×

Discreteness of subgroups of SL(2,\(\mathbf C\)) containing elliptic elements. (English) Zbl 1017.30053

The authors consider anticonformal automorphims of closed Riemann surfaces and Schottky groups. A conformal or anticonformal automorphims \(f:S\rightarrow S\) of a closed Riemann surface \(S\) is called of Schottky type if there is a Schottky uniformization of \(S,\) say \((\Omega,G,P:\Omega\rightarrow S),\) 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 \(f:S\rightarrow S\) be a conformal automorphism of a closed Riemann surface \(S.\) If there is an anticonformal automorphism \(g:S\rightarrow S\) 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 \(\sigma:S\rightarrow S\) such that \(f = \sigma^{2}.\) Theorem 3. Let \(f:S\rightarrow S\) be a conformal automorphism of Schottky type of odd order. Then there is an orientation-reversing homeomorphism \(g:S\rightarrow S\) such that \(f = g^{2}.\) Theorem 6. Let \(\sigma:S\rightarrow S\) 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 \(\sigma:S\rightarrow S\) be an anticonformal automorphism of order 6 of a closed Riemann surface \(S.\) Then \(\sigma\) is of Schottky type.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
PDFBibTeX XMLCite
Full Text: DOI