The prescription property of a presented pro-p group

Let G = ⟨X ∣ rels⟩ be the pro-p group presented on a finite type X by a set of relators rels in the free pro-p group F = freeProP p X, and let χ : G →ₜ* ℤ_pˣ be a continuous character. Labute's prescription property of χ (TauCeti.HasPrescriptionProperty) says that every reduction H¹(G, I(χ)/pⁱ) → H¹(G, I(χ)/p) is surjective; for a topologically finitely generated pro-p group it says that continuous crossed homomorphisms G → ℤ_p for χ take any prescribed values on a minimal generating tuple.

For a presented group this becomes a condition on the relators alone. A continuous crossed homomorphism F → ℤ_p for the character χ ∘ mk of the free group is determined by its values on the generators and takes any prescribed values there, and it descends to G exactly when it kills the relators. Hence χ has the prescription property if and only if every continuous crossed homomorphism F → ℤ_p for χ ∘ mk vanishes on every relator (TauCeti.presentedProP.hasPrescriptionProperty_iff_forall_isCrossedHom_eq_zero), when the presentation is minimal, that is when the relators lie in the Frattini subgroup of F; the sufficiency of the condition needs no minimality. For a one-relator group this is the vanishing of a single ℤ_p-linear form in the values on the generators, whose coefficients are polynomial in the values of χ; computing it on the normal-form relators of the Demushkin groups is how the canonical character of each normal form is found (Labute, Theorem 4).

Main results

• TauCeti.presentedProP.exists_continuous_isCrossedHom_comp_mk_eq: a continuous crossed homomorphism of the free group for χ ∘ mk that kills the relators descends to a continuous crossed homomorphism of the presented group for χ.
• TauCeti.presentedProP.hasPrescriptionProperty_of_forall_isCrossedHom_eq_zero: if every continuous crossed homomorphism of the free group for χ ∘ mk kills the relators, then χ has the prescription property.
• TauCeti.presentedProP.hasPrescriptionProperty_iff_forall_isCrossedHom_eq_zero: for a minimal presentation, the converse holds as well.
• TauCeti.HasPrescriptionProperty.exists_continuous_isCrossedHom_comp_mk_forall_apply_of_eq: for a minimal presentation, a character with the prescription property admits a continuous crossed homomorphism of the free group for χ ∘ mk with any prescribed values on the generators that kills the relators; …_forall_freeProPGen_eq_ite is the Kronecker case on a presentation on Fin n, the form in which the relator computations read off one coefficient at a time.

References

• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §2, Proposition 6 and Theorem 4.

theorem TauCeti.presentedProP.exists_continuous_isCrossedHom_comp_mk_eq {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} {rels : Set (freeProP p X)} {χ : presentedProP p X rels →ₜ* ℤ_[p]ˣ} {F : freeProP p X → ℤ_[p]} (hFc : Continuous F) (hF : IsCrossedHom (⇑(χ.comp (mk p rels))) F) (h : ∀ r ∈ rels, F r = 0) : ∃ (F' : presentedProP p X rels → ℤ_[p]), Continuous F' ∧ IsCrossedHom (⇑χ) F' ∧ F' ∘ ⇑(mk p rels) = F

Descent of crossed homomorphisms along a presentation. A continuous crossed homomorphism F : freeProP p X → ℤ_p for the character χ ∘ mk that vanishes on every relator is constant on the cosets of the relation subgroup, so it is F' ∘ mk for a continuous crossed homomorphism F' : ⟨X ∣ rels⟩ → ℤ_p for χ.

theorem TauCeti.presentedProP.hasPrescriptionProperty_of_forall_isCrossedHom_eq_zero {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} {rels : Set (freeProP p X)} {χ : presentedProP p X rels →ₜ* ℤ_[p]ˣ} [Finite X] (h : ∀ (F : freeProP p X → ℤ_[p]), Continuous F → IsCrossedHom (⇑(χ.comp (mk p rels))) F → ∀ r ∈ rels, F r = 0) : HasPrescriptionProperty χ

Crossed homomorphisms killing the relators give the prescription property. If every continuous crossed homomorphism freeProP p X → ℤ_p for χ ∘ mk vanishes on every relator, then χ has the prescription property: any values on the generators are attained by a continuous crossed homomorphism of the free group, which descends to the presented group. No minimality of the presentation is needed.

