Assume $\Gamma$ is a discrete subgroup of some $GL_n$, and let $G$ be its Zariski closure. Let $H$ an algebraic cocompact normal subgroup of $G$. Do we have that $H\cap \Gamma$ is of finite index in $\Gamma$. I'm not sure this is true (actually i think it should be false), i'm far from being an expert in this field, and i don't have any couterexample yet.

Many thanks