Cusps of conjugate Fuchsian groups #
Let Γ ≤ PSL(2, ℝ), g ∈ PSL(2, ℝ), and let Γ' = g Γ g⁻¹, written
ConjAct.toConjAct g • Γ = Γ'. Conjugation by g carries the parabolic elements of Γ fixing a
boundary point c to the parabolic elements of Γ' fixing g • c, so translation by g carries
the cusp points and the cusp orbits of Γ onto those of Γ'.
It also carries normalized cusp data. If D has cusp c, scaling σ, generator γ and width
w, then D.conj h has cusp g • c, scaling σ g⁻¹, generator g γ g⁻¹ and the same width
w. Since (σ g⁻¹) • (g • z) = σ • z, the horodiscs of D.conj h are the translates by g of
those of D, and the q-coordinate of D.conj h at g • z is the q-coordinate of D at z.
This is what makes conjugation compatible with the cusp charts of the compactified quotients.
The conjugate is passed as a subgroup Γ' together with the equation ConjAct.toConjAct g • Γ = Γ'
rather than as the expression ConjAct.toConjAct g • Γ: the inverse transport is then the same
construction for g⁻¹, and an element of the normalizer of Γ acts on the cusps of Γ itself.
Main declarations #
Subgroup.isCuspPoint_smul_iff_of_conjAct_smul_eq:g • cis a cusp point ofg Γ g⁻¹exactly whencis a cusp point ofΓ.Subgroup.cuspOrbitConjEquiv: the induced bijection of cusp orbits, with the identity and composition lawsSubgroup.cuspOrbitConjEquiv_oneandSubgroup.cuspOrbitConjEquiv_trans.Subgroup.CuspDatum.conj: the transported cusp datum, withSubgroup.CuspDatum.cuspOrbit_conjand the identity and composition lawsSubgroup.CuspDatum.conj_oneandSubgroup.CuspDatum.conj_conj.TauCeti.Subgroup.CuspDatum.horodisc_conjandTauCeti.Subgroup.CuspDatum.coordinate_conj_smul: transport of horodiscs and of the q-coordinate.
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.
Cusp points of conjugate groups correspond. If Γ' = g Γ g⁻¹, then g • c is a cusp
point of Γ' exactly when c is a cusp point of Γ: conjugation by g carries the parabolic
elements of Γ fixing c to the parabolic elements of Γ' fixing g • c.
The cusp orbits of conjugate groups correspond: if Γ' = g Γ g⁻¹, translation by g
sends the cusp orbit of c under Γ to the cusp orbit of g • c under Γ'.
Equations
Instances For
The inverse of the bijection of cusp orbits induced by g is the one induced by g⁻¹.
Conjugation by 1 induces the identity of the cusp orbits.
Conjugating by g and then by g' induces the same bijection of cusp orbits as conjugating
by g' * g.
Transport of a normalized cusp datum to a conjugate group. If Γ' = g Γ g⁻¹ and D is a
cusp datum of Γ with cusp c, scaling σ, generator γ and width w, then Γ' has the cusp
datum with cusp g • c, scaling σ g⁻¹, generator g γ g⁻¹ and the same width w.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The transported cusp datum represents the image of the cusp orbit of D.
Transport of a cusp datum by conjugation by 1 is the identity.
Transporting a cusp datum by conjugation by g and then by g' is transporting it by
conjugation by g' * g.
The translate by g of a point lies in a horodisc of the transported cusp datum exactly when
the point lies in the horodisc of the same height of the original datum.
The horodiscs of the transported cusp datum are the translates by g of the horodiscs of the
original datum.
The q-coordinate is compatible with conjugation: the q-coordinate of the transported cusp
datum at g • z is the q-coordinate of the original datum at z.