What is 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} :

More precisely, G {\displaystyle G} acts on Hom ⁡ {\displaystyle \operatorname {Hom} } by conjugation, and two homomorphisms are defined to be equivalent if and only if their orbit closures intersect. This is the weakest equivalence relation on the set of conjugation orbits, Hom ⁡ / G {\displaystyle \operatorname {Hom} /G} , that yields a Hausdorff space.