Character Variety - Variants

Variants

This is not necessarily the same construction as the Culler-Shalen character variety (generated by evaluations of traces), although when the they do agree since Procesi has shown that in this case the ring of invariants is in fact generated by only traces. Since trace functions are invariant by all inner automorphisms, the Culler-Shalen construction essentially assumes that we are acting by

on

even if

For instance, for a free group of rank 2 and the conjugation action is trivial and the -character variety is the torus

But the trace algebra is a strictly small subalgebra (there are less invariants). This provides an involutive action on the torus that needs to be accounted for to yield the Culler-Shalen character variety. The involution on this torus yields a 2-sphere. The point is that up to -conjugation all points are distinct, but the trace identifies elements with differing anti-diagonal elements (the involution).