Leone Slavich (Università di Pavia)

The study of totally geodesic immersions between (complete, finite-volume) hyperbolic manifolds is a classical problem in the field of hyperbolic geometry. There are two main approaches to this problem which often interplay with each other:
1) Given a hyperbolic manifold N, determine the hyperbolic manifolds in which N can be immersed geodesically;2) Given a hyperbolic manifold, determine its totally geodesic immersed submanifolds.
We will show how it is possible to build totally geodesic immersed submanifolds in a hyperbolic manifold M using finite subgroups in the commensurator of M.
We will then focus on the class of arithmetic manifolds i.e. those whose fundamental groups is commensurable with the integral points of some k-form of Isom(H^n)=PO(n,1,R), for some real algebraic number field k. We will show how to characterise all totally geodesic immersions in this setting through the analysis of Vinberg's commensurability invariants: the adjoint trace field (which is an algebraic number field) and the ambient group (an algebraic group defined over the adjoint trace field). This is joint work with Mikhail Belolipetski, Nikolay Bogachev and Alexander Kolpakov.