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 $F_2 \times F_2.$

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