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Hyperelliptic compact non-orientable Klein surfaces without boundary. (English) Zbl 0688.14024
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).
Reviewer: A.Duma

##### MSC:
 14H05 Algebraic functions and function fields in algebraic geometry 20H15 Other geometric groups, including crystallographic groups 30F10 Compact Riemann surfaces and uniformization
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