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Let $\Gamma$ be a discrete subgroup of $PSL(2, \mathbb{R}) = Aut(H) \simeq Aut(D)$.

I want to prove that

  1. $\Gamma$ acts on upper half plane $H$ / unit disk $D$ properly discontinuously.

  2. For all $p \in D$, Stabilizer subgroup of $\Gamma$ at p, $Stab_p \cap \Gamma$ is a finite cyclic group.

  3. Orbit space $D/\Gamma$ has a riemann surface structure that makes $\pi :D \to D/\Gamma$ a holomorphic map.

Proof of 2.

$0 \in D$ $Stab_0 = S^1 $. Since $\Gamma$ is discrete, $Stab_0 \cap \Gamma $ is discrete subgroup of $S^1$, hence a finite cyclic group. Since $Aut(D)$ acts transitively on $D$, $\forall p \in D, \exists g \in Aut(D) s.t. g(0) = p. $ Thus $Stab_p \cap \Gamma = g ( Stab_0 \cap g ^{-1} \Gamma g ) g ^{-1}$ is also a finite cyclic group.

Is there something wrong with my proof of 2? Also I am lost about what to do next for proof of 1 and 3. I would appreciate any reference or hints.

GouldBach
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  • For 3: https://math.stackexchange.com/questions/160170/riemann-surface-and-discontinuous-group-action?rq=1. For 1: https://math.stackexchange.com/questions/2183326/katoks-fuchsian-groups-properly-discontinuous-action-homeomorphisms-vs-isome?noredirect=1&lq=1 – Moishe Kohan Feb 05 '18 at 14:43

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