Dessins d’enfants: bipartite maps and Galois groups

Gareth Jones, Southampton, U. K.

Abstract. Belyĭ’s Theorem implies that the Riemann surfaces defined over the field of algebraic numbers are precisely those which support bipartite maps; this provides a faithful representation of the Galois group of this field on these combinatorial objects.

My aim in this note is to show how combinatorics can play a central role in uniting such topics as Galois theory, algebraic number theory, Riemann surfaces, group theory and hyperbolic geometry. The relevant combinatorial objects are maps on surfaces, often called dessins d’enfants in view of the rather naïve appearance of some of the most common examples. For simplicity I will restrict attention to bipartite maps, though triangulations and hypermaps also play an important role in this theory. More detailed surveys can be found in [2, 5, 7], and for recent progress see [8].

A bipartite map B consists of a bipartite graph G imbedded (without crossings) in a compact, connected, oriented surface X, so that the faces (connected components of X \ G) are simply connected. One can describe B by a pair of permutations g_{0} and g_{1} of its edge-set E: the vertices can be coloured black or white, so that each edge joins a black and a white vertex; the orientation of X then determines a cyclic ordering of the edges around each black or white vertex, and these are the disjoint cycles of g_{0} and g_{1} respectively. These two permutations generate a subgroup G = ⟨g_{0}, g_{1}⟩ of the symmetric group S^{E} of all permutations of E, called the monodromy group of B; the topological hypotheses imply that G has to be connected, so G acts transitively on E. Conversely, every 2-generator transitive group arises in this way from some bipartite map B: the edges are the symbols permuted, the black and white vertices correspond to the cycles of the two generators g_{0} and g_{1}, and the faces correspond to the cycles of g_{∞} = (g_{0}g_{1})^{−1}. Isomorphism of maps (preserving orientation and vertex-colours) corresponds to conjugacy of pairs (g_{0}, g_{1}) in S^{E}, and the automorphism group of B can be identified with the centraliser of G in S^{E}, that is, the group of all permutations which commute with G.

(Historical note: A slight modification of these ideas allows one to describe any oriented map, whether bipartite or not, by a pair of permutations [6]. Although generally regarded as a modern development, this use of permutations can be traced back at least as far as Hamilton’s construction of what we now call Hamiltonian cycles in the icosahedron [4].)

Every bipartite map B is a quotient of the universal bipartite map B̂, drawn on the upper half-plane U = { z ∈ C | Im(z) > 0 }; the vertices of B̂ are the rational numbers r/s (in reduced form) with s odd, coloured black or white as r is even or odd, and the edges are the hyperbolic geodesics (euclidean semicircles) joining vertices r/s and x/y with ry − sx = ±1. The automorphisms of B̂ are the Möbius transformations

z ↦ \frac{az + b}{cz + d}  (a, b, c, d ∈ Z, ad − bc = 1)  (∗)

such that a ≡ d ≡ 1 and b ≡ c ≡ 0 mod (2); these form a normal subgroup Γ(2) of index 6 in the modular group Γ = P SL_{2}(Z) of all transformations (∗), called the principal congruence subgroup of level 2 in Γ. Now Γ(2) is a free group of rank 2, generated by the transformations

T_{0} : z ↦ \frac{z}{−2z + 1}  and  T_{1} : z ↦ \frac{z − 2}{2z − 3} ,

so there is an epimorphism θ : Γ(2) → G, T_{i} ↦ g_{i}, which gives a transitive permutation representation of Γ(2) on E. If B denotes the subgroup θ^{−1}(G_{e}) of Γ(2) fixing an edge e of B then B acts as a group of automorphisms of B̂, and one can show that B ≅ B̂/B. The underlying surface of B̂/B is now a compact Riemann surface \overline{U/B} = (U ∪ Q ∪ \{∞\})/B, formed by compactifying U/B with finitely many points, corresponding to the orbits of B on the extended rationals Q∪{∞}. One can regard B̂/B as a rigid, conformal model of B, with a complex structure induced from that of U: for instance, the edges are geodesics, the angles between edges around any vertex are equal, and the automorphisms of B are conformal isometries of the Riemann surface.

