Labute's normal form for the dyadic Demushkin groups of odd rank

Let F = freeProP 2 (Fin n) be the free pro-2 group on an odd number n of generators and let r ∈ Φ(F) be a relator presenting a Demushkin group G = ⟨x₁, …, x_n ∣ r⟩. Since n is odd, the cup form of G is not alternating and q(G) = 2. Labute's Theorem 3 says that a continuous automorphism of F carries r to one of the words

x₁² x₂^{2^f} (x₂, x₃)(x₄, x₅) ⋯ (x_{n-1}, x_n), 2 ≤ f < ∞, or x₁² (x₂, x₃)(x₄, x₅) ⋯ (x_{n-1}, x_n), the value f = ∞,

so that G is presented by that word. This file proves it, by Labute's route (pp. 118–119). The normal form modulo λ_2(F) carries the class of r to that of the level-∞ word (TauCeti.freeProP.exists_continuousMulEquiv_gradedMap_eq_gradedMk_demushkinWordTwoOddTop), the successive approximation with tails carries r itself to the intermediate form x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n) t₂ ⋯ t_n, with t_i = x_i^{a_i} a 2-adic power of x_i whose exponent a_i ∈ ℤ_2 is divisible by 4 (TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_demushkinWordTwoOddTop_mul_padicPow). At rank n = 1 there is no tail: the intermediate form is already x₁², the level-∞ word, and the theorem is proved. For n > 1, the tail relator r' = (x₂, x₃) ⋯ (x_{n-1}, x_n) t₂ ⋯ t_n is a word in x₂, …, x_n alone. Read in the free pro-2 group on those n - 1 ≥ 2 generators, r' has the class of (x₂, x₃) ⋯ (x_{n-1}, x_n) in gr_1, so it presents a Demushkin group whose q-invariant is not 2, and the normal-form theorem for q ≠ 2 (freeProP.exists_continuousMulEquiv_apply_eq_demushkinWordNeTwo_of_demushkinQ_ne) carries r' to x₂^{q'} (x₂, x₃) ⋯ (x_{n-1}, x_n) with q' = 0 or q' = 2^f, f ≥ 2. Extending that automorphism to F by fixing x₁ (TauCeti.freeProP.finSuccExtend) gives the theorem.

The level f is an invariant of G, the level of the image {±1} × U^(f) of its canonical character; this file proves only the existence of the normal form, and does not identify f.

Main results

• TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_demushkinWordTwoOdd_of_odd: Labute's Theorem 3 for q = 2 and odd rank: a relator in Φ(F) presenting a Demushkin group on an odd number of generators is carried by a continuous automorphism of F to x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n) or to x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n) for some f ≥ 2.
• TauCeti.freeProP.exists_continuousMulEquiv_presentedProP_demushkinWordTwoOdd_of_odd and its intrinsic form IsDemushkin.exists_continuousMulEquiv_presentedProP_demushkinWordTwoOdd_of_odd_demushkinRank: a Demushkin group at p = 2 of odd rank n is topologically isomorphic to ⟨x₁, …, x_n ∣ x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n)⟩ or to ⟨x₁, …, x_n ∣ x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n)⟩ for some f ≥ 2.

References

• J. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §3, Theorem 3, case (2).
• J.-P. Serre, Galois Cohomology, Chapter I, §4.5.

theorem TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_demushkinWordTwoOdd_of_odd {n : ℕ} (hn : Odd n) {r : freeProP 2 (Fin n)} (hr : r ∈ proPFrattini 2 (freeProP 2 (Fin n))) (hG : IsDemushkin 2 (presentedProP 2 (Fin n) {r})) : ∃ (e : freeProP 2 (Fin n) ≃ₜ* freeProP 2 (Fin n)), e r = demushkinWordTwoOddTop n (freeProPGen 2 n) ∨ ∃ (f : ℕ), 2 ≤ f ∧ e r = demushkinWordTwoOdd f n (freeProPGen 2 n)

Labute's normal form for q = 2 and odd rank (Labute, Theorem 3, case (2)). Let r ∈ Φ(F) be a relator of the free pro-2 group on an odd number n of generators presenting a Demushkin group G = ⟨x₁, …, x_n ∣ r⟩. Then a continuous automorphism of F carries r either to x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n), Labute's level f = ∞, or to x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n) for some f ≥ 2.

theorem TauCeti.freeProP.exists_continuousMulEquiv_presentedProP_demushkinWordTwoOdd_of_odd {n : ℕ} (hn : Odd n) {r : freeProP 2 (Fin n)} (hr : r ∈ proPFrattini 2 (freeProP 2 (Fin n))) (hG : IsDemushkin 2 (presentedProP 2 (Fin n) {r})) : Nonempty (presentedProP 2 (Fin n) {r} ≃ₜ* presentedProP 2 (Fin n) {demushkinWordTwoOddTop n (freeProPGen 2 n)}) ∨ ∃ (f : ℕ), 2 ≤ f ∧ Nonempty (presentedProP 2 (Fin n) {r} ≃ₜ* presentedProP 2 (Fin n) {demushkinWordTwoOdd f n (freeProPGen 2 n)})

Labute's normal form for a one-relator Demushkin group of odd rank. Let r ∈ Φ(F) be a relator of the free pro-2 group on an odd number n of generators presenting a Demushkin group. Then ⟨x₁, …, x_n ∣ r⟩ is topologically isomorphic to ⟨x₁, …, x_n ∣ x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n)⟩ or to ⟨x₁, …, x_n ∣ x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n)⟩ for some f ≥ 2.

theorem TauCeti.IsDemushkin.exists_continuousMulEquiv_presentedProP_demushkinWordTwoOdd_of_odd_demushkinRank {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin 2 G) (hn : Odd (demushkinRank hG)) : Nonempty (G ≃ₜ* presentedProP 2 (Fin (demushkinRank hG)) {demushkinWordTwoOddTop (demushkinRank hG) (freeProPGen 2 (demushkinRank hG))}) ∨ ∃ (f : ℕ), 2 ≤ f ∧ Nonempty (G ≃ₜ* presentedProP 2 (Fin (demushkinRank hG)) {demushkinWordTwoOdd f (demushkinRank hG) (freeProPGen 2 (demushkinRank hG))})

Labute's normal form for a Demushkin group of odd rank, intrinsic form (Labute, Theorem 3, case (2)). A Demushkin group G at p = 2 of odd rank n = demushkinRank hG is topologically isomorphic to ⟨x₁, …, x_n ∣ x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n)⟩ or to ⟨x₁, …, x_n ∣ x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n)⟩ for some f ≥ 2.