  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 $H^n_Q$. In particular, we show that
quaternionic bisectors enjoy various decompositions by totally geodesic
submanifolds of $H^n_Q$. 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 $H^n_Q$ by totally geodesic
submanifolds of $H^n_Q$ isometric to complex hyperbolic space $H^n_C$ which we
call the fan decompositions. Also, we derive a formula for the orthogonal
projection onto totally geodesic submanifolds in $H^n_Q$ isometric to $H^n_C$.
Using this, we introduce a new class of hypersurfaces in $H^n_Q$, which we call
complex hyperbolic packs in $H^n_Q$. We hope that the complex hyperbolic packs
will be useful for constructing fundamental polyhedra for discrete groups of
isometries of quaternionic hyperbolic space.
