Deformations of discrete subgroups of Lie groups: rigidity vs. flexibility

Given a discrete subgroup Gamma of a Lie group G, can one deform Gamma inside G? The motivation for this question is geometric, in terms of deforming a geometric structure on a certain manifold/orbifold. Examples include Euclidean structures on tori, hyperbolic structures on surfaces and higher-dimensional manifolds as well as more exotic structures. We will discuss these examples and review the main results on this question in the past 60 years, including the Weil and Mostow rigidity theorems, Thurston's Hyperbolic Dehn Surgery theorem and eventually some more recent results.