Riemann surfaces with maximal real symmetry

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• 作者：E. Bujalance^a ; ¹ ; ^{eb@mat.uned.es" class="auth_mail" title="E-mail the corresponding author} ; F.J. Cirre^a ; ¹ ; ^{jcirre@mat.uned.es" class="auth_mail" title="E-mail the corresponding author} ; M.D.E. Conder^b ; ² ; ^{m.conder@auckland.ac.nz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Riemann surface ; Klein surface ; Automorphisms ; Symmetry type
• 刊名：Journal of Algebra
• 出版年：2015
• 出版时间：1 December 2015
• 年：2015
• 卷：443
• 期：Complete
• 页码：494-516
• 全文大小：467 K

文摘

Let S   be a compact Riemann surface of genus g>1$g>1$, and let τ:S→S$\tau :S\to S$ be any anti-conformal automorphism of S, of order 2. Such an anti-conformal involution is known as a symmetry of S, and the species of all conjugacy classes of all symmetries of S constitute what is known as the symmetry type of S. The surface S is said to have maximal real symmetry   if it admits a symmetry τ:S→S$\tau :S\to S$ such that the compact Klein surface S/τ$S/\tau$ has maximal symmetry (which means that S/τ$S/\tau$ has the largest possible number of automorphisms with respect to its genus). If τ   has fixed points, which is the only case we consider here, then the maximum number of automorphisms of S/τ$S/\tau$ is 12(g−1)$12(g-1)$. In the first part of this paper, we develop a computational procedure to compute the symmetry type of every Riemann surface of genus g   with maximal real symmetry, for given small values of g>1$g>1$. We have used this to find all of them for 1<g≤101$1<g\le 101$, and give details for 1<g≤25$1<g\le 25$ (in an appendix). In the second part, we determine the symmetry types of four infinite families of Riemann surfaces with maximal real symmetry. We also determine the full automorphism group of the Klein surface S/τ$S/\tau$ associated with each symmetry τ:S→S$\tau :S\to S$.