On bisectors in quaternionic hyperbolic space

TL;DR

This paper explores the geometry of bisectors in quaternionic hyperbolic space, revealing richer decompositions than in complex hyperbolic geometry and introducing new structures called complex hyperbolic packs.

Contribution

It develops the theory of bisectors in quaternionic hyperbolic space, introduces fan decompositions, and defines complex hyperbolic packs, expanding understanding of quaternionic hyperbolic geometry.

Findings

Quaternionic bisectors have multiple decompositions by totally geodesic submanifolds.

Introduces fan decompositions of bisectors by complex hyperbolic subspaces.

Defines complex hyperbolic packs for potential use in constructing fundamental polyhedra.

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 $H^n_Q$. 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.