A Klein surface with boundary of algebraic genus , has at most automorphisms. The groups attaining this upper bound are called -groups, and the corresponding surfaces are said to have maximal symmetry. The -groups are characterized by a partial presentation by generators and relators. The alternating groups were proved to be -groups when by M. Conder. In this work we prove that is an -group if and only if or . In addition, we describe topologically the surfaces with maximal symmetry having as automorphism group, in terms of the partial presentation of the group. As an application we determine explicitly all such surfaces for . Each finite group acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, . If is an -group then . We end our work by calculating the real genus of the alternating groups which are not -groups.
Cite this article
José Javier Etayo Gordejuela, Ernesto Martínez, The real genus of the alternating groups. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 865–894DOI 10.4171/RMI/558