- 09 Feb 2022
Totally geodesic immersions of hyperbolic manifolds
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 call the subspaces arising from this construction “finite centralise subspaces (or fc-subspaces)” and use them to provide an arithmeticity criterion in terms of their finiteness/non-finiteness. In the case of arithmetic hyperbolic manifolds we will show how to characterise all totally geodesic immersions through the analysis of certain algebraic invariants: the adjoint trace field (which is an algebraic number field) and the ambient group (an algebraic group defined over the adjoint trace field).