Some results on abelian groups of automorphisms of compact Riemann surfaces

A compact Riemann surface is called pseudo-real if it admits orientation-reversing automorphisms but none of them has order two. In this paper, we find necessary and sufficient conditions for the existence of an action on a pseudo-real surface of genus $$g\geqslant 2$$ g ⩾ 2 of an abelian group containing orientation-reversing automorphisms. Several consequences are obtained, such as the solution of the minimum genus problem for such abelian actions.

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Abelian actions on compact nonorientable Riemann surfaces

Rodríguez

2021

Glasgow Math. J.

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Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

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Some results on abelian groups of automorphisms of compact Riemann surfaces

Cited by 4 publications

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Abelian actions on compact bordered Klein surfaces

Abelian actions on compact bordered Klein surfaces

Abelian Actions on Pseudo-real Riemann Surfaces

Abelian actions on compact nonorientable Riemann surfaces

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