Contact topology and hyperbolic geometry are two well-established, yet so far largely unrelated subfields of 3-manifold topology. We will discuss a recent result relating phenomena in these two fields. Specifically, we will demonstrate that tightness of certain contact structures on hyperbolic manifolds is detected by the behaviour of geodesics in the underlying hyperbolic geometry. A key geometric tool we will discuss is the deformation theory for hyperbolic manifolds. 

I will construct a fully-faithful functor from the category of co-admissible D-cap modules of Ardakov—Wadsley, to the category of quasi-coherent sheaves on the "analytic de Rham space”, at least in the case when the rigid variety is affinoid and étale over a polydisk. 

to follow