Basis modifications of a free pro-`p` group and the maps `δ` #

Let F = freeProP p X be the free pro-p group on a finite linearly ordered type X, with canonical generators x_i = freeProP.of i, and let λ_k = λ_k(F) be its lower p-series. A family w : X → λ_m(F) defines the basis modification θ_w : F → F, x_i ↦ x_i * w_i (TauCeti.freeProP.basisModification). It is congruent to the identity modulo λ_m, so for a relator r ∈ λ_1(F) it moves r inside its coset by the element r⁻¹ * θ_w r ∈ λ_{m+1}(F), whose class in gr_{m+1}(F) is the graded deviation D_1 ρ of θ_w (TauCeti.gradedDeviation) on the class ρ ∈ gr_1(F) of r.

For m ≥ 1 that class is given by the basis-modification map δ = TauCeti.freeProP.basisModificationDelta: writing ρ in the standard basis TauCeti.freeProP.degreeOneBasis as ρ = Σ_i c_i π ξ_i + Σ_{i<k} a_{ik} [ξ_i, ξ_k], where ξ_i ∈ gr_0(F) is the class of x_i, and writing ω_i ∈ gr_m(F) for the class of w_i, the class of r⁻¹ * θ_w r in gr_{m+1}(F) is

δ(ω) = Σ_i c_i (π ω_i + (p choose 2) • [ω_i, ξ_i]) + Σ_{i<k} a_{ik} ([ω_i, ξ_k] - [ω_k, ξ_i]),

an 𝔽_p-linear function of the classes ω_i alone, and 𝔽_p-linear in ρ as well. The bracket part is the derivative of the commutator part of ρ in the direction ω, and for odd p the p-power part contributes Σ_i c_i π ω_i. For p = 2 the p-power part contributes in addition the brackets Σ_i c_i [ω_i, ξ_i]: the square of x_i * w_i is x_i ^ 2 * w_i ^ 2 * ⁅w_i, x_i⁆ up to λ_{m+2}(F), and the commutator ⁅w_i, x_i⁆ lies in λ_{m+1}(F) and may have a nonzero class in gr_{m+1}(F). That term is the trace, in every degree, of the failure of additivity of π on gr_0(F) at p = 2.

At level m = 0, where θ_w is an arbitrary continuous endomorphism of F, the class of r⁻¹ * θ_w r in gr_1(F) is still a function of the classes ω_i ∈ gr_0(F) alone, but a quadratic one; that map and its polarization identity are in TauCeti.Topology.Algebra.Group.Profinite.Free.BasisModification.LevelZero.

The image of δ is the subspace of gr_{m+1}(F) that the successive-approximation arguments of the classification of Demushkin groups compare with gr_{m+1}(F); there m + 1 is the modulus of the normal-form congruence, and the classes ω_i are the level-m basis corrections.

For p = 2 those arguments compare gr_j(F) with the image of δ enlarged by one further subspace, the span in gr_j(F) of the iterated p-powers π^j ξ_i over a set S of generators (TauCeti.freeProP.gradedPowIterSpan). The tail T_j(ρ) (TauCeti.freeProP.basisModificationTail) is the instance at the generators whose coefficient c_i in ρ vanishes, which are the generators contributing no π-term to δ. For the dyadic relator x₁² x₂^{2^f} ⁅x₂, x₃⁆ ⋯ with f ≥ 2 these are x₂, …, x_n; for the even-rank relator x₁^{2+α} ⁅x₁, x₂⁆ x₃^{2^f} ⋯ the right index set is instead the complement of x₂, which is not a tail.

