Involve: A Journal of Mathematics

  • Involve
  • Volume 2, Number 3 (2009), 323-340.

The genus level of a group

Matthew Arbo, Krystin Benkowski, Ben Coate, Hans Nordstrom, Chris Peterson, and Aaaron Wootton

Full-text: Open access

Abstract

We introduce the notion of the genus level of a group as a tool to help classify finite conformal group actions on compact Riemann surfaces. We classify all groups of genus level 1 and use our results to outline an algorithm to classify actions of p-groups on compact Riemann surfaces. To illustrate our results, we provide a number of detailed examples.

Article information

Source
Involve, Volume 2, Number 3 (2009), 323-340.

Dates
Received: 5 November 2008
Revised: 2 July 2009
Accepted: 6 July 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799166

Digital Object Identifier
doi:10.2140/involve.2009.2.323

Mathematical Reviews number (MathSciNet)
MR2551129

Zentralblatt MATH identifier
1181.14032

Subjects
Primary: 14H37: Automorphisms 30F20: Classification theory of Riemann surfaces

Keywords
automorphism groups of compact Riemann surfaces hyperelliptic surfaces genus zero actions

Citation

Arbo, Matthew; Benkowski, Krystin; Coate, Ben; Nordstrom, Hans; Peterson, Chris; Wootton, Aaaron. The genus level of a group. Involve 2 (2009), no. 3, 323--340. doi:10.2140/involve.2009.2.323. https://projecteuclid.org/euclid.involve/1513799166


Export citation

References

  • R. Benim, “Classification of quasplatonic abelian groups and signatures”, Rose-Hulman Undergraduate Math. J. 9:1 (2008).
  • T. Breuer, Characters and automorphism groups of compact Riemann surfaces, London Math. Soc. Lecture Note Ser. 280, Cambridge University Press, Cambridge, 2000. http:www.ams.org/mathscinet-getitem?mr=2002i:14034MR 2002i:14034
  • S. A. Broughton, “Classifying finite group actions on surfaces of low genus”, J. Pure Appl. Algebra 69:3 (1991), 233–270.
  • E. Bujalance, F. J. Cirre, J. M. Gamboa, and G. Gromadzki, “On compact Riemann surfaces with dihedral groups of automorphisms”, Math. Proc. Cambridge Philos. Soc. 134:3 (2003), 465–477.
  • Y. Fuertes and G. González-Diez, “On unramified normal coverings of hyperelliptic curves”, J. Pure Appl. Algebra 208:3 (2007), 1063–1070.
  • W. J. Harvey, “Cyclic groups of automorphisms of a compact Riemann surface”, Quart. J. Math. Oxford Ser. $(2)$ 17 (1966), 86–97.
  • S. Kallel and D. Sjerve, “Genus zero actions on Riemann surfaces”, Kyushu J. Math. 55:1 (2001), 141–164.
  • R. Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics 5, American Mathematical Society, Providence, RI, 1995.
  • D. Robinson, A course in the theory of groups, Grad. Texts in Math. 80, Springer, 1995.
  • D. Singerman, “Subgroups of Fuschian groups and finite permutation groups”, Bull. London Math. Soc. 2 (1970), 319–323.
  • E. Tyszkowska and A. Weaver, “Exceptional points in the elliptic-hyperelliptic locus”, J. Pure Appl. Algebra 212:6 (2008), 1415–1426. http://www.emis.de/cgi-bin/MATH-item?1137.14020Zbl 1137.14020
  • A. Wootton, “Non-normal Belyĭ $p$-gonal surfaces”, pp. 95–108 in Computational aspects of algebraic curves (Moscow, ID, 2005), edited by T. Shaska, Lecture Notes Ser. Comput. 13, World Sci. Publ., Hackensack, NJ, 2005.
  • A. Wootton, “Defining equations for cyclic prime covers of the Riemann sphere”, Israel J. Math. 157 (2007), 103–122.
  • A. Wootton, “The full automorphism group of a cyclic $p$-gonal surface”, J. Algebra 312:1 (2007), 377–396.