DOI 10.15507/2079-6900.23.202102.147–158

Original article

ISSN 2587-7496 (Print)

ISSN 2079-6900 (Online)

MSC2020 37E30

Classification of periodic transformations of an orientable surface of genus two

D. A. Baranov, O. V. Pochinka

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. In this paper, we find all admissible topological conjugacy classes of periodic transformations of a two-dimensional surface of genus two. It is proved that there are exactly seventeen pairwise topologically non-conjugate orientation-preserving periodic pretzel transformations. The implementation of all classes by lifting the full characteristics of mappings from a modular surface to a surface of genus two is also presented. The classification results are based on Nielsen’s theory of periodic surface transformations, according to which the topological conjugacy class of any such homeomorphism is completely determined by its characteristic. The complete characteristic carries information about the genus of the modular surface, the ramified bearing surface, the periods of the ramification points and the turns around them. The necessary and sufficient conditions for the admissibility of the complete characteristic are described by Nielsen and for any surface they give a finite number of admissible collections. For surfaces of a small genus, one can compile a complete list of admissible characteristics, which was done by the authors of the work for a surface of genus 2.

Key Words: periodic homeomorphisms of surfaces, Nielsen-Thurston theory, orientable surface, topological conjugacy

For citation: D. A. Baranov, O. V. Pochinka. Classification of periodic transformations of an orientable surface of genus two. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:2(2021), 147–158. DOI: https://doi.org/10.15507/2079-6900.23.202102.147–158

Submitted: 25.03.2021; Revised: 26.04.2021; Accepted: 24.05.2021

Information about the authors:

Denis A. Baranov, student of the Faculty of Informatics, Mathematics and Computer Science, Higher School of Economics (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0002-6587-5305, denbaranov0066@gmail.com

Olga V. Pochinka, Professor of the Department of Fundamental Mathematics, National Research University Higher School of Economics (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603150, Russia), D. Sci. (Physics and Mathematics), ORCID: https://orcid.org/0000-0002-4274-8215, olga-pochinka@yandex.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

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