The McCullough-Miller complex for right angled Artin groups

TL;DR

This paper generalizes the McCullough-Miller complex to pure symmetric automorphism groups of arbitrary right-angled Artin groups, enabling new computations and insights into their algebraic and topological properties.

Contribution

The authors extend the McCullough-Miller complex construction from free groups to all RAAGs, providing a new tool for studying their automorphism groups.

Findings

Constructed a generalized complex for arbitrary RAAGs.

Applied the complex to compute cohomological properties of automorphism groups.

Demonstrated the complex's utility in various algebraic and topological analyses.

Abstract

McCullough and Miller constructed a contractible complex on which the pure symmetric automorphism group of a free group acts with free abelian stabilizers. This complex has been used for computations such as the cohomological dimension of these groups, their cohomology rings, and results about $\ell^2$-Betti numbers or BNRS-invariants. We generalize this construction to pure symmetric automorphism groups of arbitrary RAAGs and exhibit applications of this generalization.