The span statement at the dyadic normal form of odd rank

Let F = freeProP 2 (Fin n) be the free pro-2 group on an odd number n of generators, and let ρ ∈ gr_1(F) be the class of Labute's normal-form word x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n) with f ≥ 2, which is π ξ₁ + [ξ₂, ξ₃] + ⋯ + [ξ_{n-1}, ξ_n]. For every m ≥ 1 the image of the basis-modification map δ_ρ : gr_m(F)^n → gr_{m+1}(F) together with the tail T_{m+1}(ρ), spanned by the 2-powers π^{m+1} ξ_i for i ≥ 2, is all of gr_{m+1}(F):

gr_{m+1}(F) = Im δ_ρ + T_{m+1}(ρ).

This is the dyadic span statement of Labute's successive-approximation argument for the relators with q = 2 of odd rank: the class in gr_{m+1}(F) of a discrepancy between two relators with class ρ is the class of a basis modification up to the classes of the powers x_i^{2^{m+1}}, i ≥ 2, which the argument carries along rather than absorbs; in Labute's proof they give rise to the factor x₂^{2^f} of the normal form. It is the instance of TauCeti.freeProP.range_basisModificationDelta_sup_gradedPowIterSpan_compl_eq_top_two at x₁: the degree-one form of ρ is nondegenerate, and its column at the first coordinate character is the vector of 2-power coefficients of ρ, since x₁ is the only generator with a 2-power coefficient and the first coordinate character is orthogonal to the others, with B_ρ(χ₁, χ₁) = c₁; the spanning powers π^{m+1} ξ_i, i ≠ 1, all lie in the tail T_{m+1}(ρ).

Main results

• freeProP.range_basisModificationDelta_sup_basisModificationTail_eq_top_demushkinWordTwoOdd: gr_{m+1}(F) = Im δ_ρ + T_{m+1}(ρ) for the class ρ of x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n).

References

• J. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §3, Proposition 5 and the proof of Theorem 3.

theorem TauCeti.freeProP.range_basisModificationDelta_sup_basisModificationTail_eq_top_demushkinWordTwoOdd {n : ℕ} (hn : Odd n) {f : ℕ} (hf : 2 ≤ f) {m : ℕ} (hm : 1 ≤ m) : ((basisModificationDelta 2 (Fin n) hm) (gradedMk 2 (freeProP 2 (Fin n)) 1 ⟨demushkinWordTwoOdd f n (freeProPGen 2 n), ⋯⟩)).range ⊔ basisModificationTail 2 (Fin n) (gradedMk 2 (freeProP 2 (Fin n)) 1 ⟨demushkinWordTwoOdd f n (freeProPGen 2 n), ⋯⟩) (m + 1) = ⊤

The dyadic span statement at the normal form of odd rank (Labute, Proposition 5, the case q = 2). Let n be odd, f ≥ 2, and let ρ ∈ gr_1(F) be the class of x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n) in the free pro-2 group F on n generators. Then for every m ≥ 1 gr_{m+1}(F) = Im δ_ρ + T_{m+1}(ρ), the tail T_{m+1}(ρ) being spanned by the 2-powers π^{m+1} ξ_i of the generator classes other than ξ₁.