The coarse quotient of a Fuchsian group is a Riemann surface #
Let Γ ≤ PSL(2, ℝ) act properly discontinuously on the upper half-plane, as every discrete
subgroup does. This file puts complex charts on the whole orbit space Γ \ ℍ, elliptic orbits
included, and proves that they make it a Riemann surface with a holomorphic orbit projection.
The atlas consists of the charts Subgroup.stabilizerBallQuotientChart at every point of the
upper half-plane and every admissible radius; the chart at an orbit is the chart at a chosen
representative with the radius Subgroup.chartRadius, which exists because the local orbit space
of a small enough disc embeds openly into Γ \ ℍ. The transition maps between these charts are
holomorphic (Subgroup.differentiableOn_stabilizerBallQuotientChart_symm_trans), so the atlas is
analytic, and the local formula for a chart along the orbit projection makes the projection
ℍ → Γ \ ℍ holomorphic.
The chart centred at an orbit of stabilizer order m is the cyclic quotient model u ↦ u ^ m:
near a point of the disc about the centre, the chart composed with the orbit projection is the
m-th power of the disc coordinate
(Subgroup.exists_stabilizerBallQuotientChart_quotientMk_eventuallyEq). At a free orbit m = 1,
so the chart centred there is a disc coordinate pushed forward along the orbit projection.
Main declarations #
Subgroup.chartRadius: a positive radius at which the chart at the orbit ofzexists.Subgroup.instChartedSpaceOrbitRelQuotient: the atlas of all chartsSubgroup.stabilizerBallQuotientChartonΓ \ ℍ, withSubgroup.chartAt_eq,Subgroup.mem_atlas_orbitRelQuotient_iffandSubgroup.stabilizerBallQuotientChart_mem_atlasdescribing it.Subgroup.instIsManifoldOrbitRelQuotient: this atlas is analytic, soΓ \ ℍis a Riemann surface. It is Hausdorff and second countable byt2Space_of_properlyDiscontinuousSMul_of_t2SpaceandSubgroup.instSecondCountableTopologyOrbitRelQuotient.Subgroup.mdifferentiable_quotientMk: the orbit projection is holomorphic.
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.
For a properly discontinuous action there is a positive radius whose local orbit space embeds
openly into Γ \ ℍ, so the chart at the orbit of z exists.
A positive radius at which the chart Subgroup.stabilizerBallQuotientChart at the orbit of z
exists.
Equations
- Γ.chartRadius z = ⋯.choose
Instances For
The atlas of Γ \ ℍ. It consists of the charts Subgroup.stabilizerBallQuotientChart at
every point and every admissible radius; the chart at an orbit is the chart at a chosen
representative with the radius Subgroup.chartRadius.
Equations
- One or more equations did not get rendered due to their size.
A chart of Γ \ ℍ belongs to the atlas exactly when it is the chart
Subgroup.stabilizerBallQuotientChart at some point and some admissible radius.
The coarse orbit quotient Γ \ ℍ is second countable, as the orbit space of a second
countable space under a continuous group action.
The coarse quotient of a Fuchsian group is a Riemann surface: the transition maps between the charts of the atlas are holomorphic.
The orbit projection is holomorphic. In the chart at the orbit of τ it is a power of a
disc coordinate.
Holomorphic descent at a free orbit. A map from the orbit space is holomorphic at the orbit of a point with trivial stabilizer as soon as its pullback along the orbit projection is holomorphic at that point.