Dehn filling in semisimple Lie groups

Thurston's Hyperbolic Dehn Filling Theorem is a seminal result
in the theory of 3-manifolds. Given a single noncompact finite-volume
hyperbolic 3-manifold M, the theorem provides a construction for a
countably infinite family of closed hyperbolic 3-manifolds converging to
M in a geometric sense. The theorem is a major source of examples of
3-manifolds admitting hyperbolic structures, and closely connects the
topology of a 3-manifold to the analysis of the PSL(2,C) character
variety of its fundamental group. In this talk, I will discuss some
analogs and generalizations of Thurston's theorem in the context of
general (arbitrary-rank) semisimple Lie groups. There are potential
applications in higher-rank and in rank one; time permitting, I will try
to discuss some upcoming joint work with Jeff Danciger, giving
applications towards convex cocompact and geometrically finite complex
hyperbolic manifolds.