Abstract
Plainly speaking, a compact Riemann surface, S, can be thought of as the layer of glaze on a g-holed doughnut. A group of symmetries of S is a group that acts on S while preserving some of its underlying structure. We provide an easily understood exposition of the modern techniques used to determine which groups can act as symmetry groups on a compact Riemann surface S of genus g ≥ 2. We then illustrate these techniques by providing necessary and sufficient conditions for the existence of an A 4-action on such a surface (in terms of a signature), and through explicit geometric construction show which of these actions are embeddable.
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Notes
- 1.
Stated more carefully, the extra structure that makes a classical surface a Riemann surface is a complex analytic structure resulting from the presence of a collection of local C ∞-charts, which make the surface a one-dimensional complex manifold. As our approach here does not make direct use of the complex analytic structure of these surfaces, however, we may safely eschew such technical details.
- 2.
Our use of the word “symmetry” here is a colloquial choice to replace the more precise but less widely known “conformal automorphism.”
- 3.
Specifically, the signatures (0; −), (1; −), (0; 2), (0; 2, 2), (0; 2, 2, 2), (0; 2, 2, 2, 2), (0; 3), (0; 3, 3), (0; 3, 3, 3), (0; 2, 3), (0; 2, 2, 3), and (0; 2, 3, 3), can all be excluded, where (h; −) denotes an action with no branch points.
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Peterson, V., Wootton, A. (2017). A Tale of Two Symmetries: Embeddable and Non-embeddable Group Actions on Surfaces. In: Wootton, A., Peterson, V., Lee, C. (eds) A Primer for Undergraduate Research. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-66065-3_2
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DOI: https://doi.org/10.1007/978-3-319-66065-3_2
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