Character variety
In the mathematics of moduli theory, given an algebraic, reductive, Lie group G {\displaystyle G} and a finitely generated group π {\displaystyle \pi } , the G {\displaystyle G} -character variety of π {\displaystyle \pi } is a space of equivalence classes of group homomorphisms from π {\displaystyle \pi } to G {\displaystyle G} : R ( π , G ) = Hom ( π , G ) / ∼ . {\displaystyle {\mathfrak {R}}(\pi ,G)=\operatorname {Hom} (\pi ,G)/\! \sim \,.} More precisely, G {\displaystyle G} acts on Hom ( π , G ) {\displaystyle \operatorname {Hom} (\pi ,G)} by conjugation, and two homomorphisms are defined to be equivalent (denoted ∼ {\displaystyle \sim } ) if and only if their orbit closures intersect.