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TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Two.Even.Twisted

Labute's normal form for the dyadic Demushkin groups of even rank with twisted image #

Let G be a Demushkin group at p = 2 of even rank n whose canonical character has image the twisted subgroup U^[f] ≤ ℤ_2ˣ, the closed subgroup generated by -1 + 2^f, for some finite f ≥ 2. Labute's Theorem 5 says that G is presented on n generators by the single relator x₁^{2 + 2^f} (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n), the normal-form word TauCeti.demushkinWordNeTwo (2 + 2^f) n; in particular two such groups with the same rank and the same f are topologically isomorphic. This file proves that theorem.

Write G = ⟨x₁, …, x_n ∣ r⟩ with r ∈ Φ(F), F = freeProP 2 (Fin n), and let χ be the canonical character read on F. The proof normalizes the basis in three steps and then runs the successive approximation of TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Kernel.Approximation.

  1. The class of r in gr_1(F) has nondegenerate degree-one form, which is not alternating: an alternating form would put the image of χ inside 1 + 4ℤ_2, which does not contain -1 + 2^f. At p = 2 the classes with nondegenerate nonalternating form form a single orbit (TauCeti.exists_continuousMulEquiv_gradedMap_eq_of_not_isAlt), so a change of basis carries r to the class of the normal-form word modulo λ_2(F).
  2. In that basis the values of χ on the generators are pinned modulo 4: χ(x_i) ∈ 1 + 4ℤ_2 for i ≠ 2, since the Kronecker crossed homomorphisms for χ kill r (TauCeti.freeProP.apply_mem_unitsPrincipal_two_of_crossedHom_single_eq_zero), and χ(x₂) ∉ 1 + 4ℤ_2, since otherwise the image of χ would lie in 1 + 4ℤ_2. Inside U^[f] this says χ(x_i) ∈ U^(f+1) for i ≠ 2 and that χ(x₂) topologically generates U^[f].
  3. A basis modification x_i ↦ x_i · x₂^{2 s_i} with 2-adic exponents s_i, which does not change the class of r modulo λ_2(F), arranges the exact values χ(x₂) = -(1 + 2^f)⁻¹ and χ(x_i) = 1 for i ≠ 2: the squares of χ(x₂) topologically generate U^(f+1).

The relator then lies in the kernel X of the exponent sum at x₂, which is ker χ, it is killed by every continuous crossed homomorphism for χ by the prescription property of the canonical character, and the successive approximation inside X carries the normal-form word to it.

The same groups are presented by Labute's words at a finite level and with any exponent of the same valuation, rank by rank. For every natural α of exact divisibility depth g and every finite f > g, the word x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{n-1}, x_n) presents a Demushkin group of rank n ≥ 4 whose canonical character has image U^[g], so by the uniqueness theorem it presents the same group as the rank-n level-∞ word x₁^{2 + 2^g} (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n). In rank two the word x₁^{2 + α} (x₁, x₂) presents a Demushkin group of rank 2 with the same image, hence the same group as the rank-two word x₁^{2 + 2^g} (x₁, x₂). Neither the exponent within its valuation nor the level above g = v₂(α) is an invariant in this branch.

Main results #

References #

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theorem TauCeti.freeProP.exists_continuousMulEquiv_apply_eq_demushkinWordNeTwo_of_range_eq {n : ℕ} {r : freeProP 2 (Fin n)} (hr : r ∈ proPFrattini 2 (freeProP 2 (Fin n))) (hG : IsDemushkin 2 (presentedProP 2 (Fin n) {r})) (hn : Even n) {f : ℕ} (hf : 2 ≤ f) {w : ℤ_[2]ˣ} (hw : ↑w = -1 + 2 ^ f) (hA : (demushkinCharacter hG).range = (Subgroup.zpowers w).topologicalClosure) :
∃ (e : freeProP 2 (Fin n) ≃ₜ* freeProP 2 (Fin n)), e r = demushkinWordNeTwo (2 + 2 ^ f) n (freeProPGen 2 n)

Labute's normal form for the dyadic Demushkin groups of even rank with twisted image (Labute, Theorem 5). Let r ∈ Φ(F) be a relator of the free pro-2 group on an even number n of generators presenting a Demushkin group G = ⟨x₁, …, x_n ∣ r⟩ whose canonical character has image the twisted subgroup U^[f], the closed subgroup generated by -1 + 2^f, for a finite f ≥ 2. Then a continuous automorphism of F carries r to x₁^{2 + 2^f} (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n).

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theorem TauCeti.freeProP.exists_continuousMulEquiv_presentedProP_demushkinWordNeTwo_of_range_eq {n : ℕ} {r : freeProP 2 (Fin n)} (hr : r ∈ proPFrattini 2 (freeProP 2 (Fin n))) (hG : IsDemushkin 2 (presentedProP 2 (Fin n) {r})) (hn : Even n) {f : ℕ} (hf : 2 ≤ f) {w : ℤ_[2]ˣ} (hw : ↑w = -1 + 2 ^ f) (hA : (demushkinCharacter hG).range = (Subgroup.zpowers w).topologicalClosure) :
Nonempty (presentedProP 2 (Fin n) {r} ≃ₜ* presentedProP 2 (Fin n) {demushkinWordNeTwo (2 + 2 ^ f) n (freeProPGen 2 n)})

Labute's normal form for a one-relator dyadic Demushkin group of even rank with twisted image. Let r ∈ Φ(F) be a relator of the free pro-2 group on an even number n of generators presenting a Demushkin group whose canonical character has image U^[f], f ≥ 2 finite. Then ⟨x₁, …, x_n ∣ r⟩ is topologically isomorphic to ⟨x₁, …, x_n ∣ x₁^{2 + 2^f} (x₁, x₂) ⋯ (x_{n-1}, x_n)⟩.

