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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.

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