I am considering finitely generated subgroup $G=\langle A,B,C\rangle$ of $GL_2(\mathbb{Q})$ such that $A,B,C$ all have the upper triangular form $$A=\begin{pmatrix}a_1 & a_2 \\ 0 & a_3\end{pmatrix},\;B=\begin{pmatrix}b_1 & b_2 \\ 0 & b_3\end{pmatrix},\;C=\begin{pmatrix}c_1 & c_2 \\ 0 & c_3\end{pmatrix}.$$
Are there known conditions on the parameters $a_i,b_i,c_i$ such that $G$ is isomorphic to the free group $F_3$? Inversely, many conditions can be established, such that this is not the case, by simple matrix multiplication to obtain, e.g., $A^2BC^{-1}=Id_2$. But I do not manage do find the correct relations to make the group free.
I look in particular at the example $$A=\begin{pmatrix}4 & 1 \\ 0 & 1\end{pmatrix},\;B=\begin{pmatrix}2 & -1 \\ 0 & 3\end{pmatrix},\;C=\begin{pmatrix}4 & -1 \\ 0 & 3\end{pmatrix}.$$ So the question is in this case: Is the group generated by these three examples isomorphic to $F_3$.
