Marked normal forms of Demushkin groups #
The normal form of a Demushkin group records more than its abstract isomorphism type: the
isomorphism can be chosen so that the canonical character takes Labute's prescribed values on the
marked generators. This file states the marked classification for the q ≠ 2 family, for the
dyadic family of odd rank, and for the dyadic family of even rank, whose orientation image is
{±1} × U^(f), a twisted subgroup U^[g] of ℤ_2ˣ, or the endpoint {±1}.
For q ≠ 2 the relator is
x₁ ^ q (x₁, x₂) (x₃, x₄) ⋯ (xₙ₋₁, xₙ),
the canonical character takes the value (1 - q)⁻¹ on x₂, and it is trivial on the other
generators. The equation is stated without choosing an inverse, as
χ(x₂) * (1 - q) = 1 in ℤ_p.
For odd rank n, where p = 2 and q = 2, the relator is
x₁² x₂^{2^f} (x₂, x₃) (x₄, x₅) ⋯ (xₙ₋₁, xₙ), 2 ≤ f < ∞, or x₁² (x₂, x₃) ⋯ (xₙ₋₁, xₙ), f = ∞,
the canonical character takes the value -1 on x₁, the value (1 - 2^f)⁻¹ on x₃ at a finite
level, and is trivial on the other generators. The level f is an invariant of the group, the
level of the image {±1} × U^(f) of its canonical character, with f = ∞ the image {±1}; the
statements take the image as a hypothesis, which is what pins f, and the finite-level statement
assumes n ≥ 3, since at rank one the marking clause on the third generator would read
1 - 2^f = 1 (TauCeti.not_marked_of_demushkinRank_le). The standard dyadic group D₀ is the
case n = 3, f = 2.
For even rank n ≥ 4, where p = 2 and q = 2, with the image of the canonical character
{±1} × U^(f), 2 ≤ f < ∞, the relator is
x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (xₙ₋₁, xₙ) for any natural exponent α with 2^f ∣ α,
including Labute's α = 0; the canonical character takes the value -(1 + α)⁻¹ on x₂, the
value (1 - 2^f)⁻¹ on x₄, and is trivial on the other generators. The image pins the level f
but not the exponent α above it: the presented groups for the different α with 2^f ∣ α all
have image {±1} × U^(f) and are isomorphic by Labute's Theorem 6
(TauCeti.IsDemushkin.exists_continuousMulEquiv_presentedProP_demushkinWordTwoEven_of_range_eq).
For even rank n, where p = 2 and q = 2, with the image of the canonical character the twisted
subgroup U^[g], the closed subgroup generated by -1 + 2^g, 2 ≤ g < ∞, the relator is
x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (xₙ₋₁, xₙ) for n ≥ 4 and any finite f > g, or
x₁^{2 + α} (x₁, x₂) for n = 2,
for any natural exponent α of exact divisibility depth g, that is 2^g ∣ α and 2^{g+1} ∤ α;
the canonical character takes the value -(1 + α)⁻¹ on x₂, the value (1 - 2^f)⁻¹ on x₄ when
n ≥ 4, and is trivial on the other generators. Here the image pins g = v₂(α) but neither α
within its valuation nor the level f, which is free above g
(TauCeti.nonempty_continuousMulEquiv_presentedProP_demushkinWordTwoEven_of_not_dvd); the
statement at rank n ≥ 4 therefore holds for every such α and every finite f > g, and the
rank-two word carries no level. The fourth-generator clause is stated only for n ≥ 4: at rank two
it would read 1 - 2^f = 1 (TauCeti.not_marked_of_demushkinRank_le), although the other
hypotheses are met there, by ⟨x₁, x₂ ∣ x₁⁶ (x₁, x₂)⟩ with α = 4 and g = 2.
The two finite-parameter branches of even rank n ≥ 4 are the two ways the image pins the pair
(α, f): either 2^f ∣ α and the image is {±1} × U^(f), or v₂(α) < f and the image is
U^[v₂(α)]. The single statement TauCeti.isDemushkin_marked_of_q_two_even takes this
disjunction as its hypothesis and is the even-rank entry of the marked classification.
For even rank n with orientation image {±1}, the endpoint f = ∞ of the even family, the
relator is x₁² (x₁, x₂)(x₃, x₄) ⋯ (xₙ₋₁, xₙ), the q ≠ 2 word read at q = 2, and the canonical
character takes the value -1 on x₂ and is trivial on the other generators.
Main results #
TauCeti.isDemushkin_marked_of_q_ne_two: the marked classification of a Demushkin group whoseq-invariant is not2.TauCeti.isDemushkin_marked_of_q_two_odd,TauCeti.isDemushkin_marked_of_q_two_odd_top: the marked classification of a Demushkin group of odd rank, at a finite level and at levelf = ∞.TauCeti.isDemushkin_marked_of_q_two_even_unitsPlusMinus: the marked classification of a Demushkin group of even rankn ≥ 4whose orientation image is{±1} × U^(f).TauCeti.isDemushkin_marked_of_q_two_even_twisted,TauCeti.isDemushkin_marked_of_q_two_rank_two: the marked classification of a Demushkin group of even rank whose orientation image is a twisted subgroupU^[g], at rankn ≥ 4and at rank2; at rank two the twisted subgroups are the only finite-parameter images.TauCeti.isDemushkin_marked_of_q_two_even: the marked classification of a Demushkin group of even rankn ≥ 4at a finite level, the two branches above as one statement whose hypothesis is the image table read on the pair(α, f).TauCeti.isDemushkin_marked_of_q_two_even_top: the marked classification of a Demushkin group of even rank whose orientation image is{±1}.
