Nikolay Bogachev (IHES, IITP RAS): From geometry to arithmetic of hyperbolic manifolds

Nikolay Bogachev (IHES; IITP RAS)

IHES; IITP RAS

Exposé  réel 

 Lieu : Salle Visio, M3

 Résumé : In a recent joint paper with Misha Belolipetsky, Sasha Kolpakov and Leone Slavich we developed a large industry connecting geometry and arithmetic of hyperbolic orbifolds and manifolds. We introduce a new class of the so-called finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic orbifolds: a hyperbolic orbifold M is arithmetic if and only if it has infinitely many fc-subspaces. We also showed that immersed totally geodesic m-dimensional suborbifolds of n-dimensional arithmetic hyperbolic orbifolds are fc-subspaces whenever m ????\ge \floor{n/2}, and we provide examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. One of the key results of our paper is a characterization of immersions of arithmetic hyperbolic orbifolds into each other. In the first part of the talk I will give an overview of this area, and the second part will be devoted to our methods.