Let $G$ by a hyperbolic group, and let $H \lhd G$ be a normal quasiconvex subgroup. Is it possible that $|H| = [G : H] = \infty$ ?



Are there quasiconvex normal subgroups?
Greenberg's theorem for hyperbolic groups, proved by Kapovich and Short, asserts that such an $H$ is finite.


