FenchelNielsenZomorrodian.Discrete.FiniteIndex.SmoothQuotientData
This module sets up the finite-stage and inverse-limit description of the construction. It records the stage maps, projections, and comparison lemmas used to pass back to the completed object.
import
abbrev FiniteSolvableSmoothQuotientData
(σ : FuchsianSignature) (m : ℕ) :=
SmoothQuotientData
(FuchsianPresentedGroup σ)
(Fin σ.numPeriods)
σ.periods
(ellipticElement σ)
mtheorem sourceSubgroup_exists_classical
{σ : FuchsianSignature} {m : ℕ}
(D : FiniteSolvableSmoothQuotientData σ m) :
∃ L : Subgroup (FuchsianPresentedGroup σ),
L.FiniteIndex ∧ IsTorsionFreeGroup L ∧
SubgroupQuotientHasDerivedLengthAtMost L mShow proof
SmoothQuotientData.sourceSubgroup_exists_of_finiteOrderClassification
D
(fun i _n hdiv => ellipticElement_zpow_eq_one_of_period_int_dvd σ i hdiv)
(finiteOrder_eq_one_or_isConj_elliptic_zpow σ)Proof. Use the Fenchel--Nielsen--Zomorrodian presentation with elliptic generators of prescribed periods, handle and boundary generators, and the defining product relation. The proof checks the images of the named generators and verifies that every presentation relator is preserved in the target quotient, abelianization, or reduced presentation. Schreier-rewriting steps are performed with the chosen transversal; the letter-by-letter formula sends each relator to the corresponding canonical word in the subgroup presentation. Kernel and normal-closure claims are proved by showing that each rewritten relator lies in the generated normal subgroup and that the quotient map kills exactly those relations required by the presentation. Finiteness and cardinality estimates use the prescribed period data and the finite quotient of the presentation determined by those periods. Because the generator images satisfy all defining relations and agree with the displayed quotient data, the presentation universal property yields the asserted identity. The reductions preserve the ordering of the canonical generators and the period data, so the image of every relator can be read directly in the target presentation. In the abelianized arguments, commutators vanish and only the period coefficients remain. The divisibility or lcm calculation then gives the required scalar multiple, while the presentation relation supplies its vanishing. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
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