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. 

Let G be an infinite discrete group; e.g. free group, surface groups, or hyperbolic 3-manifold group.

Finite dimensional unitary representations of G of fixed dimension are usually very hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps. 

The talk is a broadly accessible discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas, R. van Handel.