Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
Future Work/References
Galois actions on regular dessins and
Fuchsian group covers
S. Allen Broughton (joint work with Aaron Wootton)
Rose-Hulman Institute of Technology
Conference on Riemann Surfaces and Dessins d’Enfants
On the Occasion of Jürgen Wolfart’s 65th Birthday
May 24, 2010
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
Outline
1
Regular Dessins and Belyi functions
Conformal actions
The triangulation structure
2
Galois Action on Surfaces
Galois action on varieties and morphisms
Rotation numbers
3
Galois Action on Generating Vectors
Setup
Results
Adjacencies
4
Future Work/References
Problems and work in progress
Future Work/References
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
Future Work/References
Conformal actions
Fuchsian group covers of conformal actions
Assume the following:
S is a closed, orientable Riemann surface.
G is a finite group acting conformally on S, namely there is
a monomorphism:
: G → Aut(S).
(1)
Then there are Fuchsian groups Π ≤ ∆ such that:
Π is torsion free and S w H/Π
ΠE∆
G acts on S w H/Π via an epimorphism
η:∆→G
such that η : ∆/Π ↔ S is the inverse of : G → Aut(S)
upon identifying S w H/Π
(2)
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Conformal actions
Belyi functions from actions
If S/G w P 1 (C) then the quotient map β : S → S/G is a
meromorphic function:
β : S → P 1 (C).
(3)
We will assume from now on that β is ramified over only
three points {0, 1, ∞}.
In this case the Fuchsian group ∆ must be a triangle group.
The map β is a so-called regular Belyi function.
Any regular Belyi function arises in exactly this way.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Conformal actions
Ramification
As noted, β is ramified only over {0, 1, ∞}.
The ramification branching orders at 0,1, and ∞ are
denoted by b0 , b1 , and b∞ , respectively.
The ramification orders are the orders of the cyclic groups
of stabilizers of the G-action on S.
The group ∆ is a (b0 , b1 , b∞ )-triangle group.
We say that G has a (b0 , b1 , b∞ )-action on S.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Conformal actions
Generation
The group ∆ has a set {γ0 , γ1 , γ∞ } of generators
satisfying
D
E
b0
b1
b∞
∆ = γ0 , γ1 , γ∞ : γ0 = γ1 = γ∞ = γ0 γ1 γ∞ = 1 .
Let
g0 = η(γ0 ), g1 = η(γ1 ), g∞ = η(γ∞ ).
The triple (g0 , g1 , g∞ ) is called a (b0 , b1 , b∞ )-generating
vector of G. The generating vector satisfies:
G = hg0 , g1 , g∞ i
(4)
b0 = o(g0 ), b1 = o(g1 ), b∞ = o(g∞ )
(5)
g0 g1 g∞ = 1
(6)
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Conformal actions
Classification of surfaces with G-action - 1
The following is mostly correct.
Proposition
The surfaces with a (b0 , b1 , b∞ )-action of G on S are in 1-1
correspondence to either of the following
quotients H/Π where Π is the kernel of an exact sequence
η
Π ,→ ∆ → G
(7)
or
generating vectors satisfying equations 4, 5 and 6 up to the
equivalence action of Aut(S) on the generating vectors,
defined next.
The action is (g0 , g1 , g∞ ) → (ω(g0 ), ω(g1 ), ω(g∞ )) for ω
∈ Aut(G).
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Conformal actions
Classification of surfaces with G-action - 2
The proposition is “mostly” correct in that we have to make
further identifications induced by the normalizer of ∆, in case
two of the ramification orders are equal.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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The triangulation structure
Lifting the triangulation structure of the sphere - 1
We are now going to put a triangulation on S that will create
three dessins on S. First we create a triangulation on the
sphere P 1 (C)
In the sphere P 1 (C) identify the equator with the real line
and the north and south poles with i and −i.
Alternatively, the upper half plane corresponds to the upper
hemisphere and the lower half plane corresponds to the
lower hemisphere
This is not the orientation used in the standard
stereographic projection.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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The triangulation structure
Lifting the triangulation structure of the sphere - 2
We may decompose the sphere into the following:
the open upper half and lower half planes,
the open intervals (∞, 0), (0, 1), (1, ∞),
the three points 0, 1, ∞.
