A section of the peripheral exponent over the principal units

Let F be a free pro-p group of positive rank with a marked basis. The exponent character exponent : peripheralAut hF (basis e) →* ℤ_[p]ˣ is continuous and surjective. This file shows that it splits continuously and homomorphically over every principal unit group U^(k) = 1 + p ^ k ℤ_p with k ≥ 1, and k ≥ 2 when p = 2.

The group peripheralAut hF (basis e) is profinite, but for odd p it is not pro-p, since its exponent maps it onto ℤ_[p]ˣ, which contains the roots of unity of order p - 1. So p-adic powers are not available in it directly. Instead, choose a peripheral automorphism φ whose exponent is a unit w of exact level k, and pass to its p-part q = φ ^ᶻ e_p, a profinite power by the p-adic idempotent of the profinite integers. The closed subgroup generated by q is pro-p, and q still has exponent w, because w lies in the pro-p group U^(k). The section sends v ∈ U^(k) to the p-adic power q ^ λ(v), where λ is the inverse of the topological isomorphism principalUnitsEquiv : Multiplicative ℤ_[p] ≃ₜ* U^(k), l ↦ w ^ l. Two continuous homomorphisms out of ℤ_p agreeing at 1 coincide, so the exponent of q ^ l is w ^ l, and the exponent of the section at v is w ^ λ(v) = v.

No section is claimed over all of ℤ_[p]ˣ. Over the roots of unity a section would be a peripheral automorphism of finite order with prescribed exponent, and nothing here constructs one compatibly.

Main results

• TauCeti.Peripheral.exists_isProP_closedZpowers_and_exponent_eq_iff: the exponents of the peripheral automorphisms generating a pro-p closed subgroup are exactly the units in 1 + pℤ_p.
• TauCeti.Peripheral.exists_section_principalUnits: the exponent character has a continuous homomorphic section over U^(k).

References

• Y. Ihara, "Braids, Galois groups, and some arithmetic functions", Proc. ICM Kyoto 1990, 99–120, for peripheral automorphisms of free pro-p groups and their common exponent.
• L. Ribes and P. Zalesskii, Profinite Groups, Sections 4.1 and 4.3, for the prime parts of elements of a profinite group.

theorem TauCeti.Peripheral.exists_isProP_closedZpowers_and_exponent_eq_iff {p r : ℕ} [Fact (Nat.Prime p)] {F : Type u_1} [Group F] [TopologicalSpace F] [IsTopologicalGroup F] [CompactSpace F] [TotallyDisconnectedSpace F] (hF : IsProP p F) (e : F ≃ₜ* freeProP p (Fin r)) (hr : 0 < r) (w : ℤ_[p]ˣ) : (∃ (q : ↥(peripheralAut hF (basis e))), IsProP p ↥(closedZpowers q) ∧ (exponent hF e hr) q = w) ↔ w ∈ unitsPrincipal p 1

The exponents of pro-p procyclic peripheral automorphisms are the principal units. A unit w is the exponent of a peripheral automorphism q whose closed procyclic subgroup is pro-p exactly when w ∈ 1 + pℤ_p.

theorem TauCeti.Peripheral.exists_section_principalUnits {p r : ℕ} [Fact (Nat.Prime p)] {F : Type u_1} [Group F] [TopologicalSpace F] [IsTopologicalGroup F] [CompactSpace F] [TotallyDisconnectedSpace F] (hF : IsProP p F) (e : F ≃ₜ* freeProP p (Fin r)) (hr : 0 < r) {k : ℕ} (hk : 0 < k) (hk₂ : p = 2 → 2 ≤ k) : ∃ (s : ↥(unitsPrincipal p k) →ₜ* ↥(peripheralAut hF (basis e))), ∀ (v : ↥(unitsPrincipal p k)), (exponent hF e hr) (s v) = ↑v

The section of the exponent over the principal units. Let F be a marked free pro-p group of positive rank, and let k ≥ 1, with k ≥ 2 when p = 2. Then the exponent character of the peripheral automorphism group has a continuous homomorphic section over the principal units U^(k) = 1 + p ^ k ℤ_p: a continuous homomorphism s : U^(k) →ₜ* peripheralAut hF (basis e) with exponent (s v) = v for every v.