Pure symmetric automorphisms, extensions of RAAGs, and Koszulness

TL;DR

The paper establishes a combinatorial criterion $(*)$ on the defining graph $\Gamma$ that exactly determines when the descending central Lie algebras $gr_\bullet(PAut(A_\Gamma))$ and $gr_\bullet(POut(A_\Gamma))$ are Koszul. It combines 1-formality/malcev theory with explicit Lie-algebra presentations and Day–Wade relative automorphism theory to show that, under $(*)$, these groups admit iterated extensions of RAAGs (i.e., are poly-RAAGs) and hence have Koszul graded Lie algebras via known RAAG/Koszul results. An obstruction is proved for $n \ge 4$ showing that $PAut(F_n)$ and $POut(F_n)$ are not poly-finitely generated free, using cohomological dimension, FP properties, Euler characteristics, and BNS-invariant arguments. The paper also develops a detailed presentation framework for Day–Wade factors, and proves exact sequences of graded Lie algebras that mirror the group decompositions, providing a coherent algebraic bridge between the graph combinatorics, formality, and Koszulness of automorphism groups of RAAGs. Together, these results illuminate how graph structure controls the homological and Lie-algebraic properties of automorphism groups in RAAGs, with implications for formality and Koszulness in this class of groups.

Abstract

We characterize in terms of a combinatorial condition on the graph $Γ$ when the group $\mathrm{PAut}(A_Γ)$ of pure symmetric automorphisms of the RAAG $A_Γ$ and its outer version $\mathrm{POut}(A_Γ)$ have a descending central Lie algebra which is Koszul. To do that, we prove that our combinatorial condition implies that these groups are iterated extensions of RAAGs; in particular, they are poly-free. On the other hand, we show that $\mathrm{PAut}(F_n)$ is not poly-finitely generated free for $n \geq 4$. We also show that groups in a certain class containing $\mathrm{PAut}(A_Γ)$ are 1-formal.

Theorems & Definitions (70)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• proof : Proof of Theorem \ref{['teoB']}