Deformations of hyperbolic 3-manifolds in the 5-dimensional hyperbolic space: a combinatorial approach featuring quaternions

By Mostow-Prasad rigidity, finite-volume hyperbolic 3-manifolds admit a unique complete hyperbolic structure. However, one can try to deform this structure in higher-dimensional spaces, in the following sense: every manifold M as above naturally induces a 5-dimensional hyperbolic manifold M0 , with infinite volume, which is diffeomorphic to M × R 2 and into which M lies geodesically as the horizontal section M × {0}. Does the new manifold M0 admit new hyperbolic structures? In this seminar, I will present an approach to this deformation problem, based on the work of Thurston on the hyperbolic Dehn filling theorem. The main idea is to equip the manifold M with an ideal triangulation, realize the tetrahedra geometrically in the 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 gluing equations. In our setting, the hyperbolic 5-space is identified with H × R +, where H is the algebra of quaternions. I will show how this leads to quaternionic shape parameters and consistency equations, which can be solved explicitly for the figure-eight knot complement.