The symmetric cross-cap number of the groups C m × D n

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by $\tilde{\sigma}(G)$ . This number is related to other parameters defined on surfaces as the symmetric...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2008-12, Vol.138 (6), p.1197-1213
Main Authors: Gordejuela, J. J. Etayo, Martínez, E.
Format: Article
Language:English
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Summary:Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by $\tilde{\sigma}(G)$ . This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus. The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups C m × D n . As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups C m × D n .
ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210507000169