The canonical characters of the Demushkin normal forms #

Labute's classification attaches to a Demushkin group G its canonical character, the unique continuous χ : G → ℤ_pˣ with the prescription property (TauCeti.HasPrescriptionProperty): every reduction H¹(G, I(χ)/pⁱ) → H¹(G, I(χ)/p) of the twisted coefficients is surjective. This file computes that character on each of the normal-form presentations of TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Basic and on the presentation by the even dyadic word with a 2-adic exponent of TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Two.Even.PadicExponent.Basic, proving Labute's Theorem 4 for the presented groups: each normal form has exactly one continuous character with the prescription property, the standard orientation of TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Orientation, whose values on the generators are

q ≠ 2, relator x₁^q (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n): χ(x₂) = (1 - q)⁻¹, all other χ(x_i) = 1;
q = 2, n odd, relator x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n): χ(x₁) = -1, χ(x₃) = (1 - 2^f)⁻¹, all other χ(x_i) = 1; at level f = ∞, on the relator x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n), χ(x₁) = -1 and all other χ(x_i) = 1;
q = 2, n even, relator x₁^{2+α} (x₁, x₂) x₃^q (x₃, x₄) ⋯ (x_{n-1}, x_n) with a 2-adic exponent α and tail exponent q: χ(x₂) = -(1 + α)⁻¹, χ(x₄) = (1 - q)⁻¹ whenever the factor x₃^q is present (2 < n), all other χ(x_i) = 1; the natural-exponent relator x₁^{2+a} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{n-1}, x_n) is the case α = a, q = 2^f, with χ(x₂) = -(1 + a)⁻¹ and χ(x₄) = (1 - 2^f)⁻¹; in rank two, on the word x₁^{2+a} (x₁, x₂) with no level, χ(x₁) = 1 and χ(x₂) = -(1 + a)⁻¹.

The values are stated as equations in ℤ_p, χ(x₂) (1 - q) = 1 and so on, so that no inverse has to be formed to state them.

The proof is the forced computation on a derivation. For a presented pro-p group the prescription property of χ says that every continuous crossed homomorphism F of the free group for χ ∘ mk, a continuous F with F (x * y) = χ x * F y + F x, kills the relator (TauCeti.presentedProP.hasPrescriptionProperty_iff_forall_isCrossedHom_eq_zero). On a relator word, F is computed from F (x ^ k) = (1 + χ x + ⋯ + χ x ^ (k-1)) F x and χ x * χ y * F (x, y) = (1 - χ y) F x + (χ x - 1) F y on Labute's commutator (x, y) = x⁻¹y⁻¹xy: its value is an ℤ_p-linear form in the values F (x_i), with coefficients polynomial in the χ(x_i). Since F takes any prescribed values on the generators, the form must vanish identically. Reading off its coefficients, one generator at a time, forces the values of χ above: the commutator factor (x_a, x_b) containing x_j gives χ(x_b) = 1 or χ(x_a) = 1, and the generator carrying the p-th power gives the equation q + χ(x₂)⁻¹ - 1 = 0, respectively 1 + χ(x₁) = 0 or 2 + a + χ(x₂)⁻¹ - 1 = 0. Conversely, at the tabulated values the form vanishes, so the standard orientation has the prescription property.

