The z-classes of quaternionic hyperbolic isometries

Let H-H(n) be the n-dimensional quaternionic hyperbolic space. The group Sp(n, 1) acts as the isometry group of H-H(n). We analyze when two isometries of H-H(n) commute. We apply this analysis to determine the conjugacy classes of centralizers or the z-classes in Sp(n, 1). Furthermore, we count the conjugacy classes of centralizers. In Appendix A, we show that our methods can be used to obtain the centralizers up to conjugacy in real and complex hyperbolic geometries as well. This provides a unified approach to the determination of the conjugacy classes of centralizers in hyperbolic geometries.