Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups

TL;DR

This paper develops a homology theory for subspace arrangements to derive new invariants from the Bieri-Neumann-Strebel invariant, characterizing when certain automorphism groups of right-angled Artin groups are themselves right-angled Artin groups.

Contribution

It introduces a novel homology framework for subspace arrangements and applies it to analyze automorphism groups of right-angled Artin groups, providing new characterizations.

Findings

New homology invariants for subspace arrangements

Characterization of pure symmetric outer automorphisms as right-angled Artin groups

Enhanced understanding of automorphism group structures

Abstract

We introduce a homology theory for subspace arrangements, and use it to extract a new system of numerical invariants from the Bieri-Neumann-Strebel invariant of a group. We use these to characterize when the set of basis conjugating outer automorphisms (a.k.a. the pure symmetric outer automorphism group) of a right-angled Artin group is itself a right-angled Artin group.