The scheme of characters in 𝑆𝐿₂

Abstract

The aim of this article is to study the SL 2 ⁡ ( C ) \operatorname {SL}_2(\mathbb {C}) –character scheme of a finitely generated group. Given a presentation of a finitely generated group Γ \Gamma , we give equations defining the coordinate ring of the scheme of SL 2 ⁡ ( C ) \operatorname {SL}_2(\mathbb {C}) –characters of Γ \Gamma (finitely many equations when Γ \Gamma is finitely presented). We also study the scheme of abelian and non-simple representations and characters. Finally we apply our results to study the SL 2 ⁡ ( C ) \operatorname {SL}_2(\mathbb {C}) –character scheme of the Borromean rings.