the stability group the action of Fuchsian group on the extended complex plane is cyclic?

Question

Let $\hat{\mathbb C}:=\mathbb C\cup \{\infty\} $ be the extended complex plane. I found the following proposition 2.2. from Iwaniec's Spectral methods in automorphic forms but I can't find a proof of it.

Any stability group $\Gamma_z:=\{\gamma\in \Gamma:\gamma.z=z\}$ the action of Fuchsian group on the extended complex plane is cyclic.

Can anyone show me a proof or a reference to a proof?

Source:

Answer 1

Everything is in your other questions.

• The action of $SL_2(\Bbb{R})$ on the upper half-plane is transitive, $z=g.i$, $Stab(z) = g Stab(i) g^{-1}$

• $\frac{ai+b}{ci+d} = i $ implies $ci+d = -a+ib$ thus $Stab(i) = SO_2(\Bbb{R})$ which is isomorphic to $\Bbb{R/Z}$.

• If $G$ is discrete in $SL_2(\Bbb{R})$ then $G \cap SO_2(\Bbb{R})$ is discrete in $SO_2(\Bbb{R})$, an infinite subgroup of $\Bbb{R/Z}$ has some elements arbitrary close $0$ thus such a subgroup is dense, thus discete subgroup of $\Bbb{R/Z}$ must be finite and cyclic.

• When you add $\infty$ you get a new kind of stabilizer $Stab(\infty) = \pmatrix{a &b\\ 0 & a^{-1}}$, you need to make clear the topology you put on it

Answer 2

My go-to reference for such questions is

A.F.Beardon, "Geometry of Discrete Groups", Springer Verlag, 1983.

See Theorem 8.2.1 in his book: If $\Gamma$ is a Fuchsian group then the $\Gamma$-stabilizer $\Gamma_z$ of any point $z\in \hat{\mathbb C}$ is cyclic.

The key to the proof is that all (nontrivial) elements of $\Gamma_z$ are of the same type (either all are elliptic: when $z$ is not on the invariant circle) or all are parabolic or all are hyperbolic (when $z$ is a limit point).