Ramification of the orbit projection of a Fuchsian group #
Let Γ ≤ PSL(2, ℝ) act properly discontinuously on the upper half-plane, as every discrete
subgroup does, so that the coarse orbit quotient Γ \ ℍ is a Riemann surface and the orbit
projection ℍ → Γ \ ℍ is holomorphic. This file computes the ramification index of that
projection: its local multiplicity at z is the order m of the stabilizer of z
(Subgroup.localMultiplicity_quotientMk). So the projection is unramified exactly at the points
of the free locus, and ramifies exactly at the elliptic points, where its local model is the
cyclic quotient map u ↦ u ^ m.
The ramification formula for descent follows: a holomorphic map F on Γ \ ℍ and its pullback
to the upper half-plane satisfy
localMultiplicity (F ∘ π) z = m * localMultiplicity F (π z)
(Subgroup.localMultiplicity_comp_quotientMk). Read from right to left, this computes the local
multiplicity of an invariant holomorphic map upstairs from that of its unique descent
(Subgroup.existsUnique_mdifferentiable_quotientMk) downstairs. Both sides are local
multiplicities: localMultiplicity F q is the vanishing order of the chart representative of F
recentred at F q, not the order of vanishing of F itself, so the formula says nothing on its
own about the zeros of F. When F is nonconstant near the orbit of z these two multiplicities
are the ramification indices of F and of F ∘ π; when F is constant there both sides vanish.
Main declarations #
Subgroup.localMultiplicity_quotientMk: the local multiplicity of the orbit projection atzis the order of the stabilizer ofz.Subgroup.localMultiplicity_quotientMk_eq_one_iffandSubgroup.one_lt_localMultiplicity_quotientMk_iff: the projection is unramified exactly at a point with trivial stabilizer and ramified exactly at an elliptic point.Subgroup.exists_injOn_nhds_quotientMk_iff_stabilizer_eq_bot: the projection is locally injective, hence a local homeomorphism, exactly at such a point.Subgroup.localMultiplicity_comp_quotientMk: the local multiplicity of a holomorphic map onΓ \ ℍis multiplied by the stabilizer order when it is pulled back to the upper half-plane.
References #
- Hershel Farkas and Irwin Kra, Riemann Surfaces, Graduate Texts in Mathematics 71, Springer, second edition, 1992, Chapter I §§4–5.
- Rick Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics 5, American Mathematical Society, 1995, Chapter III §§3–4.
- Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992, §2.4.
The orbit projection of a Fuchsian group has ramification index the stabilizer order. Its
local multiplicity at z is the order of the stabilizer of z.
The orbit projection has positive local multiplicity: it is nowhere locally constant.
The orbit projection is not constant on any neighbourhood of a point.
The ramification index of the orbit projection depends only on the orbit.
The orbit projection is unramified exactly on the free locus: its local multiplicity at
z is one exactly when the stabilizer of z is trivial, that is, when z lies in
TauCeti.freeLocus Γ ℍ.
The orbit projection is locally injective exactly on the free locus. Near an elliptic point every neighbourhood contains a pair of distinct points of one stabilizer orbit, so the projection is not a local homeomorphism, hence not a covering map, there.
The orbit projection ramifies exactly at the elliptic points: its local multiplicity at
z exceeds one exactly when the stabilizer of z is nontrivial.
The ramification formula for descent. Pulling a holomorphic map on the coarse quotient back
to the upper half-plane multiplies its local multiplicity by the order of the stabilizer. Applied
to the unique descent of an invariant holomorphic map
(Subgroup.existsUnique_mdifferentiable_quotientMk), this computes the local multiplicity of the
map upstairs from that of its descent downstairs. Recall that the local multiplicity of F at q
is the vanishing order of the chart representative of F recentred at F q, so this is a
statement about local multiplicities, not about the zeros of F; they are the ramification indices
of F and of F ∘ π when F is nonconstant near the orbit of z, and both are 0 when F is
constant there.