Genus Zero Actions on Riemann Surfaces

1999/12/21 by Sadok Kallel, Kallel, Sadok, Denis Sjerve +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.math/9912176

21 pages, 5 figures

arxiv created 1999/12/21 · openalex publication_date 1999/12/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we determine all finite groups G that can act on some compact Riemann surface M with the property that if H is any non-trivial subgroup of G, then the orbit surface M/H is the Riemann sphere. The idea is to look at the induced action on the vector space of holomorphic differentials on M (in the positive genus case) and then use the old-known (Wolf) classification of groups admitting fixed point-free linear actions. A description of the corresponding group actions is given in terms of Fuchsian representations.

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