Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces $\mathbb{C} H^n$, $n \geq 3$. For the quaternionic hyperbolic spaces $\mathbb{H} H^n$, $n \geq 3$, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.