Ihara’s theorem

Let $\mathrm{\Gamma }$ be a discrete, torsion-free subgroup of ${\mathrm{SL}}_2{\mathbb{Q}}_p$ (where ${\mathbb{Q}}_p$ is the field of $p$-adic numbers (http://planetmath.org/PAdicIntegers)). Then $\mathrm{\Gamma }$ is free.

Proof, or a sketch thereof.

There exists a $p+1$ regular tree $X$ on which ${\mathrm{SL}}_2{\mathbb{Q}}_p$ acts, with stabilizer ${\mathrm{SL}}_2{\mathbb{Z}}_p$ (here, ${\mathbb{Z}}_p$ denotes the ring of $p$-adic integers (http://planetmath.org/PAdicIntegers)). Since ${\mathbb{Z}}_p$ is compact in its profinite topology, so is ${\mathrm{SL}}_2{\mathbb{Z}}_p$. Thus, ${\mathrm{SL}}_2{\mathbb{Z}}_p\cap \mathrm{\Gamma }$ must be compact, discrete and torsion-free. Since compact and discrete implies finite, the only such group is trivial. Thus, $\mathrm{\Gamma }$ acts freely on $X$. Since groups acting freely on trees are free, $\mathrm{\Gamma }$ is free.∎