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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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Future Work/References 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 Galois Action on Generating Vectors Future Work/References 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 Future Work/References 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 Future Work/References 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 Galois Action on Generating Vectors Future Work/References 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 Future Work/References 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.