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A survey on the minimum genus and maximum order problems for bordered Klein surfaces

Published online by Cambridge University Press:  05 July 2011

By
E. Bujalance ,
F. J. Cirre ,
J. J. Etayo ,
G. Gromadzki  and
E. Martínez
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
E. Bujalance
Affiliation:
UNED, Spain
F. J. Cirre
Affiliation:
UNED, Spain
J. J. Etayo
Affiliation:
Universidad Complutense, Spain
G. Gromadzki
Affiliation:
Gdańsk University, Poland
E. Martínez
Affiliation:
UNED, Spain
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

Every finite group acts as a group of automorphisms of some compact bordered Klein surface of algebraic genus g ≥ 2. The same group G may act on different genera and so it is natural to look for the minimum genus on which G acts. This is the minimum genus problem for the group G. On the other hand, for a fixed integer g ≥ 2, there are finitely many abstract groups acting as a group of automorphisms of some compact bordered Klein surface of algebraic genus g. The condition g ≥ 2 assures that all such groups are finite. So it makes sense to look for the largest order of groups G acting on some surface of genus g when g is fixed and G runs over a prescribed family F of groups. This is the maximum order problem for the family F. There is a significant amount of research dealing with these two problems (or with some of their variations), and the corresponding results are scattered in the literature. The purpose of this survey is to gather some of these results, paying special attention to important families of finite groups.

Introduction

A natural extension of the definition of a compact Riemann surface, which is orientable and has no boundary, is to allow dianalytic transition functions, that is, functions which are either analytic or the composite of complex conjugation with an analytic function.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 161 - 182
Publisher: Cambridge University Press
Print publication year: 2011

