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

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