On the space of discrete linear groups.

I will discuss a work in progress with Emmaneul Breuillard. Let Hom(F_k, G) be the set of homomorphisms from a free group F_k into a Lie group G. We study the subset of this space consisting of homomorphisms with discrete image and prove several results generalizing classical results of Jorgensen in the case where G is SL(2,R) or SL(2,C). As additional motivation for this work, we use an idea of Lubotzky to produce an interesting new class of locally rigid subgroups of Lie groups.