In 2009, Kahn and Markovic proved the Surface Subgroup Theorem, and they constructed a large family of \pi_1-injective immersed surfaces in closed hyperbolic 3-manifolds. Hamenstadt later showed that any cocompact lattice of some simple rank 1 Lie group other than SO(2m,1) (m >= 1) has a surface subgroup. Recently, we constructed \pi_1-injective immersed surfaces in closed hyperbolic 2n-manifolds, which addresses the cases missing from Hamenstadt's work. This is joint work with Jeremy Kahn.
