Hypermaps: constructions and operations

Abstract

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 Δ = Δ (l, m, n) = ⟨X,Y,Z|X ^l = Y ^m =
Z ⁿ = X Y Z = 1⟩ contains infinitely many subgroups of finite 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 infinitely 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 classification of all operations of finite 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 defined as the minimal subgroup D(H) ≤¦ M on (H) such that
H = D (H) is a self-dual hypermap. We prove that for any positive integer d,
we can find 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 find a
non self-dual hypermap such that |Mon (H) | = 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.

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