The scheme of characters in SL 2

TL;DR

This paper investigates the algebraic structure of the SL(2,C)-character scheme associated with finitely generated groups, providing explicit equations and analyzing special cases like abelian and nonsimple representations, with applications to the Borromean rings.

Contribution

It offers explicit equations for the SL(2,C)-character scheme of finitely generated groups and studies special representation schemes, including applications to topological links.

Findings

Derived equations defining the coordinate ring of the scheme.

Analyzed the scheme of abelian and nonsimple representations.

Applied results to the SL(2,C)-character scheme of the Borromean rings.

Abstract

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 $\Gamma$, we give equations defining the coordinate ring of the scheme of SL(2,C)-characters of $\Gamma$ (finitely many equations when $\Gamma$ 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.