MR0548436 (81k:20064) Reviewed
Margulis, G. A.; Soĭfer, G. A.
Nonfree maximal subgroups of infinite index of the group SLn(Z). (Russian)
Uspekhi Mat. Nauk 34 (1979), no. 4(208), 203–204.
20H05 (22E40)
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In previous work [Dokl. Akad. Nauk SSSR 234 (1977), no. 6, 1261–1264; MR0466412 (57 #6292)], the authors showed among other things that for n≥3, SL(n,Z) has a maximal subgroup of infinite index. Independently, V. P. Platonov and G. Prasad asked whether such a subgroup must possess a free subgroup of finite index. Here the authors provide a negative answer: at least when n≥4, SL(n,Z) has a maximal subgroup of infinite index containing a free abelian group of rank 2. Besides their own previous techniques, the authors make use of the methods of J. Tits [J. Algebra 20 (1972), 250–270; MR0286898 (44 #4105)].
{English translation: Russian Math. Surveys 34 (1979), no. 4, 178–179.}
Reviewed by James E. Humphreys

**EDIT** to address Khalid's and Yves' comments: First, it is a result of yours truly that a RANDOM (two-generator) subgroup of a Zariski-dense subgroup is Zariski dense, and result of R. Aoun that such a random subgroup is free (he proves it in slightly less generality, but the result is true, essentially by ping-pong combined with the work of Guivarc'h and Goldsheid. Note that free is NOT what Khalid wants. However, if you read Margulis-Soifer (not the paper I cited, but the one in Journal of Algebra, which actually contains complete proofs), they construct a finitely generated Zariski-dense subgroup which contains a $\mathbb{Z} \oplus \mathbb{Z}.$ They then take the maximal subgroup containing it, but you don't care about maximality, so don't do that part. Unfortunately, the Margulis-Soifer example is constructed only for $n\geq 4,$ not $n=3.$

However, progress marched on, and Venkataramana (who, I believe is a frequent contributor here), in this paper:

@article {MR892908,
AUTHOR = {Venkataramana, T. N.},
TITLE = {Zariski dense subgroups of arithmetic groups},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {108},
YEAR = {1987},
NUMBER = {2},
PAGES = {325--339},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {20G30 (20H05 22E40)},
MRNUMBER = {892908 (88i:20068)},
MRREVIEWER = {Alexander Lubotzky},
DOI = {10.1016/0021-8693(87)90106-2},
URL = {http://dx.doi.org/10.1016/0021-8693(87)90106-2},
}

provided examples of (finitely generated, infinite index) zariski-dense subgroups which contain unipotent subgroups, thus bearing out Yves' conjecture (more or less) about the Heisenberg group.