Let $M_1$ and $M_2$ be two matrices as follows:
$$ M_1= \left( \begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 5 & 5 & 1 & 0 \\ 0 & -5 & -1 & 1\\ \end{array} \right)$$ and $$M_2= \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\\ \end{array} \right). $$
One may easily check that $(M_1M_2)^5=I, $ where $I$ is the identity matrix. A well known problem is the following:
Is the relation $(M_1M_2)^5=I $, the only relation between $M_1$ and $M_2$?
Now my question is the following:
Does the group generated by $M_1$ and $M_2$ contain a free subgroup?
Is it known?