Belyi Functions, Hypermaps and Galois Groups

Mathematics is full of dictionaries, one of the most well-thumbed being that between compact Riemann surfaces and complex algebraic curves; for example, branched coverings of surfaces and their groups of covering transformations can be translated into extensions of rational function fields and their Galois groups. An important problem is that of determining, for a given subfield K <= C, those surfaces defined over K, that is, given by a polynomial equation with coefficients in K. This is particularly interesting when K is the field Q of algebraic numbers, the algebraic closure in C of the rational field Q; being the union of all the algebraic number fields, Q has a particularly rich Galois theory, full of explicitly computable examples and tantalisingly difficult challenges, such as the Inverse Galois Problem, Hilbert's conjecture that every finite group is a Galois group over Q (see [35]). It follows from Weil's rigidity theorem [45] that a sufficient condition for a curve X to be defined over Q is that there should be a meromorphic function /? from X to the Riemann sphere (or projective line) £ = P*(Q = C U {oo} with at most three critical values (which we can take to be 0, 1 and oo). As a lemma for realising certain Chevalley groups as Galois groups over cyclotomic fields, Bely! [3] showed in 1979, by a simple but ingenious argument given in Section 2, that this condition is also necessary. Such a branched covering /?: Z ^ I , called a Belyi function, is completely determined by three permutations g0, g1 and g^ describing how the sheets are permuted by analytic continuation around 0, 1 and oo. They satisfy g0£i£oo = 1 a d generate a finite transitive permutation group, the monodromy group of /?, and conversely every such triple of permutations arises in this way. This has important consequences for uniformisation: for example, X is defined over Q if and only if X ^ %IK where °U is the upper half-plane and K is a subgroup of finite index in a triangle group A(/, m, n), or equivalently X = %/H where H has finite index in the principal congruence subgroup F(2) of the modular group (see Section 4). Since the time of Dyck [10] and Heffter [12,13] it has been known that maps (2-cell imbeddings of graphs) on compact orientable surfaces also correspond to such triples, satisfying the additional relation g\ = 1; the vertices, edges and faces correspond to the cycles of g0, gl and gx, with incidence given by non-empty intersection and orientation given by cyclic ordering. Thus maps correspond to subgroups of finite index in triangle groups A(/, 2, n); for regular maps, corresponding to normal subgroups, this is implicit in Chapter 8 of [8], and the general theory was developed independently in the 1970s by Malgoire and Voisin [24] and the authors [18,38].