Let $f_1,\dots,f_n$ be maps from $\mathbb{R}$ to $\mathbb{R}$ of the form $f_i(x) := a_ix + b_i$ with $a_i,b_i \in \mathbb{Q}$. We construct the transformation group $G = \langle f_1, \dots, f_n \rangle$ via composition, which can also be realized as a subgroup of $\mathrm{AGL}_2(\mathbb{Q})$ where $f_i \mapsto \begin{bmatrix} a_i & b_i \\ 0 & 1 \end{bmatrix}$.
For most choices of $f_1,\dots,f_n$ with $n > 1$, $G$ has some relators, which is to say non-trivial compositions which reduce to the identity. How can these relators be found in general? I've seen that the freeness of $G$ is at least decidable since $G$ is virtually free, but no explicit description of computational methods.