New sequences of non-free rational points

Abstract

We exhibit some new infinite families of rational values of $\tau$, some of them squares of rationals, for which the group or even the semigroup generated by the matrices $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ and $\left(\begin{array}{cc}1& 0\\ \tau & 1\end{array}\right)$ is not free.

Fixed a typo: I had written "free" instead of "non-free" in a few places; and made a few other minor edits