References #
- J. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106--132, Theorems 3, 4, 5 and 6.
The marked classification at q ≠ 2. A Demushkin group with q-invariant different
from 2 is isomorphic to the standard one-relator presentation on its generator rank. Under this
isomorphism the canonical character takes the value (1 - q)⁻¹ on the second marked generator
and is trivial on every other marked generator.
The marked classification at q = 2 with n odd, finite level. A Demushkin group at
p = 2 of odd rank n ≥ 3 whose canonical character has image {±1} × U^(f), with f ≥ 2, is
isomorphic to ⟨x₁, …, xₙ ∣ x₁² x₂^{2^f} (x₂, x₃) ⋯ (xₙ₋₁, xₙ)⟩. Under this isomorphism the
canonical character takes the value -1 on the first marked generator, the value (1 - 2^f)⁻¹ on
the third, and is trivial on every other marked generator.
The marked classification at q = 2 with n odd, level f = ∞. A Demushkin group at
p = 2 of odd rank n whose canonical character has image {±1} is isomorphic to
⟨x₁, …, xₙ ∣ x₁² (x₂, x₃) ⋯ (xₙ₋₁, xₙ)⟩. Under this isomorphism the canonical character takes
the value -1 on the first marked generator and is trivial on every other marked generator. At
rank one this is ℤ/2 = ⟨x₁ ∣ x₁²⟩ with χ(x₁) = -1.
The marked classification at q = 2 with n ≥ 4 even and orientation image
{±1} × U^(f). A Demushkin group at p = 2 of even rank n ≥ 4 whose canonical character has
image {±1} × U^(f), with f ≥ 2, is isomorphic to
⟨x₁, …, xₙ ∣ x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (xₙ₋₁, xₙ)⟩ for every natural exponent α
with 2^f ∣ α, including α = 0. Under this isomorphism the canonical character takes the value
-(1 + α)⁻¹ on the second marked generator, the value (1 - 2^f)⁻¹ on the fourth, and is trivial
on every other marked generator. The image pins the level f; the exponent α above it is not an
invariant of G.
The marked classification at q = 2 with n ≥ 4 even and twisted orientation image. A
Demushkin group at p = 2 of even rank n ≥ 4 whose canonical character has image the twisted
subgroup U^[g], the closed subgroup generated by -1 + 2^g, with g ≥ 2, is isomorphic to
⟨x₁, …, xₙ ∣ x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (xₙ₋₁, xₙ)⟩ for every natural exponent α
of exact divisibility depth g, that is 2^g ∣ α and 2^{g+1} ∤ α, and every finite level
f > g. Under this isomorphism the canonical character takes the value -(1 + α)⁻¹ on the
second marked generator, the value (1 - 2^f)⁻¹ on the fourth, and is trivial on every other
marked generator. The image pins g = v₂(α) only: the exponent within its valuation and the level
above it are not invariants of G.
The marked classification at q = 2 with n ≥ 4 even. A Demushkin group at p = 2 of
even rank n ≥ 4 is isomorphic to
⟨x₁, …, xₙ ∣ x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (xₙ₋₁, xₙ)⟩, for a natural exponent α with
4 ∣ α and a finite level f ≥ 2, as soon as the image of its canonical character is the one
the table of Labute's Theorem 4 assigns to the pair (α, f): {±1} × U^(f) when 2^f ∣ α, and
the twisted subgroup U^[v₂(α)], the closed subgroup generated by -1 + 2^{v₂(α)}, when
v₂(α) < f. Under this isomorphism the canonical character takes the value -(1 + α)⁻¹ on the
second marked generator, the value (1 - 2^f)⁻¹ on the fourth, and is trivial on every other
marked generator. The image equation is what pins the parameters to G: in the first branch it
pins f and leaves α free above it, in the second it pins v₂(α) and leaves f free above it.
The rank-two family is TauCeti.isDemushkin_marked_of_q_two_rank_two, and the image {±1}, the
level f = ∞, is TauCeti.isDemushkin_marked_of_q_two_even_top.
The marked classification at q = 2 and rank two. A Demushkin group at p = 2 of rank 2
whose canonical character has image the twisted subgroup U^[g], the closed subgroup generated by
-1 + 2^g, with g ≥ 2, is isomorphic to ⟨x₁, x₂ ∣ x₁^{2 + α} (x₁, x₂)⟩ for every natural
exponent α of exact divisibility depth g, that is 2^g ∣ α and 2^{g+1} ∤ α. Under this
isomorphism the canonical character is trivial on the first marked generator and takes the value
-(1 + α)⁻¹ on the second. The image pins g = v₂(α) only, and the exponent within its valuation
is not an invariant of G. At rank two these twisted subgroups are the only images with a finite
parameter: the image {±1}, the word x₁² (x₁, x₂) with α = 0, is the level f = ∞ of
TauCeti.isDemushkin_marked_of_q_two_even_top, and {±1} × U^(f) with f < ∞ is not procyclic,
while a rank-two image is topologically generated by χ(x₂).
The marked classification at q = 2 with n even, level f = ∞. A Demushkin group at
p = 2 of even rank n whose canonical character has image {±1} is isomorphic to
⟨x₁, …, xₙ ∣ x₁² (x₁, x₂)(x₃, x₄) ⋯ (xₙ₋₁, xₙ)⟩. Under this isomorphism the canonical character
takes the value -1 on the second marked generator and is trivial on every other marked
generator.