Arithmetic Orr invariants of absolute Galois groups

Based on the analogies between mapping class groups and absolute Galois groups, we introduce an arithmetic pro-$\mathrm{\ell }$ analogue of Orr invariants for a Galois element associated with Galois action on 'etale fundamental groups of punctured projective lines. At the same time, we also introduce pro-$\mathrm{\ell }$ 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 ${\mathbb{Z}}_{\mathrm{\ell }}$-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.