| Generalized quasi-dihedral group acting on pseudo-real Riemann surfaces |
| Yerika Marín Montilla |
| Universidad de La Frontera (Chile) |
| Abstract: |
| A closed Riemann surface of genus \[g\geq 2\] is called pseudo-real if it has anticonformal automorphisms but no anticonformal involutions. These Riemann surfaces, together with real Riemann surfaces, form the real locus of the moduli space \[\mathcal{M}_g\] of closed Riemann surfaces of genus \[g\geq 2\] . On the other hand, pseudo-real Riemann surfaces are examples of Riemann surfaces which cannot be defined over their field of moduli [1]. |
| In general, a finite group might not be realized as the group of conformal/anticonformal automorphisms, admitting anticonformal ones, of a pseudo-real Riemann surface, for instance, in [2], it was observed that a necessary condition for that to happen is for the group to have order a multiple of \[4\] . In this talk, we consider conformal/anticonformal actions of the generalized quasi-dihedral group of order \[8n\] , \[G_n=\langle x, y:\ x^{4n}=y^2=1, yxy=x^{2n-1}\rangle \quad (\text{for $n\geq 2$})\] on pseudo-real Riemann surfaces. We consider two cases either \[G_n\] has anticonformal elements or \[G_n\] only contains conformal elements [3]. |
| This is part of my Ph.D. Thesis, under the adviser Saúl Quispe and Rubén A. Hidalgo. |
| References |
| [1] M. Artebani, S. Quispe and C. Reyes. Automorphism groups of pseudoreal Riemann surfaces. Journal of Pure and Applied Algebra 221 (2017), 2383–2407. |
| [2] E. Bujalance, M. D. E. Conder and A. F. Costa. Pseudo-real Riemann surfaces and chiral regular maps, Trans. Am. Math. Soc. 362 (7) (2010), 3365–3376. |
| [3] R. A. Hidalgo, Y. Marín Montilla and S. Quispe. Generalized quasi-dihedral group as automorphism group of Riemann surfaces, Preprint 2022. |
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