The stability group of the action of fuchsian group on the upper half plane of is finite?

Question

Let $\Gamma$ be a fuchsian group acting on the upper half plane $\mathbb H$, I wonder if for any $z\in \mathbb H$, the stability group of the action of fuchsian group on the upper half plane $$\Gamma_z:=\{\gamma\in \Gamma:\gamma.z=z\}$$ is necessarily finite.

I need this result to show that the subgroup of $\Gamma$ generated by an elliptic element is finite. If the above assertion is wrong I wish someone could show me how to prove this one. Thanks in advance!

Answer 1

Yes, this is true, here's a proof.

By definition, a Fuchsian group $\Gamma$ is a discrete subgroup of $\operatorname{PSL(2,\mathbb R)} = \operatorname{SL(2,\mathbb R)} / \pm \text{Id}$ which is the full group of orientation preserving isometries of $\mathbb H$ (where a matrix representing an element of $\operatorname{PSL(2,\mathbb R)}$ acts on $\mathbb H$ by a fractional linear transformation. The topology on the group $\operatorname{PSL(2,\mathbb R)}$ can be described either as the quotient of the matrix topology on $\operatorname{SL(2,\mathbb R)}$.

Consider $z \in \mathbb H$. I'll use your notation $\Gamma_z$ for the stabilizer subgroup of $z$ in $\Gamma$. I also need a notation $Stab(z)$ for the stabilizer group of $z$ in the full isometry group $\operatorname{PSL(2,\mathbb R)}$.

The two key facts needed are:

• $Stab(z)$ is a compact group, in fact it is isomorphic to the circle group $S^1$ under complex multiplication.
• $\Gamma_z$ is a discrete subgroup of $Stab(z)$, because the intersection of a discrete subgroup of $Stab(z)$ and an arbitrary subgroup $H < Stab(z)$ is a discrete subgroup of $H$.
• Every discrete subgroup of a compact group is finite, and easy exercise in topological groups.

So yes, $\Gamma_z$ is finite. Furthermore, $\Gamma_z$ is a finite subgroup of the circle group, which implies that $\Gamma_z$ is a finite cyclic group.