  We study the identity problem for matrices, i.e., whether the identity matrix
is in a semigroup generated by a given set of generators. In particular we
consider the identity problem for the special linear group following recent
NP-completeness result for ${\rm SL}(2,\mathbb{Z})$ and the undecidability for
${\rm SL}(4,\mathbb{Z})$ generated by $48$ matrices. First we show that there
is no embedding from pairs of words into $3\times3$ integer matrices with
determinant one, i.e., into ${\rm SL}(3,\mathbb{Z})$ extending previously known
result that there is no embedding into $\mathbb{C}^{2\times 2}$. Apart from
theoretical importance of the result it can be seen as a strong evidence that
the computational problems in \slthreez are decidable. The result excludes the
most natural possibility of encoding the Post correspondence problem into ${\rm
SL}(3,\mathbb{Z})$, where the matrix products extended by the right
multiplication correspond to the Turing machine simulation. Then we show that
the identity problem is decidable in polynomial time for an important subgroup
of ${\rm SL}(3,\mathbb{Z})$, the Heisenberg group ${\rm H}(3,\mathbb{Z})$.
Furthermore, we extend the decidability result for ${\rm H}(n,\mathbb{Q})$ in
any dimension $n$. Finally we are tightening the gap on decidability question
for this long standing open problem by improving the undecidability result for
the identity problem in ${\rm SL}(4,\mathbb{Z})$ substantially reducing the
bound on the size of the generator set from $48$ to $8$ by developing a novel
reduction technique.
