Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T07:55:55.147Z Has data issue: false hasContentIssue false
  • English
  • Français

REGULAR HYPERMAPS OVER PROJECTIVE LINEAR GROUPS

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  01 October 2008

MARSTON CONDER ,
PRIMOŽ POTOČNIK  and
JOZEF ŠIRÁŇ
MARSTON CONDER*
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand (email: m.conder@auckland.ac.nz)
PRIMOŽ POTOČNIK
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia (email: primoz.potocnik@fmf.uni-lj.si)
JOZEF ŠIRÁŇ
Affiliation:
Open University, Milton Keynes, MK7 6AA, UK, and Slovak Technical University, Radlinskeho 11, 81368, Bratislava, Slovakia (email: j.siran@open.ac.uk)
*
✉For correspondence; e-mail: m.conder@auckland.ac.nz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An enumeration result for orientably regular hypermaps of a given type with automorphism groups isomorphic to PSL(2,q) or PGL(2,q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to nonorientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.

Keywords

hypermapregular hypermaptriangle groupprojective linear group

MSC classification

Secondary: 57M15: Relations with graph theory 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 155 - 175
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The authors gratefully acknowledge the support from the NZIMA programme on Combinatorics and Its Applications. The first and third authors wish to acknowledge the partial support from the Marsden Fund, and the third author would like to acknowledge the partial support from APVV Grants Nos. 20-000704 and 0040-06.

References

[1]Adrianov, N. M., ‘Regular maps with automorphism groups PSL2(q)’, Russian Math. Surveys 52(4) (1997), 819–821.CrossRefGoogle Scholar
[2]Breda d’Azevedo, Antonio J. and Jones, Gareth A., ‘Platonic hypermaps’, Beiträge Algebra Geom. 42(1) (2001), 1–37.Google Scholar
[3]Conder, Marston, Potočnik, Primož and Širáň, Jozef, Regular maps with almost Sylow-cyclic automorphism groups, and classification of regular maps with Euler characteristic −p 2, Preprint.Google Scholar
[4]Cori, Robert and Machì, Antonio, ‘Maps, hypermaps and their automorphisms: a survey, I, II, III’, Exposition. Math. 10 (1992), 403–427,  429–447, 449–467.Google Scholar
[5]Dickson, Leonard Eugene, Linear Groups: With an Exposition of the Galois Field Theory (Dover, New York, 1958).Google Scholar
[6]Downs, Martin, ‘Some enumerations of regular hypermaps with automorphism group isomorphic to PSL2(q)’, Quart. J. Math. Oxford Ser. (2) 48 (1997), 39–58.CrossRefGoogle Scholar
[7]Glover, Henry and Sjerve, Denis, ‘The genus of PSl2(q)’, J. Reine Angew. Math. 380 (1987), 59–86.Google Scholar
[8]Jones, Gareth A., ‘Ree groups and Riemann surfaces’, J. Algebra 165 (1994), 41–62.CrossRefGoogle Scholar
[9]Jones, Gareth A. and Silver, Stephen A., ‘Suzuki groups and surfaces’, J. London Math. Soc. (2) 48 (1993), 117–125.CrossRefGoogle Scholar
[10]Jones, Gareth and Singerman, David, ‘Maps, hypermaps and triangle groups’, in: The Grothendieck Theory of Dessins d’Enfants (Luminy, 1993), London Mathematical Society Lecture Note Series, 200 (Cambridge University Press, Cambridge, 1994), pp. 115–145.CrossRefGoogle Scholar
[11]Jones, Gareth and Singerman, David, ‘Belyĭ functions, hypermaps, and Galois groups’, Bull. London Math. Soc. 28 (1996), 561–590.CrossRefGoogle Scholar
[12]Langer, Ulrich and Rosenberger, Gerhard, ‘Erzeugende endlicher projektiver linearer Gruppen’, Results Math. 15 (1989), 119–148.CrossRefGoogle Scholar
[13]Levin, Frank and Rosenberger, Gerhard, ‘Generators of finite projective linear groups II’, Results Math. 17 (1990), 120–127.CrossRefGoogle Scholar
[14]Macbeath, A. M., ‘Generators of the linear fractional groups’, in: Number Theory, Proc. Sympos. Pure Math., Houston, TX, Vol. XII (American Mathematical Society, Providence, RI, 1967), pp. 14–32.Google Scholar
[15]Sah, Chih-han, ‘Groups related to compact Riemann surfaces’, Acta Math. 123 (1969), 13–42.CrossRefGoogle Scholar