On Bisectors in Quaternionic Hyperbolic Space

Abstract

In this paper, we study a problem related to geometry of bisectors in quaternionic hyperbolic geometry. We develop some of the basic theory of bisectors in quaternionic hyperbolic space ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {Q}}}}$. In particular, we show that quaternionic bisectors enjoy various decompositions by totally geodesic submanifolds of ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {Q}}}}$. In contrast to complex hyperbolic geometry, where bisectors admit only two types of decomposition (described by Mostow and Goldman), we show that in the quaternionic case geometry of bisectors is more rich. The main purpose of the paper is to describe an infinite family of different decompositions of bisectors in ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {Q}}}}$ by totally geodesic submanifolds of ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {Q}}}}$ isometric to complex hyperbolic space ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {C}}}}$ which we call the fan decompositions. Also, we derive a formula for the orthogonal projection onto totally geodesic submanifolds in ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {Q}}}}$ isometric to ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {C}}}}$. Using this, we introduce a new class of hypersurfaces in ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {Q}}}}$, which we call complex hyperbolic packs in ${{\textrm{H}}^{\textrm{n}}_{{\mathbb {Q}}}}$. We hope that the complex hyperbolic packs will be useful for constructing fundamental polyhedra for discrete groups of isometries of quaternionic hyperbolic space.