# Extensions of finite cyclic group actions on bordered surfaces

### Emilio Bujalance

UNED, Madrid, Spain### Francisco Javier Cirre

UNED, Madrid, Spain### Marston D. E. Conder

University of Auckland, New Zealand

## Abstract

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.

## Cite this article

Emilio Bujalance, Francisco Javier Cirre, Marston D. E. Conder, Extensions of finite cyclic group actions on bordered surfaces. Rev. Mat. Iberoam. 31 (2015), no. 1, pp. 349–372

DOI 10.4171/RMI/837