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theorem TauCeti.IsDemushkin.exists_continuousMulEquiv_presentedProP_demushkinWordNeTwo_of_range_eq {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin 2 G) (hn : Even (demushkinRank hG)) {f : ℕ} (hf : 2 ≤ f) {w : ℤ_[2]ˣ} (hw : ↑w = -1 + 2 ^ f) (hA : (demushkinCharacter hG).range = (Subgroup.zpowers w).topologicalClosure) :
Nonempty (G ≃ₜ* presentedProP 2 (Fin (demushkinRank hG)) {demushkinWordNeTwo (2 + 2 ^ f) (demushkinRank hG) (freeProPGen 2 (demushkinRank hG))})

Labute's normal form for a dyadic Demushkin group of even rank with twisted image, intrinsic form (Labute, Theorem 5). A Demushkin group G at p = 2 of even rank whose canonical character has image the twisted subgroup U^[f], the closed subgroup generated by -1 + 2^f, for a finite f ≥ 2, is topologically isomorphic to ⟨x₁, …, x_n ∣ x₁^{2 + 2^f} (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n)⟩ on n = demushkinRank hG generators.

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theorem TauCeti.IsDemushkin.nonempty_continuousMulEquiv_of_even_demushkinRank_of_range_eq {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {H : Type v} [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [CompactSpace H] [TotallyDisconnectedSpace H] (hG : IsDemushkin 2 G) (hH : IsDemushkin 2 H) (hn : demushkinRank hG = demushkinRank hH) (heven : Even (demushkinRank hG)) {f : ℕ} (hf : 2 ≤ f) {w : ℤ_[2]ˣ} (hw : ↑w = -1 + 2 ^ f) (hA : (demushkinCharacter hG).range = (Subgroup.zpowers w).topologicalClosure) (hB : (demushkinCharacter hH).range = (Subgroup.zpowers w).topologicalClosure) :
Nonempty (G ≃ₜ* H)

Uniqueness of the dyadic Demushkin groups of even rank with twisted image (Labute, Theorem 5). Two Demushkin groups at p = 2 of the same even rank whose canonical characters have the same twisted image U^[f], f ≥ 2 finite, are topologically isomorphic.

The exponent within its valuation and the level above v₂(α) #

In the even-rank dyadic word x₁^{2+α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{n-1}, x_n) on n ≥ 4 generators with a natural exponent α of exact divisibility depth g ≥ 2, that is 2^g ∣ α and 2^{g+1} ∤ α, every finite level f > g presents the group of the rank-n level-∞ word with α = 2^g, x₁^{2 + 2^g} (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n): both are Demushkin groups of rank n whose canonical characters have image U^[g], so the uniqueness theorem identifies them. In rank two the word x₁^{2+α} (x₁, x₂) likewise presents the group of the rank-two word x₁^{2 + 2^g} (x₁, x₂), both Demushkin groups of rank 2 with image U^[g].

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theorem TauCeti.nonempty_continuousMulEquiv_presentedProP_demushkinWordTwoEven_of_not_dvd {n : ℕ} (hn : Even n) (hn₃ : 3 < n) {a g f : ℕ} (hg : 2 ≤ g) (hag : 2 ^ g ∣ a) (hag' : ¬2 ^ (g + 1) ∣ a) (hgf : g < f) :
Nonempty (presentedProP 2 (Fin n) {demushkinWordTwoEven a f n (freeProPGen 2 n)} ≃ₜ* presentedProP 2 (Fin n) {demushkinWordNeTwo (2 + 2 ^ g) n (freeProPGen 2 n)})

The exponent within its valuation and the level above it are free (corollary to Labute, Theorems 4 and 5). For n ≥ 4 even, a natural exponent α of exact divisibility depth g ≥ 2 and a finite level f > g, the pro-2 group presented on n generators by x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{n-1}, x_n) is topologically isomorphic to the one presented by x₁^{2 + 2^g} (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n): both are Demushkin groups of rank n whose canonical characters have image the twisted subgroup U^[g].

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theorem TauCeti.nonempty_continuousMulEquiv_presentedProP_demushkinWordTwoRankTwo_of_not_dvd {a g : ℕ} (hg : 2 ≤ g) (hag : 2 ^ g ∣ a) (hag' : ¬2 ^ (g + 1) ∣ a) :
Nonempty (presentedProP 2 (Fin 2) {demushkinWordTwoRankTwo a (freeProPGen 2 2)} ≃ₜ* presentedProP 2 (Fin 2) {demushkinWordNeTwo (2 + 2 ^ g) 2 (freeProPGen 2 2)})

The exponent within its valuation is free in rank two (corollary to Labute, Theorems 4 and 5). For a natural exponent α of exact divisibility depth g ≥ 2, the pro-2 group presented on two generators by x₁^{2 + α} (x₁, x₂) is topologically isomorphic to the one presented by x₁^{2 + 2^g} (x₁, x₂): both are Demushkin groups of rank 2 whose canonical characters have image the twisted subgroup U^[g].