Peripheral automorphisms up to a permutation of the peripheral classes #
Let F be a pro-p group and x : Fin r → F a family, with peripheral tuple
t = (x_0, …, x_{r-1}, z), where z = (x_0 ⋯ x_{r-1})⁻¹ is the cusp. A continuous automorphism
φ of F is peripheral for the permutation σ of Fin (r + 1) with exponent u ∈ ℤ_pˣ when
it carries every t_i to a conjugate of the p-adic power t_{σ i} ^ u. For σ = 1 this is
TauCeti.Peripheral.IsPeripheralAut. The automorphisms admitting such data form a subgroup
peripheralPermAut: if φ has data (σ, u) and ψ has data (τ, v), then φ * ψ has data
(σ * τ, u * v) and φ⁻¹ has data (σ⁻¹, u⁻¹).
For the basis of a free pro-p group of rank r ≥ 2 the data are determined by the
automorphism. In the abelianization ℤ_p ^ r the classes of the peripheral tuple are the
coordinate vectors e_0, …, e_{r-1} and -(e_0 + ⋯ + e_{r-1}), and when r ≥ 2 no unit multiple
of one of them is a unit multiple of another. So the data define a homomorphism permData from
peripheralPermAut to Equiv.Perm (Fin (r + 1)) × ℤ_pˣ, whose kernel consists of the
automorphisms that are peripheral of exponent one. In rank one the data are not determined: the
peripheral tuple is (x, x⁻¹), and the automorphism x ↦ x ^ u has both data (1, u) and
(swap 0 1, -u).
Two automorphisms realise nontrivial permutations. In every rank the rotation x_i ↦ t_{i+1}
sends the cusp to x_0, so it permutes the peripheral tuple cyclically, on the nose. In rank two
the swap x_0 ↦ x_1, x_1 ↦ x_0 sends the cusp (x_0 x_1)⁻¹ to (x_1 x_0)⁻¹ = x_0⁻¹ z x_0,
so it is peripheral for the transposition of the first two classes, with exponent one.
Main definitions #
TauCeti.Peripheral.IsPeripheralPermAut: a continuous automorphism carries eacht_ito a conjugate oft_{σ i} ^ u.TauCeti.Peripheral.peripheralPermAut: the subgroup of continuous automorphisms that are peripheral for some permutation and some unit exponent.TauCeti.Peripheral.permData: for the basis of a free pro-pgroup of rank at least two, the homomorphism recording the permutation and the exponent.
Main results #
TauCeti.Peripheral.isPeripheralPermAut_iff: the predicate on the basis elements and the cusp.TauCeti.Peripheral.IsPeripheralPermAut.mul,TauCeti.Peripheral.IsPeripheralPermAut.inv: the data of a composite and of an inverse.TauCeti.Peripheral.IsPeripheralPermAut.unique: in rank at least two the permutation and the exponent are determined by the automorphism.TauCeti.Peripheral.permData_eq_iff,TauCeti.Peripheral.mem_ker_permData_iff:permDatais characterized by the predicate, and its kernel is the exponent-one peripheral part.TauCeti.Peripheral.exists_isPeripheralPermAut_swap_rank_one: the data are not determined in rank one.TauCeti.Peripheral.exists_rotation,TauCeti.Peripheral.exists_rotation_two: the rotation of the peripheral tuple, and its rank-two casex_0 ↦ x_1 ↦ z ↦ x_0.TauCeti.Peripheral.exists_swap_two: the rank-two swap.
References #
- Y. Ihara, "Braids, Galois groups, and some arithmetic functions", Proc. ICM Kyoto 1990,
99–120, for automorphisms of free pro-
pgroups of rank two permuting the three peripheral conjugacy classes.
A continuous automorphism φ of a pro-p group is peripheral for the permutation σ with
exponent u for a family x when it carries every element t_i of the peripheral tuple
x_0, …, x_{r-1}, cusp x to a conjugate of the p-adic power t_{σ i} ^ u.
Instances For
An automorphism is peripheral for σ with exponent u exactly when it carries each x_i
to a conjugate of t_{σ i} ^ u and the cusp to a conjugate of t_{σ r} ^ u.
For the trivial permutation, the predicate is peripherality of exponent u.
An automorphism permuting the peripheral tuple exactly, t_i ↦ t_{σ i}, is peripheral for
σ with exponent one.
Composition of permutation-peripheral automorphisms composes the permutations and multiplies the exponents.
The inverse of a permutation-peripheral automorphism has the inverse permutation and the inverse exponent.
The continuous automorphisms peripheral for x up to some permutation of the peripheral tuple,
with some common unit exponent.
Instances For
Membership in peripheralPermAut means admitting a permutation and a unit exponent.
The permutation and the exponent are determined in rank at least two. For the basis of a
free pro-p group of rank r ≥ 2, an automorphism peripheral for (σ, u) and for (σ', u')
has σ = σ' and u = u'.
The permutation and the exponent of a permutation-peripheral automorphism, for the basis
of a free pro-p group of rank at least two, as a homomorphism.
Instances For
permData supplies a permutation and an exponent for its argument.
permData of an automorphism is (σ, u) exactly when the automorphism is peripheral for
σ with exponent u.
The kernel of permData consists of the automorphisms that are peripheral of exponent one.
The data are not determined in rank one. In rank one the peripheral tuple is (x, x⁻¹),
and an automorphism with x ↦ x ^ u is peripheral both for the trivial permutation with
exponent u and for the transposition with exponent -u.
The rotation. In every rank some continuous automorphism sends x_i to t_{i+1}, the
next element of the peripheral tuple; it then sends the cusp to x_0, so it permutes the
peripheral tuple cyclically, t_i ↦ t_{i+1} with indices modulo r + 1, and it is peripheral for
this cyclic permutation with exponent one.
The rank-two rotation. In rank two some continuous automorphism cycles the peripheral
tuple x_0 ↦ x_1 ↦ z ↦ x_0, where z = (x_0 x_1)⁻¹ is the cusp, so it is peripheral for the
three-cycle finRotate 3 with exponent one.
The rank-two swap. In rank two some continuous automorphism exchanges the two basis
elements. It sends the cusp z = (x_0 x_1)⁻¹ to (x_1 x_0)⁻¹ = x_0⁻¹ * z * x_0, so it is
peripheral for the transposition of the first two peripheral classes, with exponent one.