Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent to the identity mod $N$). Let $\overline{\Gamma(N)}$ denote its closure in $\widehat{SL(2,\mathbb{Z})}$.


Can we describe generators for $\bigcap\_{N\ge 1}\overline{\Gamma(N)}$? (At first I thought this intersection is trivial, but since $SL(2,\mathbb{Z})$ has noncongruence subgroups, the congruence subgroups do not form a fundamental system of neighborhoods of the identity in $\widehat{SL(2,\mathbb{Z})}$, so now I'm rather uncertain...)


What about $\bigcap\_{N\ge 1}\overline{\Gamma\_1(N)}$?


($\Gamma\_1(N)$ is the subgroup consisting of matrices which mod $N$ are upper triangular unipotent).



what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?
It is a [result of Melnikov](http://www.ams.org/mathscinet-getitem?mr=466341) that the congruence kernel $ker\{ \widehat{SL\_2(\mathbb{Z})}\to SL\_2(\hat{\mathbb{Z}})\} \cong \hat{F}\_\omega$, the free profinite group on a countable number of generators. 


