Successive approximation of a relator carrying the p-power tails #
Let F = freeProP p X be the free pro-p group on a finite linearly ordered type X, with lower
p-series λ_k = λ_k(F), and let r, w ∈ λ_1(F) be relators with the same class ρ ∈ gr_1(F).
When the basis-modification map δ_ρ is not onto gr_{m+1}(F), the successive-approximation
argument of TauCeti.Topology.Algebra.Group.Profinite.Free.SuccessiveApproximation.Basic still
runs as soon as the span statement holds up to the tail of a set S of generators, the span
TauCeti.freeProP.gradedPowIterSpan of the iterated p-powers π^{m+1} ξ_i of the generators
x_i with i ∈ S:
gr_{m+1}(F) = Im δ_ρ + ⟨π^{m+1} ξ_i : i ∈ S⟩ for every m ≥ 1.
The set S is a parameter: for the odd-rank dyadic relators it is the set of generators whose
coefficient c_i in ρ vanishes, so that the span is the tail T_{m+1}(ρ) of
TauCeti.freeProP.basisModificationTail, while for the even-rank dyadic relators it also contains
the generator carrying the square.
At each level the discrepancy between the moved relator and the target is the class of a basis
modification up to a tail class Σ_{i ∈ S} c_i π^{m+1} ξ_i, which is the class of the product of
the powers x_i^{p^{m+1} c_i}. The basis modification is applied and the powers are absorbed into
per-generator tail elements t_i, which are carried along instead of being killed: the
approximation at level m is a congruence φ r ≡ t_{i₁} ⋯ t_{i_a} * w * t_{j₁} ⋯ t_{j_b}
modulo λ_{m+2}(F), where the two lists of indices fix where the tail of each generator is placed,
and the tail elements lie in the closed procyclic subgroups ⟨x_i⟩ and in λ_2(F). The tail of
each generator is placed at a single position, so the lists are required to be disjoint and
duplicate-free and to cover the set S.
The limit is taken through the levelwise comparison schema TauCeti.PLowerCentralSeriesComparison
with the finite tail data carried in the comparison data, and the tail elements are recovered from
the compatible sequence of their classes by the inverse-limit description of F. Since the closed
subgroups ⟨x_i⟩ ∩ λ_2(F) are detected on the finite quotients
(TauCeti.IsProP.mem_of_forall_mk_mem_map_pLowerCentralSeries), the limits stay in them. The
conclusion is an exact equation e r = t_{i₁} ⋯ t_{i_a} * w * t_{j₁} ⋯ t_{j_b} for a continuous
automorphism e of F.
This is the first half of Labute's treatment of the relators with q = 2. For the odd-rank
dyadic normal-form word x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n) the tails are the 2-powers
x_i^{2^{m+1}} of the generators x₂, …, x_n other than the one carrying the square, and the
argument yields the relator in the intermediate form x₁² r₀(x) x₂^{α₂} ⋯ x_n^{α_n} with 2-adic
exponents α₂, …, α_n divisible by 4
(TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Two.Odd.Approximation), before the
tail relator r₀(x) x₂^{α₂} ⋯ x_n^{α_n} is normalised in its own right. For the even-rank word
x₁² (x₁, x₂) (x₃, x₄) ⋯ (x_{n-1}, x_n) the tails are the 2-powers of every generator but x₂,
and the tail of x₁ is placed in front of the word, which is why the theorem takes two lists of
tail positions; the intermediate form is x₁^{2+α} (x₁, x₂) r₀(x) x₃^{α₃} ⋯ x_n^{α_n}
(TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Two.Even.Approximation).
Main results #
exists_continuousMonoidHom_inv_mul_apply_mem_of_range_sup_gradedPowIterSpan_eq_top, in the namespaceTauCeti.freeProP: the finite approximations, one basis modification and one tail correction per level.TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_of_range_sup_gradedPowIterSpan_eq_topis the successive-approximation theorem with tails: a continuous automorphism ofFcarriesrtowup to tail elements of the closed procyclic subgroups⟨x_i⟩ ∩ λ_2(F),i ∈ S, placed at the prescribed positions.
References #
- J. Labute, Classification of Demushkin groups, Canadian J. Math. 19 (1967), §3, Proposition 5 and the proof of Theorem 3, cases (2) and (3).
Finite successive approximation with tails. Let r, w ∈ λ_1(F) have the same class
ρ ∈ gr_1(F), let S be a set of generators and l₁, l₂ disjoint duplicate-free lists of
generators containing S, and suppose gr_{m+1}(F) = Im δ_ρ + ⟨π^{m+1} ξ_i : i ∈ S⟩ for
1 ≤ m ≤ k. Then there are a continuous endomorphism φ of F, congruent to the identity modulo
λ_1(F), and tail elements t_i ∈ ⟨x_i⟩ ∩ λ_2(F) with
φ r ≡ (∏_{i ∈ l₁} t_i) * w * (∏_{i ∈ l₂} t_i) mod λ_{k+2}(F).
The successive-approximation theorem with tails. Let r, w ∈ λ_1(F) be relators of the
free pro-p group F on a finite linearly ordered type with the same class ρ ∈ gr_1(F), let
S be a set of generators and l₁, l₂ disjoint duplicate-free lists of generators containing
S, and suppose gr_{m+1}(F) = Im δ_ρ + ⟨π^{m+1} ξ_i : i ∈ S⟩ for every m ≥ 1. Then a
continuous automorphism e of F carries r to (∏_{i ∈ l₁} t_i) * w * (∏_{i ∈ l₂} t_i) for
tail elements t_i of the closed procyclic subgroups ⟨x_i⟩, all lying in λ_2(F).