Complete nonorientable minimal surfaces with the highest symmetry group

In this paper we present a family of complete nonorientable highly symmetrical minimal surfaces with arbitrary topology and one end. For each topology we construct the most symmetrical example with one end. Furthermore, if the Euler characteristic of the closed associated surface is even, our examples minimize the energy (or the degree of the Gauss map) among the surfaces with their symmetry, topology and one end. Finally, we characterize these surfaces in terms of their topology, symmetry, total curvature and number of ends.