SL(3,C)-Character Varieties and RP2-Structures on a Trinion

Abstract

Denote the free group on two letters by F2 and the SL(3,C)-representation variety of F2 by R = Hom(F2, SL(3, C)). There is a SL(3,C)-action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. We determine explicit minimal generators and defining relations for the subring of invariants and show X is a hyper-surface in C9. Our choice of generators exhibit Out(F2) symmetries which allow for a succinct expression of the defining relations. We then show C[X] is a Poisson algebra with respect to a presentation of F2 imposed by a punctured surface. We work out the bracket on all generators when the surface is a thrice punctured sphere, or a trinion. The moduli space of convex real projective structures on a trinion, denoted by P,is a subset of X. Lastly, we determine explicit conditions in terms of C[X] that distinguish this moduli space.