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The pseudo-real genus of a group
Journal of Algebra ( IF 0.9 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.jalgebra.2019.11.032
Marston D.E. Conder , Stephen Lo

Abstract 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 ) .

中文翻译:

群的伪真属

摘要 如果紧凑黎曼曲面允许反共形(方向反转)自同构,但不允许 2 阶反共形自同构,或者等效地,如果表面是可反折的但不能在实数上定义,则称为伪实面。已知对于每个整数 g ≥ 2 都存在类 g 的伪实曲面,并且任何这样的曲面的自同构数都以 12 ( g − 1 ) 为界。在本文中,我们通过定义和研究两个新参数来扩展群的对称属、强对称属和对称交叉盖属的概念,如下: (1) 有限元的伪实属 ψ ( G )群 G 是那些伪真实曲面中最小的一个类,在这些伪真实曲面上,G 忠实地充当一组自同构,其中一些可能会反转方向,(2) G 的强伪实属 ψ ⁎ ( G ) 是那些伪实表面中最小的属,G 在这些表面上忠实地充当一组自同构,当存在这样的自同构时,其中一些反同构G 的曲面。我们的主要定理是,对于每个整数 g ≥ 2 ,存在一个有限群 G,其中 ψ ( G ) = ψ ⁎ ( G ) = g ,因此每个函数 ψ 和 ψ 的范围⁎ 是所有整数 g ≥ 2 的集合。我们还给出了一个群 G 的例子,其中定义了 ψ ⁎ ( G ) 但 ψ ( G ) ψ ⁎ ( G) 。因此,每个函数 ψ 和 ψ ⁎ 的范围是所有整数 g ≥ 2 的集合。我们还给出了一个群 G 的例子,其中定义了 ψ ⁎ ( G ) 但 ψ ( G ) ψ ⁎ ( G) 。因此,每个函数 ψ 和 ψ ⁎ 的范围是所有整数 g ≥ 2 的集合。我们还给出了一个群 G 的例子,其中定义了 ψ ⁎ ( G ) 但 ψ ( G ) ψ ⁎ ( G) 。
更新日期:2020-11-01
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