×

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.

MSC:

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
PDF BibTeX XML Cite
Full Text: Euclid

References:

[1] N.L. Alling and N. Greenleaf: Foundations of the Theory of Klein Surfaces, Lecture Notes in Mathematics 219 , Springer, Berlin, 1971. · Zbl 0225.30001
[2] T. Arakawa: Automorphism groups of compact Riemann surfaces with invariant subsets , Osaka J. Math. 37 (2000), 823-846. · Zbl 0981.30029
[3] E. Bujalance: Cyclic groups of automorphisms of compact nonorientable Klein surfaces without boundary , Pacific J. Math. 109 (1983), 279-289. · Zbl 0545.30033
[4] E. Bujalance, J.J. Etayo, J.M. Gamboa and G. Gromadzki: Automorphism Groups of Compact Bordered Klein Surfaces, Lecture Notes in Mathematics 1439 , Springer, Berlin, 1990. · Zbl 0709.14021
[5] J.A. Bujalance and B. Estrada: \(q\)-hyperelliptic compact nonorientable Klein surfaces without boundary , Int. J. Math. Math. Sci. 31 (2002), 215-227. · Zbl 1010.30028
[6] T.C. Chau: A note concerning Fox’s paper on Fenchel’s conjecture , Proc. Amer. Math. Soc. 88 (1983), 584-586. · Zbl 0497.20035
[7] C. Corrales, J.M. Gamboa and G. Gromadzki: Automorphisms of Klein surfaces with fixed points , Glasg. Math. J. 41 (1999), 183-189. · Zbl 0927.30027
[8] B. Estrada and E. Martí nez: \(q\)-trigonal Klein surfaces , Israel J. Math. 131 (2002), 361-374. · Zbl 1013.30026
[9] H.M. Farkas and I. Kra: Riemann Surfaces, Graduate Texts in Mathematics 71 , Springer, New York, 1980.
[10] N. Greenleaf and C.L. May: Bordered Klein surfaces with maximal symmetry , Trans. Amer. Math. Soc. 274 (1982), 265-283. · Zbl 0504.14020
[11] R.H. Fox: On Fenchel’s conjecture about \(F\)-groups , Mat. Tidsskr. B. 1952 (1952), 61-65. · Zbl 0049.15404
[12] G. Gromadzki: On conjugacy of \(p\)-gonal automorphisms of Riemann surfaces , Rev. Mat. Complut. 21 (2008), 83-87. · Zbl 1148.14013
[13] A. Hurwitz: Über algebraische Gebilde mit eindeutigen Transformationen in sich , Math. Ann. 41 (1893), 402-442. · JFM 24.0380.02
[14] A.M. Macbeath: The classification of non-euclidean plane crystallographic groups , Canad. J. Math. 19 (1967), 1192-1205. · Zbl 0183.03402
[15] C.L. May: Automorphisms of compact Klein surfaces with boundary , Pacific J. Math. 59 (1975), 199-210. · Zbl 0422.30037
[16] K. Oikawa: Notes on conformal mappings of a Riemann surface onto itself , Kōdai Math. Sem. Rep. 8 (1956), 23-30. · Zbl 0072.07702
[17] T. Szemberg: Automorphisms of Riemann surfaces with two fixed points , Ann. Polon. Math. 55 (1991), 343-347. · Zbl 0767.30035
[18] Á.L. Pérez del Pozo: Automorphism groups of compact bordered Klein surfaces with invariant subsets , Manuscripta Math. 122 (2007), 163-172. · Zbl 1129.30028
[19] D. Singerman: Automorphisms of compact non-orientable Riemann surfaces , Glasgow Math. J. 12 (1971), 50-59. · Zbl 0232.30012
[20] H.C. Wilkie: On non-Euclidean crystallographic groups , Math. Z. 91 (1966), 87-102. · Zbl 0166.02602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.