Arithmetic Orr Invariants of Absolute Galois Groups

Abstract

AbstractBased on the analogies between mapping class groups and absolute Galois groups, we introduce an arithmetic pro-$\ell $ analogue of Orr invariants for a Galois element associated with Galois action on étale fundamental groups of punctured projective lines. At the same time, we also introduce pro-$\ell $ Orr space as an arithmetic analogue of Orr space whose third homotopy group is a target group of Orr invariant. We then determine its rank as $\mathbb {Z}_{\ell }$-module following Igusa–Orr’s computation. Moreover, we investigate its relation with Ellenberg’s obstruction to $\pi _1$-sections associated with lower central series filtration in the context of Grothendieck’s section conjecture.