Cohomology of the pure symmetric automorphisms of right-angled Artin groups
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910134505046016 |
|---|---|
| author | Galé, Peio Ardaiz |
| author_facet | Galé, Peio Ardaiz |
| contents | 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 $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13749 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cohomology of the pure symmetric automorphisms of right-angled Artin groups Galé, Peio Ardaiz Group Theory 20J06, 20F36 (Primary) 57M07, 55P20 (Secondary) 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 $2$. |
| title | Cohomology of the pure symmetric automorphisms of right-angled Artin groups |
| topic | Group Theory 20J06, 20F36 (Primary) 57M07, 55P20 (Secondary) |
| url | https://arxiv.org/abs/2604.13749 |