| Résume | t is well known that any negtaively curved manifold has
infinitely many closed geodesics.
In recent work with Bader, Miller and Stover we show that any finite
volume real or complex
hyperbolic manifold with infinitely many closed maximal totally geodesic
submanifolds of dimension 2
or higher is arithmetic. Just as geodesics come from orbits of a one
parameter subgroup giving
the geodesic flow, higher dimensional totally geodesic manifolds come
from orbits of larger
subgroups and the proof of this geometric sounding result is almost
entirely dynamical.
I will describe the results, give some history and motivation and try to
outline some of the
main ideas in the proofs. |
