Motivation: I have a family of curves obtained from a single curve by repeatedly applying two automorphisms of the surface (Dehn twists to be specific). I am interested in the images of these curves under a covering map. In particular, I want to know which curves are null-homologous. The automorphisms give rise to automorphisms $B,C$ of the homology group, while the covering gives rise to a (non-injective) homomorphism $A$ from one homology group to the other. I represent all of these as matrices and let $x$ be the homology class of the starting curve.
Let $A$ be a fixed $m\times n$ matrix, $B,C$ fixed invertible $n\times n$ matrices and $x$ a fixed vector. Is there any way to determine all integers $i,j$ such that $AB^iC^jx=0$? If $B,C$ were diagonalizeable then it would at least be possible to place bounds on $i,j$ by converting the matrix exponentiation to exponentiation on the diagonal. Also if we had some $m\times m$ matrix $B'$ such that $B'A=AB$, or similarly for $C$, this would simplify matters. Sadly neither matrix is diagonal or has such a $B'$ or $C'$. Is there a way to solve this problem in general?