Chirality groups of maps and hypermaps

Although the phenomenon of chirality appears in many investigations of maps and hypermaps, no detailed study of chirality seems to have been carried out. Chirality of maps and hypermaps is not merely a binary invariant but can be quantified by two new invariants—the chirality group and the chirality index, the latter being the size of the chirality group. A detailed investigation of the chirality groups of orientably regular maps and hypermaps will be the main objective of this paper. The most extreme type of chirality arises when the chirality group coincides with the monodromy group. Such hypermaps are called totally chiral. Examples of these are constructed by considering appropriate “asymmetric” pairs of generators of certain non-abelian simple groups. We also show that every finite abelian group is the chirality group of some hypermap, whereas many non-abelian groups, including symmetric and dihedral groups, cannot arise as chirality groups. A. Breda D’Azevedo supported in part by UI&D Matemática e aplicações of University of Aveiro, through Program POCTI of FCT co-financed by the European Community fund FEDER. R. Nedela supported in part by the Ministry for Education of the Slovak Republic, grant no. APVT 51 027 604. M. Škoviera supported in part by APVT grant 51-027604 and by VEGA grant 1/3022/06. A. Breda D’Azevedo ( ) Departamento de Matematica, Universidade de Aveiro, 3810-193 Aveiro, Portugal e-mail: [email protected] G. Jones School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK R. Nedela Department of Mathematics, Matej Bel University, Banská Bystrica, Slovakia M. Škoviera Department of Informatics, Comenius University, 842 48 Bratislava, Slovakia 338 J Algebr Comb (2009) 29: 337–355