A GAP package for braid orbit computation and applications. Let G be a finite group. By Riemann’s existence theorem, braid orbits of generating systems for G with product 1 correspond to irreducible familes of coverings of the Riemann sphere with monodromy group G. In this article, the authors describe a GAP package for computing the braid orbits. They then apply these techniques to several interesting cases – in particular to the problem of trying to classify indecomposable rational functions (in characteristic zero) of degree n whose monodromy group is not the alternating or symmetric group of degree n. The group theoretic possibilities have almost been completely worked out by various authors and this program should be able to describe the various possibilities up to equivalence of covers. They also consider the case of indecomposable covers of degree n from the generic Riemann surface of genus g when g=2 or 3 (if g≥4, by results of the reviewer and various authors – see the references– only symmetric and alternating groups of degree n can occur).

References in zbMATH (referenced in 18 articles , 1 standard article )

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  1. van Hoeij, Mark; Kunwar, Vijay Jung: Classifying (almost)-Belyi maps with five exceptional points (2019)
  2. König, Joachim: Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups (2017)
  3. König, Joachim: On rational functions with monodromy group (M_11) (2017)
  4. James, Adam; Magaard, Kay; Shpectorov, Sergey: The lift invariant distinguishes components of Hurwitz spaces for (A_5) (2015)
  5. Costa, Antonio F.; Izquierdo, Milagros: Equisymmetric strata of the singular locus of the moduli space of Riemann surfaces of genus 4 (2010)
  6. Fried, Michael D.: Alternating groups and moduli space lifting invariants (2010)
  7. Cadoret, Anna; Tamagawa, Akio: Stratification of Hurwitz spaces by closed modular subvarieties (2009)
  8. Magaard, Kay; Shaska, Tanush; Völklein, Helmut: Genus 2 curves that admit a degree 5 map to an elliptic curve (2009)
  9. Vidunas, Raimundas; Kitaev, Alexander V.: Computation of highly ramified coverings (2009)
  10. Cadoret, Anna: Lifting results for rational points on Hurwitz moduli spaces (2008)
  11. Accola, Robert D. M.; Previato, Emma: Covers of tori: genus two (2006)
  12. Fried, Michael D.: The Main Conjecture of modular towers and its higher rank generalization (2006)
  13. Glass, D.; Pries, R.: On the moduli space of Klein four covers of the projective line (2005)
  14. Staszewski, R.; Völklein, H.; Wiesend, G.: Counting generating systems of a finite group from given conjugacy classes. (2005)
  15. Bouw, Irene I.; Wewers, Stefan: Reduction of covers and Hurwitz spaces (2004)
  16. Dettweiler, Michael: Plane curve complements and curves on Hurwitz spaces (2004)
  17. Magaard, Kay; Völklein, Helmut: The monodromy group of a function on a general curve (2004)
  18. Magaard, Kay; Shpectorov, Sergey; Völklein, Helmut: A GAP package for braid orbit computation and applications (2003)