Precisely invariant horodiscs at a cusp #
Let D be a normalized cusp datum of Γ ≤ PSL(2, ℝ), with scaling σ and width w. The
horodisc of height A at the cusp of D is the set {z | A < (σ • z).im}: in the scaling
coordinate it is the half-plane above height A, and it is exactly the preimage of a punctured
disc under the exponential coordinate
(TauCeti.Subgroup.CuspDatum.norm_qCoordinate_lt_iff).
The cusp stabilizer permutes each horodisc, because in the scaling coordinate it acts by
translations. The point of this file is the converse for a discrete Γ: as soon as the height is
at least the width, nothing else does. An element of Γ moving a point of the horodisc back into
the horodisc lies in the cusp stabilizer, so two elements of Γ in different cosets of the
stabilizer carry the horodisc to disjoint sets. This is the precise invariance that makes the
horodisc descend to a punctured-disc neighbourhood of the cusp in the quotient.
The same argument separates horodiscs at two different cusps. If g ∈ Γ does not carry the cusp
of a second datum D' to the cusp of D, then the heights of g • z above the cusp of D and
of z above the cusp of D' have product at most D.width * D'.width. Hence horodiscs whose
heights are at least the widths never meet across such an element, and at two cusps that are not
Γ-equivalent their images in the orbit space Γ \ ℍ are disjoint. These disjoint punctured-disc
neighbourhoods of inequivalent cusps are what separate distinct cusp points once they are adjoined
to the quotient, while at two Γ-equivalent cusps the horodisc images agree up to a rescaling of
the height, so that the neighbourhoods adjoined at a cusp orbit do not depend on the chosen
representative. The same inequality shows that orbits stay uniformly low near any point of ℍ,
which separates a point of the orbit space from every adjoined cusp.
All of these bounds are Shimizu's lemma
(Subgroup.im_smul_mul_im_le_abs_mul_of_upperRightHom_mem), applied to the conjugate group
σ Γ σ⁻¹, which contains the translation by w and is again discrete.
Main results #
TauCeti.Subgroup.CuspDatum.smul_horodisc_of_mem_stabilizer: the cusp stabilizer preserves every horodisc.TauCeti.Subgroup.CuspDatum.im_scaling_smul_smul_mul_im_scaling_smul_le: Shimizu's inequality comparing the heights above two cusps.TauCeti.Subgroup.CuspDatum.disjoint_smul_horodisc_horodisc: an element ofΓnot carrying one cusp to the other carries a high horodisc at the first off a high horodisc at the second.TauCeti.Subgroup.CuspDatum.mem_stabilizer_of_mem_horodisc_of_smul_mem_horodisc: precise invariance.TauCeti.Subgroup.CuspDatum.disjoint_smul_horodiscandTauCeti.Subgroup.CuspDatum.disjoint_smul_horodisc_self: the translates of a high horodisc by two elements lying in different cosets of the cusp stabilizer are disjoint.TauCeti.Subgroup.CuspDatum.exists_image_quotientMk_horodisc_eq: horodiscs at twoΓ-equivalent cusps have the same images in the orbit space up to a fixed rescaling of the height.TauCeti.Subgroup.CuspDatum.disjoint_image_quotientMk_horodisc_iff: high horodiscs at two cusps have disjoint images in the orbit space exactly when the cusps are notΓ-equivalent.TauCeti.Subgroup.CuspDatum.exists_isOpen_mem_disjoint_image_quotientMk_horodisc: near any point ofℍ, orbits stay uniformly low, so a neighbourhood of a point of the orbit space misses the image of a high horodisc.
References #
- Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992, §4.2.
- Fred Diamond and Jerry Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics 228, Springer, 2005, §2.4.
The horodisc of height A at the cusp represented by D, in its normalized scaling
coordinate.
Instances For
Membership in a horodisc is the corresponding lower bound on the scaled imaginary part.
A horodisc is the inverse image of the corresponding punctured disc under the normalized q-coordinate.
