It is known that the congruence subgroup $\Gamma_p$ of $SL_2(\mathbb{Z})$, that is the kernel of the epimorphism $SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}_p)$ (with $p$ a prime number), is a free group.
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It is known that the congruence subgroup $\Gamma_p$ of $SL_2(\mathbb{Z})$, that is the kernel of the epimorphism $SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}_p)$ (with $p$ a prime number), is a free group.
Have you a reference for this result?
If $\Gamma_p$ is torsion free (which will be the case provided $p > 2$), then it acts freely and properly diconstinuously on the upper half-plane $H$, and so is identified with the fundamental group of the quotient $H/\Gamma_p$. But this quotient is a punctured Riemann surface, and hence its $\pi_1$ is free. Thus $\Gamma_p$ is free. (And it is not difficult to compute the number of generators, since this is just a matter of determining the genus and number of punctures of $H/\Gamma_p$.)