Detecting geodesic circles in hyperbolic surfaces with persistent homology

Jelenc, Blaž (author), Virk, Žiga (author)

In this paper we provide conditions under which a geodesic circle on a hyperbolic surface admits arbitrarily small geodesically convex neighborhoods. This implies that persistent homology using selective Rips complexes detects the length and the position of such a loop via persistent homology in dimensions one, two, or three. In particular, if a surface has a unique systole, then the systole can always be detected with persistent homology. The existential results of the paper are complemented by the corresponding quantitative treatments which explain the choice of parameters of selective Rips complexes as well as conditions, under which the detection occurs via the standard Rips complexes. In particular, if a surface has a unique systole, then the parameters depend on the first spectral gap in the length spectrum.

• Language: English
• Keywords: simple closed geodesic, Rips complexes, persistent homology, hyperbolic surfaces, systole
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics; FRI - Faculty of Computer and Information Science
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 19 str.
• Numbering: Vol. 119, iss. 2, art. 32
• PID: 20.500.12556/RUL-166525
• UDC: 515.1
• ISSN on article: 1578-7303
• DOI: 10.1007/s13398-024-01699-5
• COBISS.SI-ID: 222367491
• Publication date in RUL: 20.01.2025

Record is a part of a journal

• Title: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas
• Shortened title: Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.
• Publisher: Springer Nature, Real Academia de Ciencias Exactas, Físicas y Naturales
• ISSN: 1578-7303
• COBISS.SI-ID: 17645351

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: J1-4001
• Name: Izbrani problemi iz uporabne in računske topologije

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0292
• Name: Topologija in njena uporaba