On the minimum genus problem on bordered Klein surfaces

Abstract

The minimum genus problem consists in determining the minimum algebraic genus of a surface on which a viven group G acts. For cyclic groups G this problem on bordered Klein surfaces was solved in 1989. The next step is to fix the number of boundary components of the surface and to obtain the minimum algebraic genus, and so the minimum topological genus. It was achieved for cyclic groups of prime and prime-power order in the nineties. In this work the corresponding results for cyclic groups of order N=pq, where p and q are different odd primes, is obtained. There appear different results depending on the orientability of the surface. Finally we obtain general results when the number of boundary components is small, which are valid for any odd N.