Compactified quotients of conjugate Fuchsian groups #
Let Γ ≤ PSL(2, ℝ) be discrete, g ∈ PSL(2, ℝ), and let Γ' = g Γ g⁻¹, written
ConjAct.toConjAct g • Γ = Γ'. Translation by g induces a map
Subgroup.compactifiedQuotientConj of compactified quotients: on the coarse quotient it is the
homeomorphism Subgroup.quotientConjHomeomorph sending the orbit of z to the orbit of g • z,
and at the cusps it is the bijection Subgroup.cuspOrbitConjEquiv of cusp orbits.
This map is a biholomorphism, Subgroup.CompactifiedQuotient.conjBiholomorph, whose inverse is the
map induced by g⁻¹. Along the coarse quotient this is holomorphic descent. At a cusp, transport
the cusp datum D to the cusp datum D.conj h of Γ': it maps the cusp neighbourhood of D at
height A into that of D.conj h at the same height, and the cusp charts of the two data agree
after the map (Subgroup.CompactifiedQuotient.cuspChart_conj_compactifiedQuotientConj), because
the q-coordinate of D.conj h at g • z is the q-coordinate of D at z. So, read in these
charts, the map is the identity of a disc. In particular conjugate Fuchsian groups have
biholomorphic compactified quotients, and every element of the normalizer of Γ acts on the
compactified quotient of Γ by biholomorphisms. The construction is functorial: g = 1 gives the
identity (Subgroup.CompactifiedQuotient.conjBiholomorph_one), and conjugating by g and then by
g' is conjugating by g' * g (Subgroup.CompactifiedQuotient.conjBiholomorph_trans). Only Γ
is assumed discrete: discreteness of Γ' follows
(TauCeti.discreteTopology_of_conjAct_smul_eq).
Main declarations #
Subgroup.compactifiedQuotientConj: the map of compactified quotients induced by conjugation, with the identity and composition lawsSubgroup.compactifiedQuotientConj_oneandSubgroup.compactifiedQuotientConj_comp.Subgroup.CompactifiedQuotient.continuous_compactifiedQuotientConjandSubgroup.CompactifiedQuotient.mdifferentiable_compactifiedQuotientConj: it is continuous and holomorphic.Subgroup.CompactifiedQuotient.conjBiholomorph: the biholomorphism of compactified quotients, withSubgroup.CompactifiedQuotient.conjBiholomorph_symm,Subgroup.CompactifiedQuotient.conjBiholomorph_oneandSubgroup.CompactifiedQuotient.conjBiholomorph_trans.
References #
- Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992, Chapter 4.
- Fred Diamond and Jerry Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics 228, Springer, 2005, §§2.4–2.5.
The map of compactified quotients induced by conjugation: if Γ' = g Γ g⁻¹, it sends the
orbit of z to the orbit of g • z, and the cusp orbit of c to the cusp orbit of g • c.
Equations
- Subgroup.compactifiedQuotientConj h (Subgroup.CompactifiedQuotient.ofQuotient p) = Subgroup.CompactifiedQuotient.ofQuotient ((Subgroup.quotientConjHomeomorph h) p)
- Subgroup.compactifiedQuotientConj h (Subgroup.CompactifiedQuotient.ofCusp C) = Subgroup.CompactifiedQuotient.ofCusp ((Subgroup.cuspOrbitConjEquiv h) C)
Instances For
The map induced by g⁻¹ is a left inverse of the map induced by g.
Conjugation by 1 induces the identity of the compactified quotient.
Conjugating by g and then by g' induces the same map of compactified quotients as
conjugating by g' * g.
The map induced by conjugation sends the cusp neighbourhood of a cusp datum D at height A
into the cusp neighbourhood of the transported datum D.conj h at the same height.
The cusp charts are compatible with conjugation: on the cusp neighbourhood of D, the cusp
chart of the transported datum D.conj h after the map induced by conjugation is the cusp chart of
D.
The map of compactified quotients induced by conjugation is continuous, including at the adjoined cusp points.
The map of compactified quotients induced by conjugation is holomorphic. Along the coarse quotient this is holomorphic descent; at a cusp, read in the cusp charts of a cusp datum and of its transport, the map is the identity.
Conjugate Fuchsian groups have biholomorphic compactified quotients. If Γ' = g Γ g⁻¹ for
a discrete Γ, translation by g induces a biholomorphism between the compactified quotients of
Γ and Γ', sending the orbit of z to the orbit of g • z and the cusp orbit of c to the
cusp orbit of g • c.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The inverse of the biholomorphism induced by g is the one induced by g⁻¹.
Conjugation by 1 induces the identity biholomorphism.
Conjugating by g and then by g' induces the same biholomorphism as conjugating by
g' * g.