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**Extra resources for Geometric Galois Actions, Volume 2: The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups**

**Example text**

For any involution u / — 1, a 48 Gareth Jones and Manfred Streit simple but tedious argument, counting cycles of go and gi, shows that p + q < N +1, so G = (go, g^) is the monodromy group of a single-faced bipartite map of genus greater than 0. Thus among the plane trees with N edges, only the paths of length N > 2 have monodromy groups of order 2N. Example 3. There are two plane trees of type (a; j3) — (4,1,1; 2, 2,1,1), shown in Figure 4, and in each case, the monodromy group G is a transitive subgroup of 5 6 .

Under these circumstances, a finer invariant is needed, and for this one can use the cartographic group C of a dessin; this transitive subgroup of S2N is the monodromy group of the Belyi function 4(3(1—[3) : X —> S, and our main theorem also shows that conjugate dessins have conjugate cartographic groups. Now conjugacy of cartographic groups implies conjugacy of monodromy groups, but the converse is false: we shall give examples of dessins 28 Gareth Jones and Manfred Streit which have conjugate monodromy groups but non-conjugate cartographic groups.

With riooj sides (j = 1 , . . ,fcoo),each containing an element of 7~ 1 (oo). 2: this is because q~l(M\) — B\, so q lifts the single vertex 0 of M\ to the two vertices 0 and 1 of B\, and then (3 lifts these to the black and white vertices of A4 = B. 36 Gareth Jones and Manfred Streit The combinatorial structures obtained from Belyi functions as above are sometimes known as dessins d'enfants. The natural action of G on Belyi pairs induces actions of G on the various types of dessins representing these pairs, and explicit examples of these actions are given in §7.