Journal of the Mathematical Society of Japan

Automorphism groups of q-trigonal planar Klein surfaces and maximal surfaces

Beatriz ESTRADA and Ernesto MARTÍNEZ

Full-text: Open access


A compact Klein surface X=D/Γ , where D denotes the hyperbolic plane and Γ is a surface NEC group, is said to be q-trigonal if it admits an automorphism ϕ of order 3 such that the quotient X/<ϕ > has algebraic genus q. In this paper we obtain for each q the automorphism groups of q-trigonal planar Klein surfaces, that is surfaces of topological genus 0 with k 3 boundary components. We also study the surfaces in this family, which have an automorphism group of maximal order (maximal surfaces). It will be done from an algebraic and geometrical point of view.

Article information

J. Math. Soc. Japan Volume 61, Number 2 (2009), 607-623.

First available in Project Euclid: 13 May 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F50: Klein surfaces 14J50: Automorphisms of surfaces and higher-dimensional varieties 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

Klein surfaces NEC groups automorphism groups fundamental polygons


ESTRADA, Beatriz; MARTÍNEZ, Ernesto. Automorphism groups of $q$ -trigonal planar Klein surfaces and maximal surfaces. J. Math. Soc. Japan 61 (2009), no. 2, 607--623. doi:10.2969/jmsj/06120607.

Export citation


  • N. L. Alling and N. Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Math., 219, Springer-Verlag, 1971.
  • A. F. Beardon, The geometry of discrete groups, Graduate Texts in Math., 91, Springer-Verlag, 1983.
  • E. Bujalance, Automorphism groups of compact planar Klein surfaces, Manuscripta Math., 56 (1986), 105–124.
  • E. Bujalance, J. A. Bujalance, G. Gromadzki and E. Martínez, Cyclic trigonal Klein surfaces, J. of Algebra, 159 (1993), 436–459.
  • E. Bujalance, J. J. Etayo, J. M. Gamboa and G. Gromadzki, Automorphism Groups of Compact Bordered Klein Surfaces, A Combinatorial Approach, Lecture Notes in Math., 1439, Springer-Verlag, 1990.
  • A. F. Costa and E. Martínez, Descripción geométrica del espacio de Teichmüller de las superficies de Klein que son esferas con tres componentes en el borde, Contribuciones Matemáticas, Estudios en honor del profesor Javier Etayo Miqueo, Ed. Complutense, Madrid, 1994.
  • B. Estrada and E. Martínez, $q$-Trigonal Klein surfaces, Israel J. Math., 131 (2002), 361–374.
  • B. Estrada and E. Martínez, Coordinates for the Teichmüller space of planar surface N.E.C. groups, Int. J. Math., 14 (2003), 1037–1052.
  • A. M. Macbeath, The classification of non-euclidean crystallographic groups, Canad. J. Math., 19 (1967), 1192–1205.
  • C. L. May, Automorphisms of compact Klein surfaces with boundary, Pacific J. Math., 59 (1975), 199–210.
  • C. L. May, Large automorphism groups of compact Klein surfaces with boundary, I, Glasgow Math. J., 18 (1977), 1–10.
  • R. Preston, Projective Structures and Fundamental Domains on Compact Klein Surfaces, Ph. D. thesis, University of Texas, 1975.
  • H. C. Wilkie, On non-Euclidean crystallographic groups, Math. Z., 91 (1966), 87–102.