Title:

Hypermaps : constructions and operations

It is conjectured that given positive integers l, m, n with l¡1 +m¡1 + n¡1 < 1 and an integer g ¸ 0, the triangle group = (l; m; n) = hX; Y;ZjXl = Y m = Zn = XY Z = 1i contains innitely many subgroups of nite index and of genus g. This conjecture can be rewritten in another form: given positive integers l, m, n with l¡1 +m¡1 +n¡1 < 1 and an integer g ¸ 0, there are innitely many nonisomorphic compact orientable hypermaps of type (l; m; n) and genus g. We prove that the conjecture is true, when two of the parameters l, m, n are equal, by showing how to construct those hypermaps, and we extend the result to nonorientable hypermaps. A classication of all operations of nite order in oriented hypermaps is given, and a detailed study of one of these operations (the duality operation) is developed. Adapting the notion of chirality group, the duality group of H can be dened as the the minimal subgroup D(H) E Mon(H) such that H=D(H) is a selfdual hypermap. We prove that for any positive integer d, we can nd a hypermap of that duality index (the order of D(H)), even when some restrictions apply, and also that, for any positive integer k, we can nd a non selfdual hypermap such that jMon(H)j=d = k. We call this k the duality coindex of the hypermap. Links between duality index, type and genus of a orientably regular hypermap are explored. Finally, we generalize the duality operation for nonorientable regular hypermaps and we verify if the results about duality index, obtained for orientably regular hypermaps, are still valid.