The map β : S → P 1 (C) is unramified over each of the
open hemispheres and the open intervals.
Therefore, via β,
each disc lifts to |G| open disjoint discs and
each interval lifts to |G| open non-intersecting arcs, each of
which is on the boundary of two discs.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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The triangulation structure
Lifting the triangular structure of the sphere - 3
Each lift of an open disc closes up to form a triangle each
of whose sides is the lift of an open interval, and whose
vertices are lifts of 0, 1, ∞.
The lifts of the triangle structure define dessins on S which
we discuss next by means of examples.
Regular Dessins and Belyi functions
Galois Action on Surfaces
The triangulation structure
Icosahedral example
Describe dessins verbally.
Galois Action on Generating Vectors
Future Work/References
Regular Dessins and Belyi functions
Galois Action on Surfaces
The triangulation structure
Torus example
Describe dessins verbally.
Galois Action on Generating Vectors
Future Work/References
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
Future Work/References
The triangulation structure
The three dessins
The three dessins defined by the triangulation structure are
D0,1 (β) = β −1 ([0, 1])
D1,∞ (β) = β −1 ([1, ∞])
D∞.0 (β) = β −1 ([∞, 0])
The standard dessin is given by
D = D(β) = D0,1 (β) = β −1 ([0, 1]).
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Galois action on varieties and morphisms
Galois action on surfaces - 1
A Belyi surface and its function is defined over a number
field.
Let σ ∈ Gal(C). A new surface and map are determined so
that the following commutes:
σ
S
−→
Sσ
↓β
↓ βσ
σ
P 1 (C) −→ P 1 (C)
We assume that the surfaces are defined by equations in
projective space and σ acts pointwise.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Galois action on varieties and morphisms
Galois action on surfaces- 2
More generally we have.
Proposition
Let f : V → W be any polynomial or rational map between
affine or projective varieties. Let f σ be the map constructed by
applying σ to the coefficients of the map f . Then we have:
σ(f (x)) = f σ (σ (x))
or in commutative diagram form.
f
V −→
↓σ
Vσ
fσ
W
↓σ
−→ W σ
(8)
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Galois action on varieties and morphisms
Galois action on surfaces- 3
Remark
If f : U → V and g : V → W are rational maps then
(f ◦ g)σ = g σ ◦ f σ
(9)
If S is a Riemann surface then so is S σ and the map f → f σ is
an isomorphism of the automorphism groups.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
Future Work/References
Rotation numbers
Rotation numbers - definition
Definition
Let f : M → M be a holomorphic map from one-dimensional
complex manifold M to itself. Let x ∈ M be a fixed point of f .
Then dfz : Tz (M) → Tz (M) is a complex number which we call
the rotation number or eigenvalue of f . We denote the
eigenvalue by λ(f , z).
Remark
If g ∈ G acting on S fixes a point P ∈ S then define
λ(g, P) = λ((g), P). The rotation numbers of group elements
are roots of unity.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
Future Work/References
Rotation numbers
Standard rotation numbers
Proposition
Let (g0 , g1 , g∞ ) be a generating vector for a G-action on S and
let P, Q, R be points fixed by g0 , g1 , g∞ , respectively. Then
2πi
λ(g0 , P) = exp
,
b
0
2πi
λ(g1 , Q) = exp
,
b1
2πi
.
λ(g∞ , R) = exp
b∞
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Setup
Cyclotomic action
Remark
Let σ ∈ Gal(C).
The cyclotomic field Q[ζ n ], where
2πi
ζn = exp n , is invariant under σ. The automorphism σ acts
on ζn by ζn → ζns for some s satisfying gcd(s, n) = 1.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Setup
The transferred action - 1
Proposition
Suppose that we have regular Belyi function β : S → P 1 (C)
arising from a conformal action : G → Aut(S).
The Belyi function β σ : S σ → P 1 (C) is also regular.
