The group $SL_2(\mathbb{Z})$ contains many free subgroups, for example all of the principal congruence subgroups for $n\geq 3$ and the subgroup $\left\langle \left(\begin{array}{cc} 1 & 2\\ 0 & 1 \end{array}\right),\left(\begin{array}{cc} 1 & 0\\ 2 & 1 \end{array}\right)\right\rangle $ which is almost the principal congruence subgroup for n=2, and more over it has a finite index in $SL_2(\mathbb{Z})$. In addition, when projecting $SL_2(\mathbb{Z})$ onto $SL_2(\mathbb{Z}/p)$, the restriction to these subgroups is also surjective as long as $p \nmid n$.

Is a similar phenomena true for $SL_2(\mathbb{Z}[i])$, or more generally when we substitute $\mathbb{Z}$ with the ring of integers $\mathcal{O}_k$ for some number field $k$?

EDIT: I look for free subgroups which are hopefully finite index (for example the congruence subgroups). Since finite index means they are a lattice, this enables me to count the number of lattice points in a ball of radius R asymptotically as R goes to infinity. If this is not possible, I would like to find free subgroups which are large in the sense that they have large growth.

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