A MORDELL–LANG-TYPE PROBLEM FOR GL₌

Abstract We prove a nonabelian variant of the classical Mordell–Lang conjecture in the context of finite- dimensional central simple algebras. We obtain the following result as a particular case of a more general statement. Let K be an algebraically closed field of characteristic zero, let B₁, , Bᵣ GLₘ (K) be matrices with multiplicatively independent eigenvalues and let V be a closed subvariety of GLₘ (K) not passing through zero. Then there exist only finitely many elements of GLₘ (K) of the form B₁^n₁ Bᵣ^nᵣ (as we vary n₁, , nᵣ in Z) lying on the subvariety V.