The inverse problem for universal deformation rings and the special linear group

We present a solution to the inverse problem for universal deformation rings of group representations. Namely, we show that every complete noetherian local commutative ring $R$ with a finite residue field $k$ can be realized as the universal deformation ring of a continuous linear representation of a profinite group. More specifically, $R$ is the universal deformation ring of the natural representation of $\textup {SL}_n(R)$ in $\textup {SL}_n(k)$, provided that $n\geq 4$. We also check for which $R$ an analogous result is true in case $n=2$ and $n=3$.