Restrictions of representations of a surface group to a pair of free subgroups (Q1972033)

The fundamental group \(\Pi\) of a compact orientable genus \(m\) surface has a standard presentation of the form \[ \Pi=\bigl \langle a_1,b_1, \dots,a_m, b_m\mid [a_1,b_1] \cdots [a_m,b_m]= 1\bigr\rangle. \] In the paper under review the author studies representations of the group \(\Pi\) and its special free subgroups \(A=\langle a_1,\dots, a_m\rangle\) and \(B=\langle b_1, \dots, b_m\rangle\) of rank \(m\) in a connected reductive algebraic group \(G\) over an algebraically closed field \(k\) of characteristic zero. Let \(C(F,G)\) denote the variety of closed conjugacy classes of representations of a group \(F\) in \(G\). Then the product of the restrictions of the representations associated with the two subgroups \(A\) and \(B\) of the fundamental group \(\Pi\) gives rise to an induced morphism \(r_{A,B}: C(\Pi,G)\to C(A,G) \times C(B,G)\). The main theorem of the paper states that if the genus \(m\) of the surface exceeds the semisimple rank \(s\) of \(G\), then this morphism is dominant and almost all fibres are finite; precisely, if \(m\geq s+1\) then there exists a non-empty open subset \(U\) of \(C(A,G)\times C(B,G)\) such that all fibres of the morphism \(r_{A,B}\) over the set \(U\) are finite and non-empty. From this theorem some related results concerning the associated varieties of representations are also derived. Finally, as an example, the case \(G=GL_4(k)\) is studied in detail.