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Let $G$ be a semisimple algebraic group, and let $\Gamma$ be a finitely generated subgroup. Given the generators of $\Gamma$, is there a good way to determine if $\Gamma$ is Zariski-dense in $G$?

For example, let $\Gamma \subseteq \mathrm{PSL}_2$ where $ \displaystyle \Gamma = \left\langle \gamma_1 , \gamma_2 \right\rangle$ and $\gamma_1 = \begin{pmatrix} 1 & i \newline 0 & 1 \end{pmatrix}$ and $\gamma_2 = \begin{pmatrix} 1 & 0 \newline 1 & 1 \end{pmatrix}$.

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  1. What do you mean by "good"? In practice, a theorem of either Lubotzky or Weigel (depending on who you ask) states that if a congruence quotient is surjective for SOME prime (bigger than 3, say), then the subgroup is Zariski-dense, so generally checking for a couple of primes gives a certificate.

  2. In $PSL_2,$ Zariski-dense is equivalent to non-elementary, so your group is Zariski-dense.

EDIT To answer the OP's comment: in this paper: Zariski density and genericity I Rivin - International Mathematics Research Notices, 2010, I show that two "random" matrices generate a Zariski dense subgroup. R. Aoun in this paper: Transience of algebraic varieties in linear groups and application to generic Zariski density R Aoun - Arxiv preprint arXiv:1103.0944, 2011 - arxiv.org

Generalized it (at the cost of losing the effectiveness). Aoun also showed (in a different paper) that a random subgroup is free (which, in higher rank) means that it is not a lattice. In rank one a corresponding result (not yet written up fully) was shown by Elena Fuchs, myself, and Peter Sarnak: there you show that generically the Hausdorff dimension of the limit set is small, so again you don't get a lattice generically (of course lattices are Zariski dense by the Borel Density Theorem).

You might also look at these slides:

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  • $\begingroup$ By "good," I meant that I was looking for some criterion for Zariski-density based on the generators of the group. I was looking for a way to create examples of Zariski-dense subgroups that are a little more interesting than $\mathrm{SL}_2(\mathbb{Z})$ in $\mathrm{SL}_2$ and most importantly, be able to prove that these examples are Zariski-dense. Thank you very much for your answer; it sounds like it's what I was looking for. $\endgroup$ – JS_UNCUVa Feb 23 '12 at 20:56
  • $\begingroup$ See the edit for more info... $\endgroup$ – Igor Rivin Feb 24 '12 at 2:56

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