The image of a prescribed character of a one-relator pro-p group

For a one-relator presentation G = ⟨X ∣ r⟩ with finite X, r in the Frattini subgroup and nondegenerate degree-one form, a character with Labute's prescription property takes values in 1 + p^kℤ_p exactly when p^k divides every exponent sum of r.

Main result

• TauCeti.HasPrescriptionProperty.range_le_unitsPrincipal_iff_forall_pow_dvd_exponentSum: for finite X, character image containment is equivalent to divisibility of every exponent sum.

References

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

theorem TauCeti.HasPrescriptionProperty.range_le_unitsPrincipal_iff_forall_pow_dvd_exponentSum {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} {r : freeProP p X} [Finite X] (hr : r ∈ proPFrattini p (freeProP p X)) (hnd : (freeProP.degreeOneForm (gradedMk p (freeProP p X) 1 ⟨r, ⋯⟩)).Nondegenerate) {χ : presentedProP p X {r} →ₜ* ℤ_[p]ˣ} (hχ : HasPrescriptionProperty χ) (k : ℕ) : χ.range ≤ unitsPrincipal p k ↔ ∀ (x : X), ↑p ^ k ∣ Multiplicative.toAdd ((freeProP.exponentSum p X) r) x

The level of the character with the prescription property is the content of the exponent vector of the relator. Let G = ⟨X ∣ r⟩ be a one-relator pro-p group with finite X and r ∈ Φ(F) whose class in gr_1(F) has nondegenerate degree-one form, and let χ : G → ℤ_pˣ be a continuous character with the prescription property. Then χ takes values in 1 + p^kℤ_p exactly when p^k divides the exponent sum of r at every generator.

theorem TauCeti.HasPrescriptionProperty.not_isAlt_degreeOneForm_of_not_range_le_unitsPrincipal_two {X : Type u} [Finite X] {r : freeProP 2 X} (hr : r ∈ proPFrattini 2 (freeProP 2 X)) (hnd : (freeProP.degreeOneForm (gradedMk 2 (freeProP 2 X) 1 ⟨r, ⋯⟩)).Nondegenerate) {χ : presentedProP 2 X {r} →ₜ* ℤ_[2]ˣ} (hχ : HasPrescriptionProperty χ) (h : ¬χ.range ≤ unitsPrincipal 2 2) : ¬(freeProP.degreeOneForm (gradedMk 2 (freeProP 2 X) 1 ⟨r, ⋯⟩)).IsAlt

At p = 2, a character with the prescription property taking a value outside 1 + 4ℤ_2 has nonalternating degree-one form. For G = ⟨X ∣ r⟩ with r ∈ Φ(F) of nondegenerate degree-one form, an alternating form has vanishing diagonal, that is, 4 divides every exponent sum of r, which puts the image of every character with the prescription property inside 1 + 4ℤ_2.