Let $\Gamma_1,\Gamma_2\leq SL_2(R)$ be two Fuchsian subgroups. Take any $S_1\in\Gamma_1,S_2\in\Gamma_2$. Clearly $<S_i>\leq\Gamma_i$ and they are discrete.
Now consider $G=<S_1,S_2>\leq SL_2(R)$.
Q1: How do I show this group $G$ is discrete? Is this even a Fuchsian subgroup? I hope it is but I do not know how to prove it.
Q2: How do I know $G$ has no elliptic elements?
The purpose is to ask if $\Gamma_i$ are fuchsian subgroups, then $<\Gamma_1,\Gamma_2>$ is fuchsian as well. This is implied by pants gluing procedure.