Fuchsian groups generated by half-turns and geometrical characterization of hyperelliptic and symmetric Riemann surfaces

Abstract

We construct a special type of fundamental regions for any Fuchsian group $F$ generated by an even number of half-turns, and for certain non-Euclidean crystallographic groups (NEC groups in short). By comparing these regions we give geometrical conditions in order to $F$ be the canonical Fuchsian subgroup of one of those NEC groups. Precisely speaking, we deal with NEC groups of algebraic genus $0$ having all periods in the signature equal to $2$. By means of these conditions we give a characterization of hyperelliptic and symmetric Riemann surfaces.