Chen ranks and resonance varieties of the upper McCool groups

TL;DR

This paper studies the algebraic invariants of the upper McCool groups, revealing differences from pure braid groups and showing that certain conjectures do not hold for these groups, with implications for their structure.

Contribution

The paper provides a new presentation for the infinitesimal Alexander invariant of upper McCool groups and analyzes their resonance varieties and Chen ranks, showing deviations from known conjectures.

Findings

Chen ranks conjecture fails for $P ext{Σ}_n^+$ when $n ext{≥}4$

Resonance schemes are not reduced for $n ext{≥}4$

$P ext{Σ}_n^+$ is not isomorphic to pure braid groups for $n ext{≥}4$

Abstract

The group of basis-conjugating automorphisms of the free group of rank $n$, also known as the McCool group or the welded braid group $P\Sigma_n$, contains a much-studied subgroup, called the upper McCool group $P\Sigma_n^+$. Starting from the cohomology ring of $P\Sigma_n^+$, we find, by means of a Gr\"obner basis computation, a simple presentation for the infinitesimal Alexander invariant of this group, from which we determine the resonance varieties and the Chen ranks of the upper McCool groups. These computations reveal that, unlike for the pure braid group $P_n$ and the full McCool group $P\Sigma_n$, the Chen ranks conjecture does not hold for $P\Sigma_n^+$, for any $n\ge 4$. Consequently, $P\Sigma_n^+$ is not isomorphic to $P_n$ in that range, thus answering a question of Cohen, Pakianathan, Vershinin, and Wu. We also determine the scheme structure of the resonance varieties $\mathcal{R}_1(P\Sigma_n^+)$, and show that these schemes are not reduced for $n\geq 4$.