#### 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

Email: mio@mdh.se, ds@maths.soton.ac.uk

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