The Galois action on geometric lattices and the mod-$\ell$ I/OM

TL;DR

This paper investigates the Galois action on a geometric lattice connected to mod-$\ell$ abelian-by-central quotients of fundamental groups, leading to a strengthened version of a conjecture by Ihara and Oda-Matsumoto.

Contribution

It introduces a new perspective on Galois actions on geometric lattices and proves a strengthened mod-$\ell$ abelian-by-central conjecture.

Findings

Established the Galois action on the lattice of geometric origin.

Proved a strengthened form of the Ihara/Oda-Matsumoto conjecture.

Connected the Galois action to mod-$\ell$ fundamental group quotients.

Abstract

This paper studies the Galois action on a special lattice of geometric origin, which is related to mod-$\ell$ abelian-by-central quotients of geometric fundamental groups of varieties. As a consequence, we formulate and prove the mod-$\ell$ abelian-by-central variant/strengthening of a conjecture due to Ihara/Oda-Matsumoto.