Σ^2, including a central criterion due to Meinert and a RAAG-based approach for chordal graphs, proving a concrete sufficient condition for a character to lie in Σ^2(PSAut_n) and showing that generic characters lie in Σ^2 in large n. Together, these results yield detailed finiteness-property consequences for kernels of generic characters and connect McCool groups to RAAG techniques, with potential extensions to related automorphism groups.

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 $Σ^m$ 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 $Σ^2$ for McCool groups using a general criterion due to Meinert for a character to lie in $Σ^2$.

Dense and empty BNSR-invariants of the McCool groups

TL;DR

The paper determines that the higher BNSR-invariants Σ^m for McCool groups PSAut_n and PSOut_n are either dense or empty on their character spheres, with precise thresholds: Σ^{n-2}(PSAut_n) is dense while Σ^{n-1}(PSAut_n) is empty, and for PSOut_n, Σ^{n-3} is dense while Σ^{n-2} is empty (for n ≥ appropriate bounds). The authors apply the Meier–Meinert–Van Wyk density criterion together with the Whitehead poset and McCullough–Miller space to establish density via ∞-generation by abelian subgroups. They also develop tools for Σ^2, including a central criterion due to Meinert and a RAAG-based approach for chordal graphs, proving a concrete sufficient condition for a character to lie in Σ^2(PSAut_n) and showing that generic characters lie in Σ^2 in large n. Together, these results yield detailed finiteness-property consequences for kernels of generic characters and connect McCool groups to RAAG techniques, with potential extensions to related automorphism groups.

Abstract

An automorphism of the free group is called pure symmetric if it sends each generator to a conjugate of itself. The group of all pure symmetric automorphisms and its quotient by the group of inner automorphisms are called the McCool groups. In this paper we prove that every BNSR-invariant 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 for McCool groups using a general criterion due to Meinert for a character to lie in .

Paper Structure

This paper contains 14 sections, 23 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. BNSR-invariants
  4. McCool groups
  5. Setting the stage for BNSR-invariants of McCool groups
  6. Empty BNSR-invariants
  7. Dense BNSR-invariants
  8. Higher generation and the Meier--Meinert--Van Wyk criterion
  9. The Whitehead poset
  10. A family of abelian subgroups
  11. McCullough--Miller space
  12. Proof of density
  13. More on $\Sigma^2$ for McCool groups
  14. A proof of Meinert's $\Sigma^2$-criterion

Key Result

Theorem 1.1

For all $n\ge 2$, the following hold: For all $n\ge 3$, the following hold: Hence, $\Sigma^k(\mathrm{PSAut}_n)$ is dense for all $0\leq k\le n-2$ and empty for all $k\ge n-1$, and $\Sigma^k(\mathrm{PSOut}_n)$ is dense for all $0\leq k\le n-3$ and empty for all $k\ge n-2$.

Figures (2)

  • Figure 1: Bipartite labeled trees on $[4]$, with the second tree obtained from the first tree by a folding at the vertex labeled $3$.
  • Figure 2: An example of a split/merge, and a drop/lift.

Theorems & Definitions (61)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: BNSR-invariants
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4: Symmetric
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • ...and 51 more