Quasi-abelian group acting on pseudo-real Riemann surfaces
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| Saúl Quispe |
| Universidad de La Frontera |
| 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. |
| 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 Bujalance-Conder-Costa, 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 quasi-abelian group of order \(2^n\), |
| \[QA_n=\langle x,y:\ x^{2^{n-1}}=y^2=1, yxy=x^{2^{n-2}+1}\rangle;\ n\geq 4\] |
| on the pseudo-real Riemann surfaces. We consider two cases \(QA_n\) has anticonformal elements or \(QA_n\) only contains conformal elements. This is joint work wit R. A. Hidalgo and Y. Marín-Montilla. |