The map σ : g → ((g))σ = g σ defines an action σ : G
→ Aut(S σ ).
The group Gσ = {g σ : g ∈ G} is the group of covering
transformations of β σ .
The map σ takes the (G)-fixed points of S to the
σ (G)-fixed points of S σ .
We also have
λ(g, σ(x)) = λ(((g))σ , σ(x)) = σ(λ(g, x)).
(10)
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Results
The transferred action - 2
Proposition
Let β : S → P 1 (C) be a regular Belyi function defined by a
conformal action : G → Aut(S), with generating vector
(g0 , g1 , g∞ ) . Let σ belong to the Galois group Gal(C) and σ : G
→ Aut(S σ ) be the corresponding conformal action with Belyi
function β σ : S σ → P 1 (C). Let s be chosen so that σ(ζn ) = ζns ,
and choose t so that st = 1 mod n. Then there is a generating
vector (h0 , h1 , h∞ ) for the σ action of the form
hi = ki git ki−1 , i = 0, 1, ∞, ki ∈ G
(11)
h0 h1 h∞ = 1.
(12)
where
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Results
The transferred action - 3
Remark
The triples (h0 , h1 , h∞ ) satisfying equations 11 and 12 are
not
unique. Indeed, for any k ∈ G, kh0 k −1 , kh1 k −1 , kh∞ k −1 is
another such triple.
Corollary
If the triples satisfying 11 and 12 are unique up to Aut(G)-action
then they must be generating vectors of σ . Furthermore, if
(g0 , g1 , g∞ ) and (h0 , h1 , h∞ ) are equivalent under the
Aut(G)-action then S and S σ are conformally equivalent.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Results
Examples
The following examples can be studied with the theory just
given.
Cyclic groups. Observe that
hi = ki git ki−1 = git = ω(gi ), i = 0, 1, ∞,
where ω(g) = g t . Therefore the actions are equivalent.
Surfaces studied by Wolfart and Streit with metacyclic
automorphism groups [3].
Special linear groups which are also Hurwitz groups.
Studied by Streit [2].
Look at low genus examples computed by Conder. Just
test the examples using equation 11
Regular Dessins and Belyi functions
Galois Action on Surfaces
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Adjacencies
Definition of adjacencies
Definition
An element σ ∈ Gal(C) preserves the adjacencies of a dessin
D = D(β) if and only if whenever P and Q are adjacent in D
then σ(P) and σ(Q) are adjacent in Dσ . The notion of
preserving the adjacencies of D1,∞ (β) and D∞,0 (β) is similarly
defined.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Adjacencies
Preserving adjacencies - 1
Proposition
If an element σ of the absolute Galois group preserves one
adjacency of a dessin D then it preserves all adjacencies. The
dessins D and Dσ are isomorphic by the map x → σ(x) on the
vertices of D.
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Adjacencies
Preserving adjacencies - 2
Proposition
If σ preserves adjacencies of the three dessins of a regular
Belyi function β then S and S σ are isomorphic.
Regular Dessins and Belyi functions
Galois Action on Surfaces
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Problems and work in progress
Deficiency from preserving adjacencies
If an automorphism does not preserve adjacencies,
determine the degree of failure. Given two points adjacent
in S which are not adjacent in S σ how far way are they?
Step along the dessin to measure this.
Describe the dessin of a transform purely in terms of the
generating vector.
Find a condition on the generating vectors so that the
transformed dessin may be constructed using Wilson map
operations (see [1]).
Regular Dessins and Belyi functions
Galois Action on Surfaces
Galois Action on Generating Vectors
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Problems and work in progress
References
1
Gareth Jones, Manfred Streit and Jurgen Wolfart,Wilson’s
Map Operations on Regular Dessins and Cyclotomic
Fields of Definition, preprint.
2
Manfred Streit, Field of definition and Galois orbits for the
Macbeath-Hurwitz curves, Archiv der Mathematik,74
(2000), 342-349.
3
Manfred Streit and Jurgen Wolfart, Characters and Galois
Invariants of Regular Dessins, Revista Mathematica
Complutense, (2000) Vol XIII, num 1, 49-81.