Colloque - Géométrie et spectre des grands objets - Jean Raimbault : A Priori Bounds for the Homology of Arithmetic Manifolds

Nalini AnantharamanGéométrie spectraleCollège de FranceAnnée 2025-2026Colloque - Géométrie et spectre des grands objets - Jean Raimbault : A Priori Bounds for the Homology of Arithmetic ManifoldsJean RaimbaultCNRS, Institut de mathématiques de MarseilleIt is well-known that the Betti numbers of nonpositively-curved manifolds are (under normalisation of curvature and some additional assumptions) linearly bounded by their volume. In a joint work with M. Frączyk and S. Hurtado we showed that for the sub-class of arithmetic locally symmetric spaces similar bounds hold for torsion homology. In most cases we also obtain sublinear bounds for the Betti numbers on terms of the volume. The main tools for both results are geometric, and i will explain our main technical result, a stronger version of the Margulis lemma specific to arithmetic manifolds.