FenchelNielsenZomorrodian.Discrete.Torsion.CompactFiniteOrder
This module develops the rewriting and basis constructions behind the subgroup calculations. It tracks words and relations through the chosen transversal to obtain the required presentation or basis statements.
theorem finiteOrder_isConj_elliptic_zpow_of_ne_one
(σ : FuchsianSignature) (g : FuchsianPresentedGroup σ)
(hg : IsOfFinOrder g) (hgne : g ≠ 1) :
∃ i : Fin σ.numPeriods, ∃ n : ℤ,
IsConj g (ellipticElement σ i ^ n)Every nontrivial finite-order element is conjugate to a power of an elliptic generator.
Show proof
by
exact
finiteOrder_isConj_elliptic_zpow_of_frontier (ellipticElement σ)
(fun K hKfinite hKne => by
haveI : Finite K := hKfinite
exact finiteSubgroup_le_conj_ellipticStabilizer σ K hKne)
g hg hgneProof. 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. For period-class statements, the relevant lcm or gcd divisibility is converted into a scalar multiple of the abelianized elliptic generator, and the defining period relation makes that multiple vanish. 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. 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.
□theorem finiteOrder_eq_one_or_isConj_elliptic_zpow
(σ : FuchsianSignature) (g : FuchsianPresentedGroup σ)
(hg : IsOfFinOrder g) :
g = 1 ∨ ∃ i : Fin σ.numPeriods, ∃ n : ℤ,
IsConj g (ellipticElement σ i ^ n)A finite-order element in the compact Fuchsian presentation is either trivial or conjugate to a power of an elliptic generator.
Show proof
by
exact
finiteOrder_eq_one_or_isConj_elliptic_zpow_of_frontier (ellipticElement σ)
(fun K hKfinite hKne => by
haveI : Finite K := hKfinite
exact finiteSubgroup_le_conj_ellipticStabilizer σ K hKne)
g hgProof. 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. For period-class statements, the relevant lcm or gcd divisibility is converted into a scalar multiple of the abelianized elliptic generator, and the defining period relation makes that multiple vanish. 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. Every map between the reduced presentations is determined by the images of elliptic, handle, and boundary generators. The period and product relators are checked explicitly after applying those images. Therefore the constructed quotient or abelianized map is well defined and has the claimed effect on the named generator or class. 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.
□