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

M Izquierdo and D Singerman

Geometry & Topology Monographs 1 (1998) 295–301

DOI: 10.2140/gtm.1998.1.295

arXiv: math.GT/9810193


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.

For David Epstein on the occasion of his sixtieth birthday.


automorphism of a surface, NEC group, universal covering transformation group, oval, fixed-point set

Mathematical Subject Classification

Primary: 20F10, 30F10

Secondary: 14H99, 30F35, 51M10


Received: 15 November 1997
Published: 27 October 1998

M Izquierdo
Department of Mathematics
Mälardalen University
721 23 Västerås
D Singerman
Department of Mathematics
University of Southampton
SO17 1BJ
United Kingdom