Lectures on Shimura Curves 3: More Fuchsian Groups

Proof: Writing Y (Γ) = Γ\H, it is natural to ask (as we did in the “parabolic” case C/Λ) which elements σ ∈ PSL2(R) = Aut(H) descend to give automorphisms of Y (Γ). The condition we need is clearly that for all z ∈ H and γ ∈ Γ, there exists γ′ ∈ Γ such that σ(γz) = γ′(σz). Of course this holds for all z ⇐⇒ σγ = γ′σ; in other words the condition is that σΓσ−1 = Γ, i.e., that σ ∈ N(Γ). This defines a natural action of N(Γ) on Y (Γ) by automorphisms, and it is easy to see that its kernel is Γ itself. Now, if Γ is of hyperbolic type, then H → Γ\H is the universal covering map, and by covering space theory, all automorphisms of Y (Γ) lift to the universal cover.