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References

[All81] N. L., Alling, Real elliptic curves, North-Holland Mathematics Studies, 54. Notas de Matemática, 81. North-Holland Publishing Co., Amsterdam-New York, 1981.Google Scholar
[AG71] N. L., Alling, N., Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Math. 219, Springer-Verlag (1971).
[BCT03] E., Bujalance, F. J., Cirre, P., Turbek, Groups acting on bordered Klein surfaces with maximal symmetry, in Groups St Andrews 2001 in Oxford, Vol. 1 (C. M., Campbell et al., eds.), London Math. Soc. Lecture Note Ser. 304 (CUP, Cambridge 2003), 50–58.Google Scholar
[BEG86] E., Bujalance, J. J., Etayo, J. M., Gamboa, Automorphism groups of real algebraic curves of genus 3, Proc. Japan Acad. 62 (1986), 40–42.Google Scholar
[BEGG90] E., Bujalance, J. J., Etayo, J. M., Gamboa, G., Gromadzki, Automorphism Groups of Compact Bordered Klein Surfaces, Lecture Notes in Math. 1439, Springer-Verlag, Berlin, 1990.
[BEGM89] E., Bujalance, J. J., Etayo, J. M., Gamboa, G., Martens, Minimal genus of Klein surfaces admitting an automorphism of a given order, Arch. Math. (Basel) 52 (1989), no. 2, 191–202.Google Scholar
[BG84] E., Bujalance, J. M., Gamboa, Automorphism groups of algebraic curves of ℝn of genus 2, Arch. Math. (Basel) 42 (1984), 229–237.Google Scholar
[BGM95] E., Bujalance, J. M., Gamboa, C., Maclachlan, Minimum topological genus of compact bordered Klein surfaces admitting a prime-power automorphism, Glasgow Math. J. 37 (1995), no. 2, 221–232.Google Scholar
[BG90] E., Bujalance, G., Gromadzki, On nilpotent groups of automorphisms of compact Klein surfaces, Proc. Amer. Math. Soc. 108 (3), 1990, 749–759.Google Scholar
[BM89] E., Bujalance, E., Martínez, A remark on NEC groups representing surfaces with boundary, Bull. London Math. Soc. 21 (1989), no. 3, 263–266.Google Scholar
[Buj88] J. A., Bujalance, Topological types of Klein surfaces with a maximum order automorphism, Glasgow Math. J. 30 (1988), no. 1, 87–96. Corrigendum: Glasgow Math. J.30, (1988), no. 3, 369.Google Scholar
[Cano] C., Cano, Ph. D. Thesis, in preparation.
[Con80] M. D. E., Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), 75–86.Google Scholar
[EM04] J. J., Etayo, E., Martínez, The real genus of the groups C2m × Dn (Spanish), Mathematical contributions in honor of Professor Enrique Outerelo Domínguez (Spanish), 171–182, Homen. Univ. Complut., Editorial Complutense, Madrid, 2004.
[EM06] J. J., Etayo, E., Martínez, The real genus of cyclic by dihedral and dihedral by dihedral groups, J. Algebra 296 (2006), 145–156.Google Scholar
[EM08] J. J., Etayo, E., Martínez, The real genus of the alternating groups, Rev. Mat. Iberoamericana 24 (2008), 865–894.Google Scholar
[GM82] N., Greenleaf, C. L., May, Bordered Klein surfaces with maximal symmetry, Trans. Amer. Math. Soc. 274 (1982), 265–283.Google Scholar
[Gro92] G., Gromadzki, On soluble groups of automorphisms of non-orientable Klein surfaces, Fund. Math. 14 (1992), 215–227.Google Scholar
[GM02] G., Gromadzki, B., Mockiewicz, The groups of real genus 6, 7 and 8, Houston J. Math. 28 (2002), 691–699.Google Scholar
[Har66] W. J., Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. 17 (1966), 86–97.Google Scholar
[KW04] J. H., Kwak, Y., Wang, Real genus of minimal non nilpotent groups, J. Algebra 281 (2004), 150–160.Google Scholar
[May75] C. L., May, Automorphisms of compact Klein surfaces with boundary, Pacific J. Math. 59 (1975), no. 1, 199–210.Google Scholar
[May77a] C. L., May, Cyclic automorphism groups of compact bordered Klein surfaces, Houston J. Math. 3 (1977), 395–405.Google Scholar
[May77b] C. L., May, A bound for the number of automorphisms of a compact Klein surface with boundary, Proc. Amer. Math. Soc. 63 (1977), 273–280.Google Scholar
[May77c] C. L., May, Large automorphisms groups of compact Klein surfaces with boundary I, Glasgow Math. J. 18 (1977), 1–10.Google Scholar
[May80] C. L., May, Maximal symmetry and fully wound coverings, Proc. Amer. Math. Soc. 79 (1980), 23–31.Google Scholar
[May84] C. L., May, The species of Klein surfaces with maximal symmetry of low genus, Pacific J. Math. 111 (1984), 371–394.Google Scholar
[May86] C. L., May, A family of M*-groups, Can. J. Math. 38 (1986), 1094–1109.Google Scholar
[May87] C. L., May, Nilpotent automorphism groups of bordered Klein surfaces, Proc. Amer. Math. Soc. 101 (1987), 287–292.Google Scholar
[May88] C. L., May, Supersolvable M*-groups, Glasgow Math. J. 30 (1988), 31–40.Google Scholar
[May91] C. L., May, Complex doubles of bordered Klein surfaces with maximal symmetry, Glasgow Math. J. 33 (1991), 61–67.Google Scholar
[May92] C. L., May, The groups of real genus 4, Michigan Math. J. 39 (1992), 219–228.Google Scholar
[May93] C. L., May, Finite groups acting on bordered surfaces and the real genus of a group, Rocky Mountain J. Math. 23 (1993), no. 2, 707–724.Google Scholar
[May94a] C. L., May, Groups of small real genus, Houston J. Math. 20 (1994), 393–408.Google Scholar
[May94b] C. L., May, Finite metacyclic groups acting on bordered surfaces, Glasgow Math. J. 36 (1994), 233–240.Google Scholar
[May94c] C. L., May, A lower bound for the real genus of a finite group, Canad. J. Math. 46 (1994), 1275–1286.Google Scholar
[May98] C. L., May, Finite 3-groups acting on bordered surfaces, Glasgow Math. J. 40 (1998), no. 3, 463–472.Google Scholar
[May01] C. L., May, Real genus actions of finite simple groups, Rocky Mountain J. Math. 31 (2001), 539–551.Google Scholar
[May07a] C. L., May, The real genus of 2-groups, J. Algebra Appl. 6 (2007), 103–118.Google Scholar
[May07b] C. L., May, Groups of even real genus, J. Algebra Appl. 6 (2007), 973–989.Google Scholar
[May07c] C. L., May, The real genus of groups of odd order, Rocky Mountain J. Math. 37 (2007), no. 4, 1251–1269.Google Scholar
[May09a] C. L., May, The real genus of direct products Zn × G, Houston J. Math. 35 (2009), 23–37.Google Scholar
[May09b] C. L., May, The groups of real genus ρ ≤ 16, Rocky Mountain J. Math., 39 (2009), no. 5, 1573–1595.Google Scholar
[McC90] D., McCullough, Minimal genus of abelian actions on Klein surfaces with boundary. Math. Z. 205 (1990), 421–436.Google Scholar
[Moc04] B., Mockiewicz, Real genus 12, Rocky Mountain J. Math. 34 (2004), 1391–1398.Google Scholar
[Per07] A. L., Pérez del Pozo, Automorphism groups of compact bordered Klein surfaces with invariant subsets, Manuscripta Math. 122 (2007), no. 2, 163–172.Google Scholar
[Sin87] D., Singerman, PSL(2,q) as an image of the extended modular group with aplications of group actions on surfaces, Proc. Edinburgh Math. Soc. (2) 30 (1987), 143–151.Google Scholar
[Suz82] M., Suzuki, Group Theory I, Springer-Verlag, 1982.Google Scholar

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