A complete hyperbolic n-manifold geodesically embeds if it realized a totally geodesic embedded hypersurface in an (n+1)-hyperbolic manifold. On one hand, we know no general obstruction to the fact that a hyperbolic manifold of dimension n>2 embeds, but showing explicitly that it does is often difficult.  Moreover, showing that a hyperbolic manifold embeds is sometimes a useful tool to construct hyperbolic manifolds with prescribed topological properties.

 We will survey some results on embedding of hyperbolic manifolds which are based on arithmetic techniques and hold in all dimensions. These apply to many examples of arithmetic manifolds of simplest type, as well as to examples of non-arithmetic manifolds such as those constructed by Gromov and Piatetski-Shapiro. This proves that the growth rate, with respect to volume V, of commensurabilty classes of hyperbolic manifolds having a geodesically embedding representative of volume <V is super-exponential.

This is joint work with Alexander Kolpakov and Stefano Riolo.