Cartan decomposition for $\mathrm {Sp}(1,n)$ and some applications to group dynamics

Abstract In this article, we present an elementary proof of the Cartan decomposition theorem for the group $ \mathrm {Sp}(1, n) $ . As an application, we determine the largest regular domain for discrete quaternionic hyperbolic groups acting on $ \mathbb {HP}^n $ . Furthermore, we demonstrate that Bers’ simultaneous uniformization and Köebe’s retrosection theorem fail to hold for higher-dimensional quaternionic Kleinian groups.