Successive approximation for the dyadic relators of even rank #
This module contains two stages of Labute's successive-approximation arguments for even-rank dyadic relators.
Intermediate form #
Let F = freeProP 2 (Fin n) be the free pro-2 group on an even number n ≥ 2 of generators and
let r ∈ λ_1(F) be a relator with the class of x₁² (x₁, x₂) (x₃, x₄) ⋯ (x_{n-1}, x_n) in
gr_1(F). The dyadic span statement
gr_{m+1}(F) = Im δ_ρ + ⟨π^{m+1} ξ_i : i ≠ 2⟩
feeds the successive approximation with tails. A continuous automorphism carries r to
t₁ * x₁² (x₁, x₂) (x₃, x₄) ⋯ (x_{n-1}, x_n) * t₃ ⋯ t_n,
with t_i in the closed procyclic subgroup generated by x_i and in λ_2(F). Writing the tails
as 2-adic powers and merging the tail of x₁ into the square gives Labute's intermediate form
x₁^{2+α} (x₁, x₂) * ((x₃, x₄) ⋯ (x_{n-1}, x_n) x₃^{α₃} ⋯ x_n^{α_n}),
where the exponents α, α₃, …, α_n are divisible by 4.
Constrained approximation for the second family #
Let F be the free pro-2 group on an even number n ≥ 4 of generators and let
r_f = x₁² (x₁, x₂) x₃^(2^f) (x₃, x₄) ⋯ (x_{n-1}, x_n).
For the orientation with values χ(x₂) = -1, χ(x₄) = (1 - 2^f)⁻¹, and 1
elsewhere, corrections are made inside the kernel of the exponent sum at x₄. Elements of this
kernel in the first term of the lower 2-central series are killed by χ. Labute's constrained
span and graded-functional statements therefore feed the general successive-approximation theorem: a
relator with the same degree-one class as r_f, lying in that exponent-sum kernel and killed by
every coordinate crossed homomorphism except the one at x₂, is the image of r_f under an
automorphism whose changes of all the free generators lie in that exponent-sum kernel and in
λ₁(F), so are killed by χ.
This is the limit step in the second even-rank family of Labute's classification, the family with
orientation image { ±1 } × U^(f). It turns the infinitesimal Lemmas 3 and 4 proved in
TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Two.Even.KernelSpan into an exact
equality of relators.
Main results #
exists_continuousMulEquiv_apply_eq_padicPow_mul_demushkinWordNeTwo_mul_padicPow: the relator is carried to the even-rank word with explicit2-adic power tails.exists_continuousMulEquiv_apply_eq_padicPow_mul_labuteComm_mul: Labute's intermediate form.exists_continuousMulEquiv_apply_demushkinWordTwoEven_zero_eq_of_forall_crossedHom_eq_zero: the constrained successive-approximation theorem atr_f.
References #
- J. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §3, proof of Theorem 3, case (3).
- J. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §4, Lemmas 3–4 and the proof of Theorem 6.
Constrained successive approximation for the second dyadic even-rank family (Labute,
§4, Lemmas 3–4 and Theorem 6). Let n ≥ 4 be even, f ≥ 2, and let χ take the
normal-form
values χ(x₂) = -1, χ(x₄)(1 - 2^f) = 1, and χ(x_i) = 1 otherwise. If a relator
r has the same class in gr₁(F) as
w = x₁² (x₁, x₂) x₃^(2^f) (x₃, x₄) ⋯, lies in the kernel of the exponent sum at
x₄, and is killed by every coordinate crossed homomorphism for χ except the one at x₂,
then an automorphism e of F carries w to r. Moreover every generator change
x_i⁻¹ e(x_i) lies both in the kernel of the exponent sum at x₄ and in λ₁(F).
The last clause records that the approximation preserves the orientation: an element of the
exponent-sum kernel which lies in λ₁(F) is killed by χ at these marked values
(ContinuousMonoidHom.apply_eq_one_of_mem_exponentSumKer_of_mem_pLowerCentralSeries_one).
The even-rank dyadic relators, up to tails. Let n ≥ 2 be even and let r ∈ λ_1(F) be a
relator of the free pro-2 group F on n generators with the class of
x₁² (x₁, x₂) (x₃, x₄) ⋯ (x_{n-1}, x_n) in gr_1(F). Then a continuous automorphism carries r
to that word preceded by a 2-adic power x₁ ^ a₁ of the first generator and followed by 2-adic
powers x_i ^ a_i of the generators x₃, …, x_n, where every exponent a_i is divisible by
4.
Labute's intermediate form for the dyadic relators of even rank. Let n ≥ 2 be even and
let r ∈ λ_1(F) be a relator of the free pro-2 group F on n generators with the class of
x₁² (x₁, x₂) (x₃, x₄) ⋯ (x_{n-1}, x_n) in gr_1(F). Then a continuous automorphism carries r
to x₁^{2+α} (x₁, x₂) * ((x₃, x₄) ⋯ (x_{n-1}, x_n) * x₃^{α₃} ⋯ x_n^{α_n}), with 2-adic exponents
α, α₃, …, α_n divisible by 4; the second factor is a word in the generators x₃, …, x_n,
written on the tuple of generators shifted by two.