Infinite families of surface symmetries
Let \(\Sigma_ g\) denote a closed orientable surface of genus \(g \geq 2\). A finite symmetry group \(G\) of \(\Sigma_ g\) is a group of orientation- preserving self-homeomorphisms. \textit{Wiman} [Bihang. Till. K. Svenska Vet.-Akad. Hadlingar 1-23 (Stockholm 1895-6)] showed that for every \(g\), the cyclic group of order \(4g+2\) is a symmetry group of \(\Sigma_ g\). R.D.M. \textit{Accola} [Trans. Am. Math. Soc. 131, 398-408 (1968; Zbl 0179.114)] and, independently the reviewer [J. Lond. Math. Soc. 44, 265-272 (1969; Zbl 0164.379)] showed the existence of symmetry groups of order \(8g+8\), for every \(g\). These groups contain cyclic subgroups of index 4. The author proves a general result which encompasses these two cases as follows: If there are infinitely many values of \(g\) such that there is a symmetry group \(G\) of \(\Sigma_ g\) of order \(ag+b\), where \(a,b\) are fixed rationals with \(a>0\), and \(a+b \neq 0\), then \(G\) contains a cyclic subgroup of bounded index, the bound depending only on \(a,b\). Further consequences on the general structure of such \(G\) are drawn, and these results are further extended in a sequel by \textit{M. D. E. Conder} and the author [Discrete groups and geometry, Lond. Math. Soc. Lect. Notes 173, pp. 47-56 (1992)].
scientific article; zbMATH DE number 78211
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