Quaternionic decorations of hyperbolic 3-manifolds

By Mostow-Prasad rigidity, the complete hyperbolic structure on an oriented finite-volume non-compact hyperbolic 3-manifold M, if any, is unique up to isometry. From M, we can construct an oriented complete 5-dimensional hyperbolic manifold M' with infinite volume, which is diffeomorphic to M × R² and into which M lies geodesically as the horizontal section M × {0}. Since M' has infinite volume, Mostow-Prasad rigidity no longer applies, raising a natural question: does the manifold M' admit new hyperbolic structures? We present a possible approach to this problem, based on Thurston's work on the hyperbolic Dehn filling theorem. The underlying idea is to equip the manifold M with an ideal triangulation T, realize the tetrahedra geometrically in hyperbolic space, and then glue them back together with isometries. In the three-dimensional case, this procedure admits an algebraic translation via complex shape parameters and consistency equations: the main focus of my thesis is the development of an analogous of Thurston's construction in dimension five. In our setting, hyperbolic 5-space is identified with H × R⁺, where H is the algebra of quaternions. Using this identification, I introduce the notion of quaternionic developing maps for (M, T), and show that every such map induces a labeling of the edges of the triangulation with quaternionic isometry invariants. This naturally leads to the definition of a quaternionic decoration of T, which assigns quaternionic parameters to the oriented edges of the triangulation, subject to two families of conditions (the edge and face relations). The central result of my thesis establishes a canonical bijection between valid quaternionic decorations on T and equivalence classes of generic quaternionic developing maps for (M, T).