In the scaling coordinate the cusp stabilizer acts by translations, so it preserves the height above the real axis.
The cusp stabilizer preserves every horodisc at its cusp.
A horodisc is open.
Horodiscs shrink as their height grows.
Every horodisc is nonempty.
Heights above equivalent cusps are proportional. If k ∈ Γ carries the cusp of D to
the cusp of D', then σ' k σ⁻¹ fixes ∞, so it is a positive real affine map, and the height
of k • z above the cusp of D' is a fixed positive multiple of the height of z above the cusp
of D.
An element of Γ carrying the cusp of D to the cusp of D' carries the horodiscs at D
onto the horodiscs at D', rescaling the height by a fixed positive factor.
Horodiscs at equivalent cusps have the same images. If the cusps of D and D' are
Γ-equivalent, then after rescaling heights by a fixed positive factor the horodiscs at the two
cusps have the same image in the orbit space Γ \ ℍ.
Shimizu's inequality at two cusps. Let D and D' be normalized cusp data of a discrete
Γ ≤ PSL(2, ℝ). If g ∈ Γ does not carry the cusp of D' to the cusp of D, then the height of
g • z above the cusp of D and the height of z above the cusp of D', each measured in the
scaling coordinate of its datum, have product at most D.width * D'.width.
This is Shimizu's lemma (Subgroup.im_smul_mul_im_le_abs_mul_of_upperRightHom_mem) for the
conjugate group σ Γ σ⁻¹, which contains the translation by D.width, applied to σ g σ'⁻¹,
which conjugates the translation by D'.width into it.
Horodiscs at two cusps. Let D and D' be normalized cusp data of a discrete
Γ ≤ PSL(2, ℝ), and let the heights satisfy 0 ≤ A and D.width * D'.width ≤ A * A', for
instance D.width ≤ A and D'.width ≤ A'. If g ∈ Γ does not carry the cusp of D' to the cusp
of D, it carries the horodisc of height A' at D' off the horodisc of height A at D.
Precise invariance of high horodiscs. Let D be a normalized cusp datum of a discrete
Γ ≤ PSL(2, ℝ) and let the height A be at least the width of D. If an element of Γ carries
a point of the horodisc of height A back into that horodisc, then it fixes the cusp.
High horodiscs lie in the free locus. For a discrete Γ, a point of a horodisc of height at
least the width has trivial stabilizer: an element fixing it lies in the cusp stabilizer, whose
nontrivial elements are translations in the scaling coordinate and fix no point.
Precise invariance of high horodiscs, disjointness form. Two elements of Γ carrying a
horodisc of height at least the width to sets that meet differ by an element of the cusp
stabilizer.
An element of Γ outside the cusp stabilizer moves every horodisc of height at least the
width off itself.
Horodiscs at two Γ-equivalent cusps have intersecting images in the orbit space, whatever
their heights: a point high enough above the first cusp is carried by Γ to a point high above
the second.
Horodiscs at inequivalent cusps have disjoint images. Let D and D' be normalized cusp
data of a discrete Γ ≤ PSL(2, ℝ), and let the heights satisfy 0 ≤ A and
D.width * D'.width ≤ A * A', for instance D.width ≤ A and D'.width ≤ A'. The images of the
horodiscs of heights A at D and A' at D' in the orbit space Γ \ ℍ are disjoint exactly
when the two cusps are not Γ-equivalent.
Orbits stay uniformly low near a point. Let D be a normalized cusp datum of a discrete
Γ ≤ PSL(2, ℝ). Every point of ℍ has an open neighbourhood S and a height A such that no
element of Γ carries a point of S above height A over the cusp of D: the cusp stabilizer
preserves heights, and Shimizu's inequality bounds the height of g • w for g outside it.
A point of the orbit space is separated from every cusp. Let D be a normalized cusp
datum of a discrete Γ ≤ PSL(2, ℝ). Every point of ℍ has an open neighbourhood whose image in
the orbit space Γ \ ℍ is disjoint from the image of a sufficiently high horodisc at D.