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On
the fixed-point set of automorphisms of non-orientable
surfaces without boundary
M Izquierdo and D Singerman |
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Geometry & Topology Monographs 1 (1998) 295–301
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DOI: 10.2140/gtm.1998.1.295
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arXiv: math.GT/9810193 |
Abstract |
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Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable surface with one exception; namely, when this element has order 2. In this case the fixed-point set may have simple closed curves (called ovals) as well as fixed points. In this note we extend Macbeath’s results to include the number of ovals and also determine whether they are twisted or not. |
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For David Epstein on the occasion of his sixtieth birthday. |
Keywords
automorphism of a surface, NEC group, universal covering
transformation group, oval, fixed-point set
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Mathematical Subject Classification
Primary: 20F10, 30F10
Secondary: 30F35, 51M10, 14H99
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References |
Publication
Received: 15 November 1997
Published: 27 October 1998
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Authors |
| M Izquierdo | |
| Department of Mathematics Mälardalen University 721 23 Västerås Sweden |
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| D Singerman | |
| Department of Mathematics University of Southampton Southampton SO17 1BJ United Kingdom |
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