The scheme of characters in SL 2

The aim of this article is to study the SL(2,C)-character scheme of a finitely generated group. Given a presentation of a finitely generated group $\mathrm{\Gamma }$, we give equations defining the coordinate ring of the scheme of SL(2,C)-characters of $\mathrm{\Gamma }$ (finitely many equations when $\mathrm{\Gamma }$ is finitely presented). We also study the scheme of abelian and nonsimple representations and characters. Finally we apply our results to study the SL(2,C)-character scheme of the Borromean rings.