I'm reading Serre's Trees recently. I'm trying to prove that
$\Gamma$ is a discrete torsion-free subgroup of $\mathrm{SL}_{2}(\mathbb{Q}_{p})$. If the quotient space $G/\Gamma$ is compact, then $S$ is finite, where $S$ is a set of double coset representatives for $\Gamma\backslash G/\mathrm{SL}_{2}(\mathbb{Z}_{p})$.
This is a part of the proof (page 83) of Ch.2 Thm.5 in Serre's Trees. At first, I was trying to prove that $\Gamma S$ is discrete in the quotient space. But I failed. Someone know how to prove it? Thanks for your attention.
The subgroup $K=SL_2(\mathbb Z_p)$ is open in $G=SL_2(\mathbb Q_p)$. The quotient $\Gamma\backslash G$ has the quotient topology, of course. Then compact $\Gamma\backslash G$ is covered by finitely many translates of $K$.