  We give a method for constructing dense and free subgroups in real Lie
groups. In particular we show that any dense subgroup of a connected semisimple
real Lie group G contains a free group on two generators which is still dense
in G, and that any finitely generated dense subgroup in a connected
non-solvable Lie group H contains a dense free subgroup of rank < 2 dim(H).
This answer a question of Carriere and Ghys, and it gives an elementary proof
of a conjecture of Connes and Sullivan on amenable actions, which was first
proved by Zimmer.
