Let $ \Gamma $ be a subgroup of $ G:=GL(n,\mathbb{C}) $ which is discrete with respect to the manifold topology on $ G $. Let $ \overline{\Gamma} $ be the Zariski closure of $ \Gamma $. Suppose that $ \Gamma $ is infinite. Can we conclude that $ \overline{\Gamma} $ is a positive dimensional Lie group?
Moreover, is there any way to determine the dimension of the manifold $ \overline{\Gamma} $ based on information about $ \Gamma $? For example is $ dim( \overline{\Gamma}) $ always greater than the free rank of the abelianization of $ \Gamma $?
It seems that What do the closures of cyclic groups in $\textrm{GL}_n$ look like? already shows that the above conjecture about free rank is false. Maybe $ dim( \overline{\Gamma}) $ is bounded above by $ n $ times the free rank of the abelianization of $ \Gamma $?