Collecting the brackets of δ by their degree-m entry gives the partial derivatives ∂_i ρ ∈ gr_0(F) (TauCeti.freeProP.degreeOneDeriv), the i-th row of the matrix (a_{ik}) completed skew-symmetrically with diagonal (p choose 2) c_i, and the formula δ_ρ(ω) = π (Σ_i c_i ω_i) + Σ_i [ω_i, ∂_i ρ]. The image of δ_ρ is then computed under the hypothesis that the derivatives ∂_i ρ span gr_0(F), which is the nondegeneracy of the form (a_{ik}) completed with that diagonal, and holds for every Demushkin relator in normal form: the coordinates of ∂_i ρ in the basis of generator classes form the i-th row of the matrix of the degree-one form TauCeti.freeProP.degreeOneForm ρ in the dual basis of the generators, so the derivatives span exactly when that form is nondegenerate. The span statements are: if all c_i = 0, then Im δ_ρ is the span of the brackets [gr_m(F), gr_0(F)] and gr_{m+1}(F) = Im δ_ρ + T_{m+1}(ρ) for every m ≥ 1, the tail being spanned by all the π^{m+1} ξ_i, so that Im δ_ρ consists exactly of the classes of the elements of λ_{m+1}(F) all of whose exponent sums are divisible by p ^ (m + 2); and if p is odd and some c_i ≠ 0, then gr_{m+1}(F) = Im δ_ρ for every m ≥ 1. The first case is that of the relators x₁^q (x₁, x₂) (x₃, x₄) ⋯ with q ≠ p, whose p-power part lies in λ_2(F), and the second that of x₁^p (x₁, x₂) (x₃, x₄) ⋯ at odd p. Both rest on the spanning of gr_{m+1}(F) by π gr_m(F) and [gr_m(F), gr_0(F)] and on the naturality π ∘ δ_ρ = δ_ρ ∘ π, which carries the image of δ_ρ in degree m into its image in degree m + 1. For p = 2 and a class with a p-power part the odd-p argument breaks down at the degree-zero defect of π. It survives when some generator class ξ_{i₁} has the column B_ρ(χ_i, χ_{i₁}) of the degree-one form at its coordinate character equal to the vector c_i of 2-power coefficients: then gr_{m+1}(F) = Im δ_ρ + ⟨π^{m+1} ξ_i : i ≠ i₁⟩. The key membership is ψ(y) • π v + [v, y] ∈ Im δ_ρ for ψ the coordinate at ξ_{i₁}, which makes every bracket [v, y] available once π v is, and the degree-zero defect of π is absorbed by the brackets [[ξ_a, y], ξ_a] with a ≠ i₁. For the dyadic relators x₁² x₂^{2^f} (x₂, x₃) ⋯ of odd rank, x₁ carries the 2-power part, ξ₁ occurs in no bracket and i₁ is the index of x₁, so the span is the tail T_{m+1}(ρ) and the level f is the free parameter it accounts for. For the even-rank relators x₁^{2+α} (x₁, x₂) x₃^{2^f} ⋯, x₁ carries the 2-power part and i₁ is the index of its bracket partner x₂, so the spanning powers include π^{m+1} ξ₁ although c₁ ≠ 0, and exclude π^{m+1} ξ₂.

Main definitions #

TauCeti.freeProP.basisModification: the endomorphism θ_w : F → F, x_i ↦ x_i * w_i.
TauCeti.freeProP.basisModificationEquiv: for m ≥ 1 and finite X, the same map as a continuous automorphism of F.
TauCeti.freeProP.basisModificationDelta: for m ≥ 1, the 𝔽_p-bilinear map δ : gr_1(F) → gr_m(F)^X → gr_{m+1}(F), (ρ, ω) ↦ δ_ρ(ω).
TauCeti.freeProP.basisModificationTail: the tail T_j(ρ) ≤ gr_j(F), the span TauCeti.freeProP.gradedPowIterSpan of the π^j ξ_i with c_i = 0.
TauCeti.freeProP.degreeOneDeriv: the partial derivative ∂_i : gr_1(F) →ₗ[𝔽_p] gr_0(F).

