Relative Untwisted Outer Space for Right-Angled Artin Groups
Adrien Abgrall
TL;DR
This work extends the geometric understanding of automorphism groups of right-angled Artin groups by constructing contractible subcomplexes of the spine $K_\Gamma$ that realize untwisted McCool subgroups $\mathcal{U}(A_\Gamma;\mathcal{G},\mathcal{H}^t)$ as groups acting properly and cocompactly, thereby establishing type $VF$. The authors develop a relative untwisted outer space framework, introduce minset and cospecial-action techniques on CAT(0) cube complexes, and prove contractibility via a relative peak-reduction strategy reminiscent of Bestvina–Feighn–Handel. They also connect these geometric models to corollaries about the untwisted Aut spine $L_\Gamma$, and provide a general program to handle finitely generated fixed subgroups, with concrete finite-index reductions to cyclic cases and a path toward a full McCool generalization. The results yield finite presentability and other homological consequences for these stabilizers, and open avenues for generalized stabilizers in $\mathsf{Out}(A_\Gamma)$. The framework significantly broadens the toolkit for analyzing automorphism groups of RAAGs through explicit, finite-dimensional geometric models.
Abstract
For G, H two finite collections of finitely generated subgroups of a right-angled Artin group A, the untwisted McCool group U(A; G, Ht) is the subgroup of untwisted outer automorphisms of A preserving the conjugacy class of each element of G and acting trivially up to conjugacy on each element of H. We prove that when the elements of G are standard subgroups of A, U(A; G, Ht) acts properly cocompactly on a finite-dimensional subcomplex of the spine of untwisted outer space for A, providing a geometric model for this group and proof that it is of type VF.