arXiv subjects

Peio Ardaiz Galé

Publications and source records attributed to Peio Ardaiz Galé.

3 recordsLinked to original sources

arXiv2026

The McCullough-Miller complex for right angled Artin groups

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.

math.GR↗

arXiv2026

Cohomology of the pure symmetric automorphisms of right-angled Artin groups

We compute the cohomology groups of the pure symmetric outer automorphism group $Σ$POut$(A_Γ)$ and the pure symmetric automorphism group $Σ$PAut$(A_Γ)$ of a right-angled Artin group $A_Γ$. Using the equivariant spectral sequence arising from the action of $Σ$POut$(A_Γ)$ on the generalized McCullough-Miller complex MM$_Γ$, we show that $H^q(Σ$POut$(A_Γ))$ is free abelian and we compute its rank in terms of the combinatorics of certain poset. Applying the Lyndon-Hochschild-Serre spectral sequence and the Leray-Hirsch theorem we do the same for $H^q(Σ$PAut$(A_Γ))$. In both cases the cohomology ring is generated in degree 1. Finally, we introduce the Generalized Brownstein-Lee Conjecture, proposing a presentation of $H^*(Σ$PAut$(A_Γ))$, and prove that it holds in dimension $2$.

math.GR↗

arXiv2026

Actions with a strong fundamental domain and Sigma invariants of groups

We show that if a group $G$ acts on a contractible $CW$-complex with a contractible strong fundamental domain and the stabilizers of the action satisfy certain properties, then the homological Sigma invariants of the group are either empty or dense in the character sphere. This was known for several families of groups like right-angled Artin groups and pure symmetric automorphisms of free groups. We also exhibit some applications including the generalization of the previous fact to pure symmetric automorphisms of right-angled Artin groups.

math.GR↗