Arithmetic Orr invariants of absolute Galois groups

Based 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 $π_1$-sections associated with lower central series filtration in the context of Grothendieck's section conjecture.

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