theorem TauCeti.presentedProP.hasPrescriptionProperty_iff_forall_isCrossedHom_eq_zero {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} {rels : Set (freeProP p X)} [Finite X] (hrels : rels ⊆ ↑(proPFrattini p (freeProP p X))) (χ : presentedProP p X rels →ₜ* ℤ_[p]ˣ) : HasPrescriptionProperty χ ↔ ∀ (F : freeProP p X → ℤ_[p]), Continuous F → IsCrossedHom (⇑(χ.comp (mk p rels))) F → ∀ r ∈ rels, F r = 0

The prescription property of a minimally presented pro-p group is a condition on the relators (Labute, Proposition 6, read on a presentation). For a presentation whose relators lie in the Frattini subgroup of the free pro-p group, a continuous character χ has the prescription property exactly when every continuous crossed homomorphism freeProP p X → ℤ_p for χ ∘ mk vanishes on every relator.

theorem TauCeti.HasPrescriptionProperty.exists_continuous_isCrossedHom_comp_mk_forall_apply_of_eq {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} {rels : Set (freeProP p X)} [Finite X] {χ : presentedProP p X rels →ₜ* ℤ_[p]ˣ} (hχ : HasPrescriptionProperty χ) (hrels : rels ⊆ ↑(proPFrattini p (freeProP p X))) (c : X → ℤ_[p]) : ∃ (F : freeProP p X → ℤ_[p]), Continuous F ∧ IsCrossedHom (⇑(χ.comp (presentedProP.mk p rels))) F ∧ (∀ (x : X), F (freeProP.of x) = c x) ∧ ∀ r ∈ rels, F r = 0

Prescribed values on the generators, killing the relators. For a minimal presentation, a character χ with the prescription property admits, for every c : X → ℤ_p, a continuous crossed homomorphism of the free group for χ ∘ mk taking the value c x at the generator x and vanishing on every relator.

theorem TauCeti.HasPrescriptionProperty.exists_continuous_isCrossedHom_comp_mk_forall_freeProPGen_eq_ite {p : ℕ} [Fact (Nat.Prime p)] {n : ℕ} {rels : Set (freeProP p (Fin n))} {χ : presentedProP p (Fin n) rels →ₜ* ℤ_[p]ˣ} (hχ : HasPrescriptionProperty χ) (hrels : rels ⊆ ↑(proPFrattini p (freeProP p (Fin n)))) {j : ℕ} (hj : j < n) : ∃ (F : freeProP p (Fin n) → ℤ_[p]), Continuous F ∧ IsCrossedHom (⇑(χ.comp (presentedProP.mk p rels))) F ∧ (∀ (i : ℕ), F (freeProPGen p n i) = if i = j then 1 else 0) ∧ ∀ r ∈ rels, F r = 0

The Kronecker crossed homomorphism of a minimal presentation on Fin n, continuous form. For j < n, a character χ with the prescription property admits a continuous crossed homomorphism of the free group for χ ∘ mk taking the value 1 at the j-th ℕ-indexed generator and 0 at every other one, and vanishing on every relator.

theorem TauCeti.HasPrescriptionProperty.exists_isCrossedHom_comp_mk_forall_freeProPGen_eq_ite {p : ℕ} [Fact (Nat.Prime p)] {n : ℕ} {rels : Set (freeProP p (Fin n))} {χ : presentedProP p (Fin n) rels →ₜ* ℤ_[p]ˣ} (hχ : HasPrescriptionProperty χ) (hrels : rels ⊆ ↑(proPFrattini p (freeProP p (Fin n)))) {j : ℕ} (hj : j < n) : ∃ (F : freeProP p (Fin n) → ℤ_[p]), IsCrossedHom (⇑(χ.comp (presentedProP.mk p rels))) F ∧ (∀ (i : ℕ), F (freeProPGen p n i) = if i = j then 1 else 0) ∧ ∀ r ∈ rels, F r = 0

The Kronecker crossed homomorphism of a minimal presentation on Fin n. For j < n, a character χ with the prescription property admits a crossed homomorphism of the free group for χ ∘ mk taking the value 1 at the j-th ℕ-indexed generator and 0 at every other one, and vanishing on every relator.