Riemann showed that a Riemann surface X is compact if and only if it is isomorphic to the Riemann surface of an algebraic curve f(x, y) = 0 for some polynomial f(x, y) ∈ C[x, y]. Computationally and theoretically, the most satisfactory polynomials are those with coefficients in the field \overline{Q} of algebraic numbers; results of Belyĭ [1] and Weil [9] imply that the Riemann surfaces corresponding to such polynomials f(x, y) ∈ \overline{Q}[x, y] are those obtained from bipartite maps by the above method. More precisely, Belyĭ showed that a compact Riemann surface X is defined over \overline{Q} if and only if there is a Belyĭ function β from X to the Riemann sphere Σ = C∪{∞}, that is, a meromorphic function on X which is unbranched over Σ \ {0, 1, ∞}. In these circumstances, X is the underlying surface of the bipartite map B = β^{−1}(B_{1}), where B_{1} is the trivial bipartite map B̂/Γ(2) on Σ with a black vertex at 0, a white vertex at 1, and a single edge along the unit interval [0, 1]. If each edge of B is identified with the sheet of the covering β : X → Σ which contains it, then the monodromy group G of B coincides with the monodromy group of β, regarded as a group of permutations of the sheets; in particular, the elements g_{0}, g_{1}, g_{∞} ∈ G describe how the sheets are permuted by lifting small loops in Σ around the branchpoints 0, 1 and ∞. Similarly, the automorphism group of B is identified with the group of covering transformations of β.

Since \overline{Q} is the union of the Galois (finite normal) extensions K ≥ Q in C, it follows that the absolute Galois group G = Gal (\overline{Q}/Q) of \overline{Q} over Q is the projective limit of the finite Galois groups Gal (K/Q) of these algebraic number fields; as such, it is an uncountable profinite group, which embodies the whole of classical Galois theory over Q. This group G is of fundamental importance in several areas of mathematics: for instance, the representation theory of G played a crucial role in Wiles’s proof of Fermat’s Last Theorem [10], and the Inverse Galois Problem (Hilbert’s still unproved conjecture that every finite group is a Galois group over Q) is equivalent to showing that every finite group is an epimorphic image of G. Fortunately, Belyĭ’s Theorem provides us with an explicit realisation of G in terms of bipartite maps, which is beginning to add to our rather meagre knowledge of this complicated group.

In [3], Grothendieck showed that the natural action of G on polynomials over \overline{Q} induces an action of G on bipartite maps (and on other similar combinatorial objects, generally known as dessins d’enfants), through the above correspondence between maps and polynomials. Although G preserves such properties of a map as its genus, the numbers and valencies of its black and white vertices, its monodromy group and its automorphism group, this action of G is nevertheless faithful, in the sense that each non-identity element of G sends some bipartite map to a non-isomorphic bipartite map. Moreover, this action remains faithful even when restricted to such simple objects as plane trees (maps on the sphere with one face). One therefore has a combinatorial approach to Galois theory, which is attracting interest from a wide spectrum of mathematicians and theoretical physicists (for whom maps are an effective discrete approximation to the compact Riemann surfaces which play a major role in quantum gravity).

References

[1] G. V. BELYĬ, ‘On Galois extensions of a maximal cyclotomic field’, Izv. Akad. Nauk SSSR 43 (1979) 269–276 (Russian); Math. USSR Izvestiya 14 (1980), 247–256 (English translation).

[2] P. B. COHEN, C. ITZYKSON and J. WOLFART, ‘Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi’, Comm. Math. Phys. 163 (1994), 605–627.

[3] A. GROTHENDIECK, ‘Esquisse d’un programme’, preprint, Montpellier, 1984.

[4] W. R. HAMILTON, Letter to John T. Graves ‘On the Icosian’ (17th October 1856), in Mathematical papers, Vol. III, Algebra (eds. H. Halberstam and R. E. Ingram), Cambridge University Press, Cambridge, 1967, pp. 612–625.

[5] G. A. JONES, ‘Maps on surfaces and Galois groups’, submitted.

[6] G. A. JONES and D. SINGERMAN, ‘Theory of maps on orientable surfaces’, Proc. London Math. Soc. (3) 37 (1978), 273–307.

[7] G. A. JONES and D. SINGERMAN, ‘Belyĭ functions, hypermaps and Galois groups’, Bull. London Math. Soc., to appear.

[8] L. SCHNEPS (ed.), The Grothendieck Theory of Dessins d’Enfants, London Math. Soc. Lecture Note Series 200, Cambridge University Press, Cambridge, 1994.

[9] A. WEIL, ‘The field of definition of a variety’, Amer. J. Math. 78 (1956), 509–524.

[10] A. WILES, ‘Modular elliptic curves and Fermat’s Last Theorem’, Ann. Math. 141 (1995), 443–551.

Department of Mathematics University of Southampton Southampton SO17 1BJ United Kingdom

e-mail: gaj@maths.soton.ac.uk