Abstract: One of the key themes in Grothendieck's "Esquisse d'un Programme" is the idea of studying absolute Galois groups via their action on objects of geometric origin. Motivated by this, the conjecture of Ihara/Oda-Matsumoto (I/OM) predicts that the Galois action on geometric fundamental groups of algebraic varieties leads to a combinatorial/topological description of absolute Galois groups. This conjecture was resolved by Pop in the 90's, and several variants/generalizations have since been formulated. In this talk, I will discuss the recent proof of the mod-ell abelian-by-central variant of this conjecture, which follows from studying the Galois action on certain lattices of geometric origin.