TITLE: Computing the Zariski closure of a subgroup of SL(n,Z)
QUESTION [20 upvotes]: Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is a (real) algebraic group. The method should be computer assisted but rigorous. 
In the cases we are considering, $\Gamma$ will usually be Zariski dense in $SL(n,\mathbb{R})$,      so the algorithm we are looking for should be optimized for that case. Also we would like just to know the answer, so having the program run forever if $\Gamma$ is not Zariski dense in $SL(n,\mathbb{R})$ is fine for us: we will just analyze that example further. 
We can probably come up with some ad-hoc method for doing this, but I was wondering if anyone on MO has some interesting ideas or references.

REPLY [3 votes]: In fact, the last word on this is my recent paper; but the punchline is that the subgroup is Zariski-dense if and only if there are two non-commuting elements, such that the Galois group of the characteristic polynomial of one of them is the full symmetric group.