TITLE: (un)decidability in matrix groups
QUESTION [14 upvotes]: Given a collection of matrices $S=\{M_1, \dots, M_k\}$ in (say) $SL(n, Z), \  n>2$ does $S$ generate $SL(n, Z)?$
Similar are questions are undecidable for $n\geq 4$ (eg, given a set $S$ as above, is a given matrix in the subgroup generated by $S$) but I cannot find any reference for the above.
For $n=2$ all questions of this sort are reasonably efficiently decidable.
EDIT (in response to @Misha's interesting comments).
It is not clear that Mihailova tells you that the generation problem is undecidable. I believe that it IS a result of Baumslag, Miller, Short that this is undecidable for some word-hyperbolic groups (see MR1246477 (94i:20053) 
Baumslag, G.(1-CCNY); Miller, C. F., III(5-MELB); Short, H.(1-CCNY)
Unsolvable problems about small cancellation and word hyperbolic groups. (English summary) 
Bull. London Math. Soc. 26 (1994), no. 1, 97–101. 
20F10 (20F06) ) [they use the Rips construction @Misha alludes to].
For $n\geq 4,$ there are the non-free Zariski-dense examples of Margulis-Soifer (1979). I haven't read their paper in detail, but it seems that their technique does not work in $SL(3, \mathbb{Z}).$ However, there is the nice result of Stephen Wang:
Wang, Stephen(1-HAV)
Representations of surface groups and right-angled Artin groups in higher rank. (English summary) 
Algebr. Geom. Topol. 7 (2007), 1099–1117. 
20F36 (20F65 57M25) 
Which can presumably be generalized to other RAAGs.
Geometric finiteness: I think the action of $SL(n)$ on the positive semidefinite cone was studied first by Minkowski (for $SL(2)$ the PSD cone is just the light cone in the usual Minkowski space), and I had actually implemented this. The program usually run forever.
AND ALSO Mihailova's counterexample for generalized word problem (AKA membership problem) uses SEVEN generators. Undoubtedly she had tried to get it down lower, but apparently failed. It turns out that in many applications, we have two generators, in which case it seems that even the generalized word problem is open even for $F_2 \times F_2.$

REPLY [3 votes]: I do not know direct answer to your problem, but I will think about this
as it is close to the area of problems which we are currently working on.
Although I would like to comment that not all questions are "reasonably efficiently decidable" for SL(2,Z) as most of known to me problems for 
SL(2,Z) are at least NP-hard, meaning that none of these problems have efficient (polynomial time) solutions unless P=NP.
The membership for Identity Matrix is NP-hard for a finitely generated matrix semigroup from SL(2,Z) and there is no even simple NP brute-force algorithm as there is a class of semigroups where the shortest identity could require exponential length of products to reach the identity. 
See: Paul C. Bell, Igor Potapov: On the Computational Complexity of Matrix Semigroup Problems. Fundam. Inform. 116(1-4): 1-13 (2012)
Similar results exist for Mortality problem (membership of the Zero matrix) Paul C. Bell, Mika Hirvensalo, Igor Potapov: Mortality for 2×2 Matrices Is NP-Hard. MFCS 2012: 148-159
Recently we proved that the Vector Reachability for SL(2, Z) is decidable (in general it is harder to check than membership as there is an infinite set of matrices that can connect two points and decidability of the membership is not lead to vector reachability problem decidability).
Potapov, Pavel Semukhin: Vector Reachability Problem in SL(2, Z). MFCS 2016: 84:1-84:14 http://drops.dagstuhl.de/opus/volltexte/2016/6492/
The other fresh 2016 result is the proof that the membership for nonsingular matrix semigroup from $Z^{2x2}$ is decidable.
http://arxiv.org/abs/1604.02303
In relation to the membership in SL(2,Z) we recently managed to improve 
the complexity for the membership problem from PSPACE to NP, the main idea of the paper that we are able to operate effectively with compressed versions of matrices from SL(2,Z) by application of number of new results about structural properties of matrix products. 

For undecidability side the membership problem for a semigroup generated by matrices from SL(2,H) or even for a semigroup of double quaternions is undecidable http://www.sciencedirect.com/science/article/pii/S0890540108000771
Moreover the membership of the identity matrix for semigroup generated by matrices from SL(4,Z) is undecidable:
Paul C. Bell, Igor Potapov: On the Undecidability of the Identity Correspondence Problem and its Applications for Word and Matrix 
Semigroups. Int. J. Found. Comput. Sci. 21(6): 963-978 (2010) 
Archive vestion: arxiv.org/abs/0902.1975
See Problem 10.3 http://press.princeton.edu/math/blondel/solutions.html
but the same problem for 3x3 is sill open.