On automorphisms of Klein surfaces with invariant subsets. (English) Zbl 1271.30014

Let \(X\) denote a Klein surface. These surfaces can be orientable or not and they can have a boundary or not. An unbordered orientable Klein surface is a classical Riemann surface and unbordered non-orientable Klein surfaces are also called non-orientable Riemann surfaces.
Let \(G\) be a group of automorphisms of \(X\). If \(X\) is an unbordered Klein surface with topological genus \(g\geq 2\) in the orientable case and \(g\geq 3\) otherwise, then \(G\) has at most \(84(g-\epsilon)\) automorphisms, where \(\epsilon=1\) and \(2\), respectively. This bound can be reduced if there are \(G\)-invariant subsets. K. Oikawa [Kodai Math. Semin. Rep. 8, 23–30 (1956; Zbl 0072.07702)] proved in the orientable case that \(|G|\leq 12(g-1)+6k\), where \(k\) is the cardinality of a \(G\)-invariant subset. T. Arakawa [Osaka J. Math. 37, No. 4, 823–846 (2000; Zbl 0981.30029)] obtained new bounds considering a number \(s=2\) or \(3\) of \(G\)-invariant subsets. In particular, he showed that the bound for \(s=3\) is sharp for infinitely many configurations.
In the paper under review, the authors generalize these results and obtain many new bounds studying different types of Klein surfaces: classical Riemann surfaces (in particular, they obtain that the Arakawa bound for \(s=2\) is never attained), bordered orientable Klein surfaces, non-orientable Riemann surfaces and bordered non-orientable Klein surfaces. Moreover, they also consider a number \(s\geq 4\) of \(G\)-invariant subsets. In the bordered cases they use results by A. L. Pérez del Pozo [Manuscr. Math. 122, No. 2, 163–172 (2007; Zbl 1129.30028)] to obtain new bounds. At the end of the paper, the authors study the particular cases of \(q\)-hyperelliptic and \(q\)-trigonal non-orientable bordered Klein surfaces.
As a simple example of the kind of bounds that the authors obtain, we consider here the case of a Riemann surface of genus \(g\geq 2\), a group of automorphisms \(G\) with \(B_1, \dots, B_n\), \(G\)-invariant irreducible subsets of cardinalities \(q_1, \dots, q_n\), where \(s\geq 4\) of these subsets are proper. In these conditions each \(q_i\) divides \(|G|\) and \[ |G|\leq \frac{2}{s-2}(g-1) + \frac{q_1+\cdots + q_s}{s-2}. \] Conversely, for each \(s\) these bounds are attained for infinitely many values of \(g\).
As a consequence of the new bounds in the paper, some known results from different authors are easily obtained.


30F10 Compact Riemann surfaces and uniformization
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
30F50 Klein surfaces
14H37 Automorphisms of curves
14H45 Special algebraic curves and curves of low genus
14H55 Riemann surfaces; Weierstrass points; gap sequences
Full Text: Euclid


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