The abelianization of a presented pro-p group, and the one-relator structure theorem #
Let G = presentedProP p X rels be the pro-p group presented on a finite type X by a set of
relators rels ⊆ F = freeProP p X. Abelianizing the presentation gives a continuous surjection
abelianizationHom rels : ℤ_p^X → G^{ab}, u ↦ ∏ x, x_x ^ (u x),
the composite of the abelianization isomorphism F^{ab} ≅ ℤ_p^X of
TauCeti.freeProP.abelianizationEquiv with the map F^{ab} → G^{ab} induced by the presentation,
whose kernel is the image of the closed normal closure of the relators.
For a single relator r the kernel is computed exactly: it is the ℤ_p-span of the exponent
vector v = exponentSum r of the relator, because the closed normal closure of r maps onto the
p-adic powers of the class of r, which are the ℤ_p-multiples of v
(abelianizationHom_ofAdd_eq_one_iff). Writing v = q • w with w x₀ = 1 — which is possible as
soon as the coordinate v x₀ divides all the others, and some coordinate does since ℤ_p is a
valuation ring — the change of basis TauCeti.LinearEquiv.piSplitAt then identifies
ℤ_p^X ⧸ ℤ_p v with ℤ_p^{X ∖ {x₀}} × ℤ_p ⧸ (q). This is the abelianization structure theorem
for one-relator pro-p groups (oneRelatorAbelianizationEquiv):
G^{ab} ≅ ℤ_p^{n-1} × ℤ_p ⧸ q ℤ_p, n = #X, q = v x₀.
Here q : ℤ_p is a coordinate of the exponent vector, determined by the relator only up to a
unit of ℤ_p; the factor ℤ_p ⧸ q ℤ_p depends only on the ideal q ℤ_p, that is on the valuation
of q. One has q = 0 exactly when the relator lies in the closed commutator subgroup, and then
G^{ab} ≅ ℤ_p^n is torsion-free. When q ≠ 0 the factor ℤ_p ⧸ q ℤ_p is finite cyclic of order
p^{v_p(q)} and is the torsion subgroup of G^{ab}. For a Demushkin group, whose minimal
presentation has a single relator, this is Labute's description G ⧸ [G, G] ≅ ℤ_p^{n-1} ⊕ ℤ ⧸ q(G)
of the abelianization: the integer invariant q(G) ∈ {0} ∪ p^ℕ of the classification is not the
coordinate q itself but the normalised generator p^{v_p(q)} of the ideal q ℤ_p, that is the
order of the torsion subgroup of G^{ab}, and it is 0 when q = 0.
Main definitions #
TauCeti.presentedProP.abelianizationHom: the continuous surjectionℤ_p^X → G^{ab}induced by a presentation.TauCeti.presentedProP.oneRelatorAbelianizationEquiv: forG = ⟨X ∣ r⟩withexponentSum r = q • wandw x₀ = 1, the topological isomorphismG^{ab} ≃ₜ* ℤ_p^{X ∖ {x₀}} × ℤ_p ⧸ (q).
Main results #
TauCeti.presentedProP.abelianizationHom_surjective,TauCeti.presentedProP.abelianizationHom_ofAdd: the map is surjective and isu ↦ ∏ x, x_x ^ (u x).TauCeti.presentedProP.abelianizationHom_ofAdd_eq_one_iff: for a single relatorr, the kernel is theℤ_p-span of the exponent vector ofr.TauCeti.presentedProP.oneRelatorAbelianizationEquiv_abelianizationHom_ofAdd: the isomorphism composed withabelianizationHomis the reductionTauCeti.LinearMap.piSplitAtQuotof the exponent vectors.TauCeti.presentedProP.oneRelatorAbelianizationEquiv_mk_of_ne,TauCeti.presentedProP.oneRelatorAbelianizationEquiv_mk_of_self,TauCeti.presentedProP.oneRelatorAbelianizationEquiv_symm_ofAdd_mk: the values of the isomorphism on the generators and of its inverse.TauCeti.presentedProP.oneRelator_q_eq_zero_iff_mem_topologicalClosure_commutator:q = 0exactly when the relator lies in the closed commutator subgroup of the free pro-pgroup.TauCeti.presentedProP.exists_nonempty_oneRelatorAbelianizationEquiv: for nonemptyX, some coordinatex₀of the exponent vector divides all the others, and the isomorphism exists withqthat coordinate.TauCeti.presentedProP.exists_continuousMulEquiv_singleton_padicPow_mul: a one-relator pro-pgroup⟨X ∣ r⟩is presented on the same generators by a relatorx₀ ^ q · cwithcin the closed commutator subgroup, for any prescribed generatorx₀andqa coordinate of the exponent vector ofrdividing all the others.
