Character variety

In the mathematics of moduli theory, given an algebraic, reductive, Lie group and a finitely generated group , the -character variety of is a space of equivalence classes of group homomorphisms from to : More precisely, acts on by conjugation, and two homomorphisms are defined to be equivalent (denoted ) if and only if their orbit closures intersect. This is the weakest equivalence relation on the set of conjugation orbits, , that yields a Hausdorff space. Formally, and when the reductive group is defined over the complex numbers , the -character variety is the spectrum of prime ideals of the ring of invariants (i.e., the affine GIT quotient). Here more generally one can consider algebraically closed fields of prime characteristic. In this generality, character varieties are only algebraic sets and are not actual varieties. To avoid technical issues, one often considers the associated reduced space by dividing by the radical of 0 (eliminating nilpotents). However, this does not necessarily yield an irreducible space either. Moreover, if we replace the complex group by a real group we may not even get an algebraic set. In particular, a maximal compact subgroup generally gives a semi-algebraic set. On the other hand, whenever is free we always get an honest variety; it is singular however. An interesting class of examples arise from Riemann surfaces: if is a Riemann surface then the -character variety of , or Betti moduli space, is the character variety of the surface group For example, if and is the Riemann sphere punctured three times, so is free of rank two, then Henri G. Vogt, Robert Fricke, and Felix Klein proved that the character variety is ; its coordinate ring is isomorphic to the complex polynomial ring in 3 variables, . Restricting to gives a closed real three-dimensional ball (semi-algebraic, but not algebraic). Another example, also studied by Vogt and Fricke–Klein is the case with and is the Riemann sphere punctured four times, so is free of rank three.