Revista Matemática Iberoamericana

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Volume 28, Issue 1, 2012, pp. 113–126
DOI: 10.4171/RMI/669

Published online: 2012-01-20

On the set of fixed points of automorphisms of bordered Klein surfaces

José Manuel Gamboa[1] and Grzegorz Gromadzki[2]

(1) Universidad Complutense de Madrid, Spain
(2) University of Gdańsk, Poland

The nature of the set of points fixed by automorphisms of Riemann or unbordered nonorientable Klein surfaces as well as quantitative formulae for them were found by Macbeath, Izquierdo, Singerman and Gromadzki in a series of papers. The possible set of points fixed by involutions of bordered Klein surfaces has been found by Bujalance, Costa, Natanzon and Singerman who showed that it consists of isolated fixed points, ovals and chains of arcs. They classified involutions of such surfaces, up to topological conjugacy in these terms. Here we give formulae for the number of elements of each type, also for non-involutory automorphisms, in terms of the topological type of the action of the group of dianalytic automorphisms. Finally we give some illustrative examples concerning bordered Klein surfaces with large groups of automorphisms already considered by May and Bujalance.

Keywords: Automorphisms of Riemann (Klein) surfaces, symmetries of Riemann surfaces, fixed points, Fuchsian, NEC groups

Gamboa José Manuel, Gromadzki Grzegorz: On the set of fixed points of automorphisms of bordered Klein surfaces. Rev. Mat. Iberoam. 28 (2012), 113-126. doi: 10.4171/RMI/669