Main results #

TauCeti.IsCrossedHom.mul_mul_map_labuteComm: the value of a crossed homomorphism on Labute's commutator; TauCeti.IsCrossedHom.map_demushkinWordNeTwo, TauCeti.IsCrossedHom.map_demushkinWordTwoOdd, TauCeti.IsCrossedHom.map_demushkinWordTwoEven, TauCeti.IsCrossedHom.map_demushkinWordTwoRankTwo, TauCeti.IsCrossedHom.map_demushkinWordTwoOddTop: its value on the five normal-form words; TauCeti.IsCrossedHom.map_demushkinWordTwoEvenPadic: its value on the even dyadic word with a 2-adic exponent; TauCeti.IsCrossedHom.map_demushkinWordNeTwo_eq_zero: at the tabulated character values the q ≠ 2 word is killed, on any tuple.
TauCeti.hasPrescriptionProperty_presentedProP_demushkinWordNeTwo_iff, TauCeti.hasPrescriptionProperty_presentedProP_demushkinWordTwoOdd_iff, TauCeti.hasPrescriptionProperty_presentedProP_demushkinWordTwoEven_iff, TauCeti.hasPrescriptionProperty_presentedProP_demushkinWordTwoRankTwo_iff, TauCeti.hasPrescriptionProperty_presentedProP_demushkinWordTwoOddTop_iff, TauCeti.hasPrescriptionProperty_presentedProP_demushkinWordTwoEvenPadic_iff: a continuous character of a normal-form presentation has the prescription property exactly when it takes the tabulated values; the _of_apply_eq versions are the existence halves, without the minimality and rank hypotheses. The two theorems for the even dyadic word with natural exponent are the case α = a, q = 2^f of those for the word with a 2-adic exponent.
TauCeti.hasPrescriptionProperty_orientationNeTwo, TauCeti.hasPrescriptionProperty_orientationTwoOdd, TauCeti.hasPrescriptionProperty_orientationTwoEven, TauCeti.hasPrescriptionProperty_orientationTwoRankTwo, TauCeti.hasPrescriptionProperty_orientationTwoOddTop: the standard orientations have the prescription property; TauCeti.eq_orientationNeTwo_of_hasPrescriptionProperty, TauCeti.eq_orientationTwoOdd_of_hasPrescriptionProperty, TauCeti.eq_orientationTwoEven_of_hasPrescriptionProperty, TauCeti.eq_orientationTwoRankTwo_of_hasPrescriptionProperty, TauCeti.eq_orientationTwoOddTop_of_hasPrescriptionProperty: a character with the prescription property is the orientation with its own marked values.
TauCeti.existsUnique_hasPrescriptionProperty_presentedProP_demushkinWordNeTwo, TauCeti.existsUnique_hasPrescriptionProperty_presentedProP_demushkinWordTwoOdd, TauCeti.existsUnique_hasPrescriptionProperty_presentedProP_demushkinWordTwoEven, TauCeti.existsUnique_hasPrescriptionProperty_presentedProP_demushkinWordTwoRankTwo, TauCeti.existsUnique_hasPrescriptionProperty_presentedProP_demushkinWordTwoOddTop: each normal form has exactly one continuous character with the prescription property; for the odd word at f = ∞ this holds in every odd rank, including rank one. For the even dyadic word with a 2-adic exponent, the orientation and the uniqueness statement are in TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Two.Even.PadicExponent.Character.
TauCeti.hasPrescriptionProperty_presentedProP_demushkinWordTwoOdd_one_iff and TauCeti.existsUnique_hasPrescriptionProperty_presentedProP_demushkinWordTwoOdd_one: in rank one, where the odd word reads x₁² and the group is ℤ/2, the prescription property means χ(x₁) = -1, and the sign character is the unique character with it.

References #

J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §2, Proposition 6 and Theorem 4, and Remark 2.
J.-P. Serre, Structure de certains pro-p-groupes, Séminaire Bourbaki 252 (1962/63).

The canonical characters of the Demushkin normal forms #

Main results #

References #

Crossed homomorphisms on Labute's commutator and on the normal-form words #

Crossed homomorphisms on the even dyadic word with a 2-adic exponent #

The q ≠ 2 normal form #

The q = 2, n odd normal form #

The q = 2, n odd normal form at level f = ∞ #

The q = 2, n even normal form with a 2-adic exponent #

The q = 2, n even normal form #

The q = 2 normal form of rank two #

Crossed homomorphisms on the even dyadic word with a `2`-adic exponent #

The `q ≠ 2` normal form #

The `q = 2`, `n` odd normal form #

The `q = 2`, `n` odd normal form at level `f = ∞` #

The `q = 2`, `n` even normal form with a `2`-adic exponent #

The `q = 2`, `n` even normal form #

The `q = 2` normal form of rank two #