Seminar

If M is a closed hyperbolic three-manifold, then the hyperbolic structure on M is unique. However, M may admit a whole family of convex projective structures, which define a kind of Teichmuller space on M. Upcoming work of Ballas--Danciger--Lee--Marquis shows that, perhaps surprisingly, there are infinitely many closed hyperbolic three-manifolds for which this space of geometric structures fails to be connected; however, their methods are nonconstructive, and there is (as yet) no specific three-manifold known to have this property. I will go over some attempts to explicitly verify this property for specific three-manifolds by effectivizing a general "higher-rank Dehn filling" theorem. The majority of the talk concerns the practical construction of a "combinatorial coding" for the action of the figure-eight knot group on CP^1, which requires no higher-rank knowledge and has possible applications in the hyperbolic setting.