Hypermaps: constructions and operations

It is conjectured that given positive integers <i>l, m, n</i> with <i>l</i>^-1 + <i>m^-</i>¹ + <i>n</i>^-1 &lt; 1<br/>and an integer <i>g</i> ≥ 0, the triangle group Δ = Δ (<i>l, m, n</i>) = ⟨<i>X,Y,Z|X</i>^{<i> l</i>} = <i>Y</i> ^<i>m</i> =<br/><i>Z ⁿ</i> = <i>X Y Z</i> = 1⟩ contains infinitely many subgroups of finite index and of genus<br/>g. This conjecture can be rewritten in another form: given positive integers l,<br/>m, n with l¡1 +m¡1 +n¡1 &lt; 1 and an integer <i>g</i> ≥ 0, there are infinitely many<br/>nonisomorphic compact orientable hypermaps of type (<i>l, m, n</i>) and genus <i>g</i>.<br/>We prove that the conjecture is true, when two of the parameters <i>l, m, n</i> are<br/>equal, by showing how to construct those hypermaps, and we extend the result<br/>to nonorientable hypermaps.<br/><br/>A classification of all operations of finite order in oriented hypermaps is<br/>given, and a detailed study of one of these operations (the duality operation)<br/>is developed. Adapting the notion of chirality group, the duality group of<br/><i>H</i> can be defined as the minimal subgroup <i>D</i>(<i>H</i>) ≤¦ <i>M on</i> (<i>H</i>) such that<br/><i>H </i>= <i>D </i>(<i>H</i>) is a self-dual hypermap. We prove that for any positive integer <i>d</i>,<br/>we can find a hypermap of that duality index (the order of <i>D</i> (<i>H</i>) ), even when<br/>some restrictions apply, and also that, for any positive integer <i>k</i>, we can find a<br/>non self-dual hypermap such that |<i>Mon</i> (<i>H</i>) | = <i>d</i> = <i>k</i>. We call this <i>k</i> the <i>duality</i><br/>coindex of the hypermap. Links between duality index, type and genus of a<br/>orientably regular hypermap are explored.<br/><br/>Finally, we generalize the duality operation for nonorientable regular hypermaps <br/>and we verify if the results about duality index, obtained for orientably <br/>regular hypermaps, are still valid.

Name dc:type.qualificationname

Ph.D.

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

University of Southampton

Year dc:date.issued

2009

Author dc:creator

• Pinto, Daniel Alexandre Peralta Marques

Advisor dc:contributor.advisor

• Jones, Gareth