Restrictions of representations of a surface group to a pair of free subgroups

Let Π be the fundamental group of a compact orientable genus m surface, and let G be a connected reductive algebraic group over an algebraically closed field of characteristic zero. Define two free rank m subgroups of Π by A=〈a1,...,am〉 and B=〈b1,...,bm〉, where Π=〈a1,...,am,b1,...,bm∏mj=1[aj,bj]〉 is the standard presentation of Π. We consider representations of Π, of A and of B into G. Restriction of representations induces a morphism from C(Π,G), the variety of closed conjugacy classes of representations of Π, to C(A,G)×C(B,G). We prove that if m is greater than the semisimple rank of G then this morphism is dominant and almost all fibers are finite. © 2000 Academic Press.