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Discreteness of subgroups of SL(2,𝐂) containing elliptic elements. (English) Zbl 1017.30053
The authors consider anticonformal automorphims of closed Riemann surfaces and Schottky groups. A conformal or anticonformal automorphims f:SS of a closed Riemann surface S is called of Schottky type if there is a Schottky uniformization of S, say (Ω,G,P:ΩS), such that f can be lifted by P· A necessary and sufficient condition for a conformal automorphism to be of Schottky type, is a special conditon (A), was given by R. A. Hidalgo. The main results are: Theorem 2. Let f:SS be a conformal automorphism of a closed Riemann surface S· If there is an anticonformal automorphism g:SS such that g 2 =f, then f is of Schottky type. If f does not satisfy the condition (A), then there is no orientation-reversing homeomorphism σ:SS such that f=σ 2 · Theorem 3. Let f:SS be a conformal automorphism of Schottky type of odd order. Then there is an orientation-reversing homeomorphism g:SS such that f=g 2 · Theorem 6. Let σ:SS be an anticonformal automorphism of a closed Riemann surface S· Set R the quotient surface obtained by the action of f=σ 2 , and τ the anticonformal involution induced by σ on R· If τ has fixed points, then σ is of Schottky type. Theorem 9. Let σ:SS be an anticonformal automorphism of order 6 of a closed Riemann surface S· Then σ is of Schottky type.
MSC:
30F40Kleinian groups