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- Title
Oriented Right-Angled Artin Pro-ℓ Groups and Maximal Pro-ℓ Galois Groups.
- Authors
Blumer, Simone; Quadrelli, Claudio; Weigel, Thomas S
- Abstract
For a prime number |$\ell $| , we introduce and study oriented right-angled Artin pro- |$\ell $| groups |$G_{\Gamma ,\lambda }$| (oriented pro- |$\ell $| RAAGs for short) associated to a finite oriented graph |$\Gamma $| and a continuous group homomorphism |$\lambda \colon{\mathbb{Z}}_{\ell }\to{\mathbb{Z}}_{\ell }^{\times }$|. We show that an oriented pro- |$\ell $| RAAG |$G_{\Gamma ,\lambda }$| is a Bloch–Kato pro- |$\ell $| group if, and only if, |$(G_{\Gamma ,\lambda },\theta _{\Gamma ,\lambda })$| is an oriented pro- |$\ell $| group of elementary type, generalizing a recent result of I. Snopce and P. Zalesskiĭ—here |$\theta _{\Gamma ,\lambda }\colon G_{\Gamma ,\lambda }\to{\mathbb{Z}}_{\ell}^{\times }$| denotes the canonical |$\ell $| -orientation on |$G_{\Gamma ,\lambda }$|. This yields a plethora of new examples of pro- |$\ell $| groups that are not maximal pro- |$\ell $| Galois groups. We invest some effort in order to show that oriented right-angled Artin pro- |$\ell $| groups share many properties with right-angled Artin pro- |$\ell $| -groups or even discrete RAAG's, for example, if |$\Gamma $| is a specially oriented chordal graph, then |$G_{\Gamma ,\lambda }$| is coherent generalizing a result of C. Droms. Moreover, in this case, |$(G_{\Gamma ,\lambda },\theta _{\Gamma ,\lambda })$| has the Positselski–Bogomolov property generalizing a result of H. Servatius, C. Droms, and B. Servatius for discrete RAAG's. If |$\Gamma $| is a specially oriented chordal graph and |$\operatorname{Im}(\lambda)\subseteq 1+4{\mathbb{Z}}_{2}$| in case that |$\ell =2$| , then |$H^{\bullet }(G_{\Gamma ,\lambda },{\mathbb{F}}_{\ell }) \simeq \Lambda ^{\bullet }(\ddot{\Gamma }^{\textrm{op}})$| generalizing a well-known result of M. Salvetti (cf. [ 39 ]). Dedicated to the memory of Avinoam Mann.
- Subjects
PRIME numbers; DIRECTED graphs; CONTINUOUS groups; HOMOMORPHISMS
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 8, p6790
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad276