Quotients of non-classical flag domains are not algebraic

A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.