Chaos and integrability in SL(2, R)-geometry

Aleksei V. Bolsinov, Aleksandr P. Veselov, Yiru Ye*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We review the integrability of the geodesic flow on a threefold M3 admitting one of the three group geometries in Thurston’s sense. We focus on the SL(2,R) case. The main examples are the quotients M3Γ = Γ\PSL(2,R), where Γ ⊂ PSL(2,R) is a cofinite Fuchsian group. We show that the corresponding phase space TM3Γ contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively. As a concrete example we consider the case of the modular threefold with the modular group Γ = PSL(2,Z). In this case M3Γ is known to be homeomorphic to the complement of a trefoil knot K in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to M3Γ produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on M3Γ. Bibliography: 60 titles.

Original languageEnglish
Pages (from-to)557-586
Number of pages30
JournalRussian Mathematical Surveys
Issue number4
Publication statusPublished - Aug 2021


  • 3D geometries in the sense of Thurston
  • Geodesic flows
  • Integrability

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