Primera Jornada de Geometría y Topología en la Frontera

Del 17 al 19 de Julio del 2024

Minimum genus problem
Yerika Marín-Montilla
Universidad de La Frontera
Abstract
It is well-known that every group of automorphisms of a compact Klein surface of algebraic genus \(g \geqslant 2\) is finite [2], and also that every finite group acts on some compact Klein surface of algebraic genus \(g\geq 1\). More precisely, every finite group acts on some closed Riemann surface [3], also on some compact Klein surface [1] (in the latter case, we can also distinguish between orientable and nonorientable surfaces). 
On the other hand, a finite group may act on Klein surfaces of different genera, so given a finite group, the minimum genus problem consists in finding the least genus on which a group acts. In connection with this minimal action, there are several natural parameters associated with a finite group such has the strong symmetric genus, pure symmetric genus, hyperbolic symmetric genus, the symmetric cross-cap number, etc. (see [4]).
In this talk, we will describe the actions of  the group \(L_p\)  (a non-abelian group of order \(4p\), with  \(p\) be an odd prime such that \(4\) divides \(p-1\)),  
\[ L_{4p}=\langle x, y:\ x^p=1, y^4=1, y^{-1}xy=x^r\rangle \]
 where \(r^4\equiv 1\ ({\rm mod}\ p)\) (and \(r\), \(r^2\) are not congruent to 1 mod \(p\)) on compact Klein surfaces, and as consequence we get some minimal genus for this group.
References
[1] E. Bujalance. A note on the group of automorphisms of a compact Klein surface. Rev. Real Acad. Cienc. Exact. F\'is. Natur. Madrid 81 (1987), no. 3, 565-569. MR 950620
[2] E. Bujalance, J. J. Etayo, J. M. Gamboa and G. Gromadzki. Automorphism groups of compact bordered Klein surfaces,  A combinatorial approach.  Lecture Notes in Mathematics 1439, Springer-Verlag, Berlin, 1990.
[3]  L. Greenberg. Maximal Fuchsian groups. Bull. Amer. Math. Soc., 69 (4) (1963), 569--573.
[4] R. A. Hidalgo, Y. Marín Montilla and S. Quispe. {Quasi-abelian group as automorphism group of Riemann surfaces}, Manuscripta math., (2024). https://doi.org/10.1007/s00229-024-01552-4