# Revista Matemática Iberoamericana

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**Volume 31, Issue 1, 2015, pp. 349–372**

**DOI: 10.4171/RMI/837**

Published online: 2015-03-05

Extensions of finite cyclic group actions on bordered surfaces

Emilio Bujalance^{[1]}, Francisco Javier Cirre

^{[2]}and Marston D. E. Conder

^{[3]}(1) UNED, Madrid, Spain

(2) UNED, Madrid, Spain

(3) University of Auckland, New Zealand

We study the question of the extendability of the action of a finite cyclic group on a compact bordered Klein surface (either orientable or non-orientable). This extends previous work by the authors for group actions on unbordered surfaces. It is shown that if such a cyclic action is realised by means of a non-maximal NEC signature, then the action always extends. For a given integer $g ≥ 2$, we determine the order of the largest cyclic group that acts as the full automorphism group of a bordered surface of algebraic genus g, and the topological type of the surfaces on which the largest action takes place. In addition, we calculate the smallest algebraic genus of a bordered surface on which a given cyclic group acts as the full automorphism group of the surface. For this, we deal separately with orientable and non-orientable surfaces, and we also determine the topological type of the surfaces attaining the bounds.

*Keywords: *Klein surfaces, finite group actions

Bujalance Emilio, Cirre Francisco Javier, Conder Marston: Extensions of finite cyclic group actions on bordered surfaces. *Rev. Mat. Iberoam.* 31 (2015), 349-372. doi: 10.4171/RMI/837