Elliptic ramification of maps between Fuchsian quotients #
For an inclusion Δ ≤ Γ of discrete subgroups of PSL(2, ℝ), the compactified
quotient map has the same elliptic power-map expression as the coarse quotient map.
The local model is the cyclic disc quotient of Farkas--Kra, Riemann Surfaces, Chapter I, §§4--5.
theorem
Subgroup.compactifiedQuotientMap_ellipticChart
{Δ Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)}
(h : Δ ≤ Γ)
(z : UpperHalfPlane)
[DiscreteTopology ↥Γ]
{ε : ℝ}
(hε : 0 < ε)
(hΔ : Topology.IsOpenEmbedding (Δ.stabilizerBallQuotientToQuotient z ε))
(hΓ : Topology.IsOpenEmbedding (Γ.stabilizerBallQuotientToQuotient z ε))
{u : ℂ}
(_hu : u ∈ (stabilizerBallQuotientChart hε hΔ).target)
:
The compactified quotient map has the same power expression at an elliptic orbit, using the charts of the coarse quotient transported into the compactification.
theorem
Subgroup.CompactifiedQuotient.ofQuotientChart_compactifiedQuotientMap_eq_pow_ellipticRamificationIndex
{Δ Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)}
(h : Δ ≤ Γ)
(z : UpperHalfPlane)
[DiscreteTopology ↥Γ]
{ε : ℝ}
(hε : 0 < ε)
(hopenΔ : Topology.IsOpenEmbedding (Δ.stabilizerBallQuotientToQuotient z ε))
(hopenΓ : Topology.IsOpenEmbedding (Γ.stabilizerBallQuotientToQuotient z ε))
{q : MulAction.orbitRel.Quotient (↥Δ) UpperHalfPlane}
:
have this := ⋯;
q ∈ (stabilizerBallQuotientChart hε hopenΔ).source →
↑(ofQuotientChart (stabilizerBallQuotientChart hε hopenΓ)) (compactifiedQuotientMap h (ofQuotient q)) = ↑(ofQuotientChart (stabilizerBallQuotientChart hε hopenΔ)) (ofQuotient q) ^ ellipticRamificationIndex h z
The compactified quotient map has elliptic local expression u ↦ u ^ e in the
transported charts at points of the uncompactified quotient.