Hyperelliptic compact non-orientable Klein surfaces without boundary
Let X be a compact non-orientable Klein surface (KS). X may be represented as \(D/\Gamma\), D being the hyperbolic plane and \(\Gamma\) a non-Euclidean crystallographic group (NEC group), i.e. a discrete subgroup of the group G of all isometries of the hyperbolic plane, including orientation-reversing ones. X is q-hyperelliptic if there exists an involution \(\Phi\) of X such that \(X/<\Phi >\) has algebraic genus q. In the theorem 2.2 a characterisation of those compact non- orientable KS's without boundary is given, which are q-hyperelliptic. If such a q-hyperelliptic KS has the algebraic genus \(p>4q+1,\) then: (1) the automorphism \(\Phi\) (the automorphism of q-hyperellipticity) is central and unique (see proposition 2.3); (2) if additionally \(X/<\Phi >\) has boundary, then \(|\text{Aut}(X)| \leq 12(p-1)\). The equality is possible only for \(p=2\) and in this case Aut\((X)\) is the dihedron group \(D_ 6\) with 12 elements (s. theorem 3.1).
scientific article; zbMATH DE number 4127384
scientific article