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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 $f:S\to S$ of a closed Riemann surface $S$ is called of Schottky type if there is a Schottky uniformization of $S,$ say $\left({\Omega },G,P:{\Omega }\to S\right),$ 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 $f:S\to S$ be a conformal automorphism of a closed Riemann surface $S·$ If there is an anticonformal automorphism $g:S\to S$ 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 $\sigma :S\to S$ such that $f={\sigma }^{2}·$ Theorem 3. Let $f:S\to S$ be a conformal automorphism of Schottky type of odd order. Then there is an orientation-reversing homeomorphism $g:S\to S$ such that $f={g}^{2}·$ Theorem 6. Let $\sigma :S\to S$ be an anticonformal automorphism of a closed Riemann surface $S·$ Set $R$ the quotient surface obtained by the action of $f={\sigma }^{2},$ and $\tau$ the anticonformal involution induced by $\sigma$ on $R·$ If $\tau$ has fixed points, then $\sigma$ is of Schottky type. Theorem 9. Let $\sigma :S\to S$ be an anticonformal automorphism of order 6 of a closed Riemann surface $S·$ Then $\sigma$ is of Schottky type.
##### MSC:
 30F40 Kleinian groups