Triangular Groups Of Quaternion Hyperbolic Spaces

Hyperbolic geometry keeps closely relation with complex dynamic system, hyperbolic manifold, etc. Real hyperbolic geometry, complex hyperbolic geometry, quaternion hyperbolic geometry are generalizations of hyperbolic geometry. Generalizing the definition of complex hyperbolic triangle groups by analogy, we define the quaternionic hyperbolic triangle groups and discuss the propositions of them. The purpose of this paper is to study the discreteness of the quaternionic hyperbolic ideal triangle groups. Three vertices of the ideal triangle groups, which are the special triangle groups, are on the boundary of the quaternionic hyperbolic space. Through analyzing possible configurations of two quaternionic lines in quaternionic hyperbolic space, we parameterize this space of such groups by an angular invariant of triangles in H_H². Then we get the relationship of between the angular invariant and a （p₁,p₂,p₃）-triangle group. Applying the propositions of the Dirichlet polyhedra, we structure the fundamental domains of these groups in order to study the discreteness. Furthermore, we find that the pavement of the main theorem is also adjust to the complex hyperbolic triangle groups.There are six parts in this thesis.In Chapter 1, we will narrate some background and the research significance of our issue.In Chapter 2, we give some background knowledge including the concepts of the quater-nionic lines, the quaternionic reflections, the angular invariant and so on.In Chapter 3 of the paper, we study possible configurations of two quaternionic lines in H_H² by using the polar vectors of these lines.In Chapter 4, we introduce four proportions on the Dirichlet polyhedra in H_H². It is to prepare for the pavement of the sufficient condition of the discrete ideal triangle groups.Thirdly, we apply an algebraic characterization of isometries of H_C², proving the necessary condition of the discrete ideal triangle groups.Finally, we get the main results in the Chapter 6. It obtains a condition of the existence of a （p₁,p₂,p₃）-triangle group in H_C². Furthermore, we study the variation of the discreteness of the quaternionic hyperbolic ideal triangle groups as the change of Cartan angle.