On the fixed-point set of automorphisms of non-orientable surfaces without boundary
On the fixed-point set of automorphisms of non-orientable surfaces without boundary
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.
Mathematical Sciences Publishers
Izquierdo, M.
08a1f820-51f7-45df-9e83-ec891d6252e3
Singerman, D.
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
1998
Izquierdo, M.
08a1f820-51f7-45df-9e83-ec891d6252e3
Singerman, D.
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
Izquierdo, M. and Singerman, D.
(1998)
On the fixed-point set of automorphisms of non-orientable surfaces without boundary
(Geometry & Topology Monographs, 1)
Irvine, US.
Mathematical Sciences Publishers
7pp.
(doi:10.2140/gtm.1998.1.295).
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Monograph
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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.
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Published date: 1998
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Local EPrints ID: 29800
URI: http://eprints.soton.ac.uk/id/eprint/29800
ISSN: 1464-8989
PURE UUID: 9e8b4a49-fa31-4f7e-9b39-d91a327cf247
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Date deposited: 19 Apr 2007
Last modified: 09 Apr 2024 09:44
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Author:
M. Izquierdo
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