Geometry & Topology Monographs 1 (1998), The Epstein Birthday Schrift, paper no. 13, pages 295-301.

On the fixed-point set of automorphisms of non-orientable surfaces without boundary

M Izquierdo, D Singerman

Abstract. 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.

Keywords. Automorphism of a surface, NEC group, universal covering transformation group, oval, fixed-point set

AMS subject classification. Primary: 20F10, 30F10. Secondary: 30F35, 51M10, 14H99.

E-print: arXiv:math.GT/9810193

Submitted: 15 November 1997. Published: 27 October 1998.

Notes on file formats

M Izquierdo, D Singerman
Department of Mathematics
Malardalen University
721 23 Vasteras, Sweden

Department of Mathematics
University of Southampton
Southampton SO17,1BJ UK


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