Consider the matrices in $\textit{PSL}_2(\mathbb{R})$ given by $T= \left(\begin{matrix} 1 & 1\\ 0 & 1 \end{matrix}\right)$ and $S= \left(\begin{matrix} 4 & 0\\ 0 & 1/4 \end{matrix}\right)$. Then is the subgroup generated by $T$ and $S$ a Fuchsian group?
I would be happy with hints. I was trying to make some diagrams to understand how the translations and dilations are working, but couldn't arrive at anything concrete. Many thanks in advance.
Let us try to compute the isometry $S^{-n}TS^n$ for some $n\in\mathbb{N}$. Then doing the product of the matrices involved, we obtain $S^{-n}TS^n = \left(\begin{matrix} 1 & 4^{-2n} \\ 0 & 1 \end{matrix}\right)$. Now, as $n\to\infty$, this goes to the identity matrix. Thus the group generated by $T$ and $S$ is not discrete. Hence, it is $\textit{not}$ a Fuchsian group.