Main results #

TauCeti.freeProP.inv_mul_basisModification_mem_pLowerCentralSeries: θ_w is congruent to the identity modulo λ_m(F).
TauCeti.freeProP.toAdd_exponentSum_basisModification: the exponent vector of θ_w g is Σ_i (exponentSum g)_i • (e_i + exponentSum w_i).
TauCeti.freeProP.exponentSum_basisModification: θ_w preserves the exponent vector of g when w_i lies in the closed commutator subgroup at every generator carrying a nonzero exponent of g.
TauCeti.freeProP.gradedDeviation_basisModification, TauCeti.freeProP.gradedMk_inv_mul_basisModification: the class of r⁻¹ * θ_w r in gr_{m+1}(F) is δ(ω); in particular it depends only on the classes ω_i.
TauCeti.freeProP.finrank_basisModificationTail: dim T_j(ρ) is the number of indices i with c_i = 0.
TauCeti.freeProP.basisModificationTail_succ: π carries T_j(ρ) onto T_{j+1}(ρ) for j ≥ 1.
TauCeti.freeProP.basisModificationDelta_eq_gradedPow_add_sum, TauCeti.freeProP.gradedPow_basisModificationDelta: δ_ρ(ω) = π (Σ_i c_i ω_i) + Σ_i [ω_i, ∂_i ρ], and π (δ_ρ(ω)) = δ_ρ(π ω).
TauCeti.freeProP.exists_basisModificationDelta_smul_eq, TauCeti.freeProP.basisModificationDelta_smul_eq_gradedBracket_of_eq_zero: on a family b • v proportional to one class, δ_ρ(b • v) = c • π v + [v, y] for any prescribed y when the derivatives span, and δ_ρ(b • v) = [v, Σ_i b_i • ∂_i ρ] when b vanishes at the only generator carrying a p-power coefficient.
TauCeti.freeProP.degreeZeroBasis_repr_degreeOneDeriv, TauCeti.freeProP.span_range_degreeOneDeriv_eq_top_iff_nondegenerate_degreeOneForm: the coordinates of ∂_i ρ are the i-th row of the matrix of the degree-one form of ρ, so the derivatives span gr_0(F) exactly when that form is nondegenerate.
TauCeti.freeProP.range_basisModificationDelta_eq_span_of_repr_inl_eq_zero, TauCeti.freeProP.range_basisModificationDelta_sup_basisModificationTail_eq_top: for a class without p-power part whose derivatives span gr_0(F), Im δ_ρ = [gr_m(F), gr_0(F)] and gr_{m+1}(F) = Im δ_ρ + T_{m+1}(ρ).
TauCeti.freeProP.gradedMk_mem_range_basisModificationDelta_iff: for a class without p-power part whose derivatives span gr_0(F), the class of z ∈ λ_{m+1}(F) lies in Im δ_ρ if and only if p ^ (m + 2) divides every exponent sum of z; so Im δ_ρ is the kernel of the map to gr_{m+1}(F^{ab}).
TauCeti.freeProP.range_basisModificationDelta_eq_top_of_odd: for odd p and a class with a p-power part whose derivatives span gr_0(F), gr_{m+1}(F) = Im δ_ρ.
TauCeti.freeProP.range_basisModificationDelta_sup_gradedPowIterSpan_compl_eq_top_two: for p = 2 and a class whose derivatives span gr_0(F), and a generator class ξ_{i₁} whose column B_ρ(χ_i, χ_{i₁}) of the degree-one form is the vector of 2-power coefficients of the class, gr_{m+1}(F) = Im δ_ρ + ⟨π^{m+1} ξ_i : i ≠ i₁⟩; for a class whose 2-power part sits on a single generator x_{i₁} not occurring in the brackets, this is gr_{m+1}(F) = Im δ_ρ + T_{m+1}(ρ).

References #

J. Labute, Classification of Demushkin groups, Canadian J. Math. 19 (1967), §3, Proposition 5 and the proof of Theorem 3.

Basis modifications of a free pro-p group and the maps δ #

Main definitions #

Main results #

References #

The basis modification x_i ↦ x_i * w_i #

The basis modification as an automorphism #

The maps δ #

The partial derivatives ∂_i #

The derivatives and the degree-one form #

The tails T_j #

The image of δ #

The image of δ and the exponent sums #

The dyadic span statements #

Basis modifications of a free pro-`p` group and the maps `δ` #

The basis modification `x_i ↦ x_i * w_i` #

The maps `δ` #

The partial derivatives `∂_i` #

The tails `T_j` #

The image of `δ` #

The image of `δ` and the exponent sums #