Maximal pro $p$-quotients of absolute Galois groups

Abstract: (Joint work with Claudio Quadrelli.)

It is well-known that the absolute Galois group $G_K = \operatorname{Gal}(\bar K^{\text{sep}}/K)$ of a field $K$ is a profinite group. However, only in very restrictive circumstances it is possible to analyze the structure of $G_K$ completely. A first approximation - which is untertaken frequently - is to investigate the maximal pro-$p$ quotient $G_K(p) = G_K/O^{p} (G_K)$ for a prime $p$. Here $O_p(\_)$ is the closed subgroup being generated by all Sylow pro-$\ell$ subgroups for $\ell \ne p$. The absolute Galois group $G_K$ comes equipped with a continuous group homomorphism $$ \theta_{K,p} : G_K \to \mathbb{Z}_p^{x} ,$$ the $p$-cyclotomic character, where $\mathbb{Z}_p^{x}$ denotes group of the invertible elements in the ring of $p$-adic integers $\mathbb{Z}_p$. In case that $K$ contains a primitive $p$-th root of unity, the homomorphism $\theta_{K,p}$ is induced from a group homomorphism $$ \hat\theta_{K,p} : G_K(p) \to \mathbb{Z}_p^{x} .$$

A pro-$p$ group $G$ together with a continuous group homomorphism $\theta : G → \mathbb{Z}_p^{x}$ is also called an oriented pro-$p$ group. Although the structure of $G_K(p)$ is in general much easier to analyze than $G_K$ there are still many open questions concerning the oriented pro-$p$ groups $(G_K(p), \hat\theta_{K,p})$. E.g., around 25 years ago it was conjectured by I. Efrat, that in case that $G_K(p)$ is a finitely generated pro-$p$ group, then $(G_K(p), \hat\theta_{K,p})$ must be of elementary type. Here one defines the class of oriented pro-$p$ groups of elementary type as the smallest class of oriented pro-$p$ groups which is closed under free products, and fibre products with $\theta$-abelian oriented pro-$p$ groups which contains $(F, \alpha)$ for all finitely generated free pro-$p$ groups $F$ and any $\alpha : F \to \mathbb{Z}_p^{x}$, as well as $(D, \eth)$ for all Demush’kin pro-$p$ groups $D$, where $\eth: D \to \mathbb{Z}_p^{x}$ is the $p$-orientation induced by the dualizing module of D. In the talk I will discuss recent developments in Field theory, which transformed I. Efrat’s elementary type conjecture into a purely group theoretic question. Recently, this question has been investigated successfully for certain classes of oriented pro-$p$ groups: 1) Right-angled Artin pro-$p$ groups with trivial orientation (I. Snopce, P. Zalesskii), 2) Generalized right-angled Artin pro-$p$ groups (S. Blumer, C. Quadrelli, T.W.).

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

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