The pseudo-real genus of a group

Autor: Marston Conder, Stephen Lo
Rok vydání: 2020
Předmět:
Zdroj: Journal of Algebra. 561:149-162
ISSN: 0021-8693
DOI: 10.1016/j.jalgebra.2019.11.032
Popis: A compact Riemann surface is called pseudo-real if it admits anti-conformal (orientation-reversing) automorphisms, but no anti-conformal automorphism of order 2, or equivalently, if the surface is reflexible but not definable over the reals. It is known that there exist pseudo-real surfaces of genus g for every integer g ≥ 2 , and the number of automorphisms of any such surface is bounded above by 12 ( g − 1 ) . In this paper, we extend the concepts of symmetric genus, strong symmetric genus and symmetric cross-cap genus of a group by defining and investigating two new parameters, as follows: (1) the pseudo-real genus ψ ( G ) of a finite group G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which might reverse orientation, and (2) the strong pseudo-real genus ψ ⁎ ( G ) of G is the smallest genus of those pseudo-real surfaces on which G acts faithfully as a group of automorphisms, some of which do reverse orientation, when there exists such a surface for G. Our main theorem is that for every integer g ≥ 2 , there exists a finite group G for which ψ ( G ) = ψ ⁎ ( G ) = g , and hence that the range of each of the functions ψ and ψ ⁎ is the set of all integers g ≥ 2 . We also give an example of a group G for which ψ ⁎ ( G ) is defined but ψ ( G ) ψ ⁎ ( G ) .
Databáze: OpenAIRE
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