Free group actions on symmetric spaces-hyperbolic geometry and Atiyah-Singer index theory

We present two important actions of free groups on hyperbolic 3-space and Hermitian symmetric space via Kleinian group theory

and surface group representations into Hermitian Lie groups. The first one is related to the Thurston's conjecture of generalization of the double limit 

theorem to 3-manifolds with compressible boundary, and the second one is related to the Atiyah-Patodi-Singer index theory on flat bundles over surfaces with boundary.