Hostname: page-component-5cf477f64f-mgq6s Total loading time: 0 Render date: 2025-04-03T06:45:02.125Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE CONNECTEDNESS OF THE BRANCH LOCI OF NON-ORIENTABLE UNBORDERED KLEIN SURFACES OF LOW GENUS

Part of: Low-dimensional topology Riemann surfaces Other groups of matrices Special aspects of infinite or finite groups

Published online by Cambridge University Press:  22 December 2014

E. BUJALANCE ,
J. J. ETAYO ,
E. MARTÍNEZ  and
B. SZEPIETOWSKI
E. BUJALANCE
Affiliation:
Departamento de Matemáticas Fundamentales, UNED, Paseo Senda del Rey 9, 28040-Madrid, Spain e-mail: ebujalance@mat.uned.es
J. J. ETAYO
Affiliation:
Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense, 28040-Madrid, Spain e-mail: jetayo@mat.ucm.es
E. MARTÍNEZ
Affiliation:
Departamento de Matemáticas Fundamentales, UNED, Paseo Senda del Rey 9, 28040-Madrid, Spain e-mail: emartinez@mat.uned.es
B. SZEPIETOWSKI
Affiliation:
Institute of Mathematics, University of Gdańsk. Ul. Wita, Stwosza 57, 80-952 Gdańsk, Poland e-mail: blaszep@mat.ug.edu.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is devoted to determine the connectedness of the branch loci of the moduli space of non-orientable unbordered Klein surfaces. We obtain a result similar to Nielsen's in order to determine topological conjugacy of automorphisms of prime order on such surfaces. Using this result we prove that the branch locus is connected for surfaces of topological genus 4 and 5.

MSC classification

Primary: 57M60: Group actions in low dimensions
Secondary: 20F05: Generators, relations, and presentations 20H10: Fuchsian groups and their generalizations 30F50: Klein surfaces
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015 , pp. 211 - 230
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Alling, N. L. and Greenleaf, N., Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics, vol. 219 (Springer-Verlag, Berlin, Germany, 1971).Google Scholar
2.Bartolini, G., Costa, A. F. and Izquierdo, M., On the connectivity of branch loci of moduli spaces, Ann. Acad. Sci. Fenn. Math. 38 (2013), 245–258.Google Scholar
3.Bartolini, G., Costa, A. F., M. Izquierdo and A. M. Porto, On the connectedness of the branch locus of the moduli space of Riemann surfaces, Rev. R. Acad. Cienc. Ser. A. Math. 104 (2010), 81–86.Google Scholar
4.Bartolini, G. and Izquierdo, M., On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus, Proc. Am. Math. Soc. 140 (2012), 35–45.Google Scholar
5.Broughton, S. A., The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology Appl. 37 (1990), 101–113.Google Scholar
6.Bujalance, E., Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary, Pac. J. Math. 109 (1983), 279–289.Google Scholar
7.Bujalance, E. and Costa, A. F., Orientation reversing automorphisms of Riemann surfaces, Illinois J. Math. 38 (1994), 616–623.CrossRefGoogle Scholar
8.Bujalance, E., Costa, A. F., S. Natanzon and D. Singerman, Involutions of compact Klein surfaces, Math. Z. 211 (1992), 461–478.Google Scholar
9.Bujalance, E., Etayo, J. J. and Martínez, E., The full group of automorphisms of non-orientable unbordered Klein surfaces of topological genus 3, 4 and 5, Rev. Mat. Complut. 27 (2014), 305–326.CrossRefGoogle Scholar
10.Bujalance, J. A., Costa, A. F. and Porto, A. M., On the connectedness of the locus of real elliptic-hyperelliptic Riemann surfaces, Int. J. Math. 20 (2009), 1069–1080.Google Scholar
11.Buser, P., Seppälä, M. and Silhol, R., Triangulations and moduli spaces of Riemann surfaces with group actions, Manuscr. Math. 88 (1995), 209–224.CrossRefGoogle Scholar
12.Costa, A. F. and Izquierdo, M., On the connectedness of the locus of real Riemann surfaces, Ann. Acad. Fenn. Math. 27 (2002), 341–356.Google Scholar
13.Costa, A. F. and Izquierdo, M., On the connectedness of the branch locus of the moduli space of Riemann surfaces of genus 4, Glasgow Math. J. 52 (2010), 401–408.CrossRefGoogle Scholar
14.Harvey, W. J., On branch loci in Teichmüller space, Trans. Am. Math. Soc. 153 (1971), 387–399.Google Scholar
15.Kulkarni, R. S., Isolated points in the branch locus of the moduli space of compact Riemann surfaces, Ann. Acad. Fenn. Sci. Math. 16 (1991), 71–81.CrossRefGoogle Scholar
16.Macbeath, A. M., The classification of non-Euclidean crystallographic groups, Can. J. Math. 19 (1967), 1192–1205.Google Scholar
17.Macbeath, A. M. and Singerman, D., Spaces of subgroups and Teichmüller space, Proc. London Math. Soc. 31 (1975), 211–256.Google Scholar
18.Natanzon, S. M., Klein surfaces, Russ. Math. Surv. 45 (1990), 43–108.CrossRefGoogle Scholar
19.Preston, R., Projective structures and fundamental domains on compact Klein surfaces, PhD Thesis (University of Texas, Austin, TX, 1975).Google Scholar
20.Seppälä, M., Moduli spaces of stable real algebraic curves, Ann. Sci. Éc. Norm. Sup. (4) 24 (1991), 519–544.CrossRefGoogle Scholar
21.Singerman, D., Symmetries of Riemann surfaces with large automorphism group, Math. Ann. 210 (1974), 17–32.Google Scholar
22.Wilkie, H. C., On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87–102.Google Scholar