Homogeneous number of free generators

First we will recall ideas, conjectures and results on free subgroups in linear groups. Then I will talk about our resent result with Menny Aka and Tsachik Gelander which gives an answer on two questions of Simon Thomas. Namely, we show that for any $n\geq 3$ one can find a four generated free subgroup of $SL_{n}(\mathbb{Z})$ which is profinitely dense. More generally, we show that an arithmetic group $\Gamma$ which admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group $\Gamma$ is uncountable.