The real genus of the alternating groups

Abstract

A Klein surface with boundary of algebraic genus $\mathfrak{p}\ge 2$, has at most $12(\mathfrak{p}-1)$ automorphisms. The groups attaining this upper bound are called $M^{\ast }$-groups, and the corresponding surfaces are said to have maximal symmetry. The $M^{\ast }$-groups are characterized by a partial presentation by generators and relators. The alternating groups $A_n$ were proved to be $M^{\ast }$-groups when $n\ge 168$ by M. Conder. In this work we prove that $A_n$ is an $M^{\ast }$-group if and only if $n\ge 13$ or $n=5,10$. In addition, we describe topologically the surfaces with maximal symmetry having $A_n$ as automorphism group, in terms of the partial presentation of the group. As an application we determine explicitly all such surfaces for $n\le 14$. Each finite group $G$ acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, $\rho (G)$. If $G$ is an $M^{\ast }$-group then $\rho (G)=\frac{o(G)}{12}+1$. We end our work by calculating the real genus of the alternating groups which are not $M^{\ast }$-groups.