Orsay and Munster

A subgroup Gamma of a semisimple algebraic group G is called strongly dense if every subgroup of Gamma is either cyclic or Zariski-dense. I will describe a method for building strongly dense free subgroups inside a given Zariski-dense subgroup  Gamma of G, thus providing a refinement of the Tits alternative. The method works for a large class of G's and Gamma's. I will also discuss connections with word maps and expander graphs. This is joint work with Bob Guralnick and Michael Larsen.