# Profinite Groups with a Cyclotomic $p$-Orientation

### Claudio Quadrelli

Department of Mathematics and Applications, Università di Milano-Bicocca, Via R. Cozzi 55 - ed. U5, 20125 Milan, Italy### Thomas S. Weigel

Department of Mathematics and Applications, Università di Milano-Bicocca, Via R. Cozzi 55 - ed. U5, 20125 Milan, Italy

## Abstract

Let $p$ be a prime. A continuous representation $\theta\colon G\to\text{GL}_1(\mathbb{Z}_p)$ of a profinite group $G$ is called a cyclotomic $p$-orientation if for all open subgroups $U\subseteq G$ and for all $k,n\geq1$ the natural maps $H^k(U,\mathbb{Z}_p(k)/p^n)\to H^k(U,\mathbb{Z}_p(k)/p)$ are surjective. Here $\mathbb{Z}_p(k)$ denotes the $\mathbb{Z}_p$-module of rank 1 with $U$-action induced by $\theta\vert_U^k$. By the Rost-Voevodsky theorem, the cyclotomic character of the absolute Galois group $G_{\mathbb{K}}$ of a field $\mathbb{K}$ is, indeed, a cyclotomic $p$-orientation of $G_{\mathbb{K}}$. We study profinite groups with a cyclotomic $p$-orientation. In particular, we show that cyclotomicity is preserved by several operations on profinite groups, and that Bloch-Kato pro-$p$ groups with a cyclotomic $p$-orientation satisfy a strong form of Tits' alternative and decompose as semi-direct product over a canonical abelian closed normal subgroup.

## Cite this article

Claudio Quadrelli, Thomas S. Weigel, Profinite Groups with a Cyclotomic $p$-Orientation. Doc. Math. 25 (2020), pp. 1881–1916

DOI 10.4171/DM/788