Homogeneous number of free generators

We address two questions of Simon Thomas. First, we show that for any n >= 3 one can find a four-generated free subgroup of SLn (Z) which is profinitely dense. More generally, we show that an arithmetic group Gamma that admits the congruence subgroup property has a profinitely-dense free subgroup with an explicit bound on its rank. Next, we show that the set of profinitely-dense, locally-free subgroups of such an arithmetic group Gamma is uncountable.