Totally geodesic subspaces and arithemeticity phenomena in hyperbolic manifolds

Abstract: In this talk I will survey a well known, still wonderful, connection between geometry and arithmetics and discuss old and new results in this topic. The starting point of the story is Cartan's discovery of the correspondence between semisimple Lie groups and symmetric spaces. Borel and Harish-Chandra, following Siegel, later realized a fantastic further relation between arithmetic subgroups of semisimple Lie groups and locally symmetric space - every arithemtic group gives a locally symmetric space of finite volume. The best known example is the modular curve which is associated in this way with the group SL_2(Z). This relation has a partial converse, going under the name "arithmeticity theorem", which was proven, under a higher rank assumption, by Margulis and in some rank one situations by Corlette and Gromov-Schoen. The rank one setting is related to hyperbolic geometry - real, complex, quaternionic or octanionic. There are several open questions regarding arithmeticity of locally hyperbolic manifolds of finite volume over the real or complex fields and there are empirical evidences relating these questions to the geometry of totally geodesic submanifolds. Recently, some of these questions were solved by Margulis-Mohammadi (real hyp. 3-dim), Baldi-Ullmo (complex hyp.) and B-Fisher-Miller-Stover. The techniques involve a mixture of ergodic theory, algebraic groups theory and hodge theory. After surveying the above story, explaining all the terms and discussing some open questions, I hope to have a little time to say something about the proofs.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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