Given a finite set $S$ of $n\times n$ integer matrices, it is known that for $k\geq 3$ it is undecidable whether some product of them (allowing repetitions) is the zero matrix (called the mortality problem). It is also known (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.448&rep=rep1&type=pdf) that if $k=2$, then it is decidable whether some product of the matrices is the identity matrix. Is it known whether for $k$ sufficiently large, it is undecidable whether some product of matrices in $S$ is the identity matrix? This has some connection with a question raised by Kontsevich.
This is undecidable in dimension 4 or up see http://cgi.csc.liv.ac.uk/~igor/papers/matrixcomp.pdf
The result is proved in Bell, P. C., Potapov, I.: On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups, International Journal of Foundations of Computer Science, 21(6), 2010, 963–978.
The problem is at least NP-hard in dimension 2.
There is a new algorithm (in manuscript) that shows that Identity Problem for SL(2,Z) is in NP, meaning that it is also NP-complete.
Also point to point reachability for SL(2,Z) is also shown recently to be decidable, see proceeding of MFCS 2016 http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=6492
Which question was raised by Kontsevich and what is the connection with identity matrix? Could you please comment.