References #
- J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, p. 106.
- J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., Chapter III, §9.
The abelianization map of a presentation. For G = presentedProP p X rels, the continuous
homomorphism ℤ_p^X → G^{ab} sending u to ∏ x, x_x ^ (u x), the product of the p-adic
powers of the classes of the generators (TauCeti.presentedProP.abelianizationHom_ofAdd). It is
the composite of the inverse of TauCeti.freeProP.abelianizationEquiv with the map
F^{ab} → G^{ab} induced by the presentation, and it is surjective.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The abelianization map of a presentation sends the coordinate vector at x to the class of the
generator at x.
The abelianization map of a presentation is surjective.
The abelianization map of a presentation is u ↦ ∏ x, x_x ^ (u x), the product of the
p-adic powers of the classes of the generators in the abelian pro-p group G^{ab}.
The kernel of the abelianization map of a one-relator presentation is the ℤ_p-span of the
exponent vector of the relator: for G = ⟨X ∣ r⟩ and u : X → ℤ_p, the element ∏ x, x_x ^ (u x)
of G^{ab} is trivial exactly when u is a ℤ_p-multiple of exponentSum r.
The abelianization structure theorem for one-relator pro-p groups. Let
G = presentedProP p X {r} be the pro-p group on a finite type X with the single relator r,
and write the exponent vector of r as exponentSum r = q • w with w x₀ = 1. Then
G^{ab} ≃ₜ* ℤ_p^{X ∖ {x₀}} × ℤ_p ⧸ q ℤ_p
as topological groups, the isomorphism sending the class of the generator at x ≠ x₀ to the
coordinate vector at x and the class of the generator at x₀ to (-w, 1). When q ≠ 0 the
factor ℤ_p ⧸ q ℤ_p is finite cyclic and is the torsion subgroup of G^{ab}; when q = 0 it is
ℤ_p and G^{ab} ≅ ℤ_p^X is torsion-free.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The abelianization isomorphism of a one-relator group sends the class of mk y to the
reduction of the exponent vector of y.
The abelianization isomorphism of a one-relator group composed with the abelianization map of
the presentation is the reduction TauCeti.LinearMap.piSplitAtQuot of the exponent vectors: the
element ∏ x, x_x ^ (u x) of G^{ab} is sent to the class of u.
The abelianization isomorphism of a one-relator group sends the class of the generator at
x ≠ x₀ to the coordinate vector at x.
The abelianization isomorphism of a one-relator group sends the class of the generator at x₀
to (-w, 1).
The inverse of the abelianization isomorphism of a one-relator group: the class of (a, b) is
sent to ∏ x, x_x ^ (u x) for u the vector with coordinates a away from x₀ and b along
w, that is u = (piSplitAt x₀ w).symm (a, b).
The torsion-free case of the structure theorem. The coordinate q of the exponent vector
exponentSum r = q • w, w x₀ = 1, of the relator vanishes exactly when r lies in the closed
commutator subgroup of the free pro-p group; then the factor ℤ_p ⧸ q ℤ_p of
oneRelatorAbelianizationEquiv is ℤ_p and G^{ab} ≅ ℤ_p^X is torsion-free.
Existence form of the abelianization structure theorem. For a one-relator pro-p group
G = ⟨X ∣ r⟩ on a finite nonempty type, some coordinate x₀ of the exponent vector v of r
divides all the others, and G^{ab} ≅ ℤ_p^{X ∖ {x₀}} × ℤ_p ⧸ v x₀ ℤ_p.
A one-relator pro-p group is presented by a relator x₀ ^ q · c with c a commutator
element. For G = ⟨X ∣ r⟩ on a finite type and a generator x₀, some coordinate q = v x₁ of
the exponent vector v of r divides all the others, and there is c in the closed commutator
subgroup of the free pro-p group with G ≃ₜ* ⟨X ∣ x₀ ^ q · c⟩. The exponent vector of
x₀ ^ q · c is q e_{x₀}, so the abelianization structure theorem oneRelatorAbelianizationEquiv
applies to the new presentation with w = e_{x₀}.