# Automorphisms and coverings of Klein surfaces

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## Abstract

In this thesis the theory of automorphisms and coverings of compact Klein surfaces is discussed by considering a Klein surface as the orbit space of a non-Euclidean crystallographic group. In chapter 1 we set out some of the well-established theory concerning these ideas. In chapter 2 maximal automorphism groups of compact Klein surfaces without boundary are considered. We solve the problem of which groups PSL (2,q) act as maximal automorphism groups of non-orientable Klein surface without boundary. In chapter 3 we discuss cyclic groups acting as automorphism groups of compact Klein surfaces without boundary. It is shown that the maximum order for a cyclic group to be an automorphism group of a compact non-orientable Klein surface without boundary of genus g ≥3 is 2g, if g is odd and 2 (g – 1) if g is even. Chapter 4 is the largest section of the thesis. It is concerned with coverings (possibly folded and ramified) of compact Klein surfaces, mainly Klein surfaces with boundary. All possible two-sheeted connected unramified covering surfaces of a Klein surface are classified and the orientability of a normal n-sheeted cover, for odd n, is determined

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