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Uspekhi Mat. Nauk, 2020, Volume 75, Issue 5(455), Pages 153–190 (Mi umn9972)  

Fenchel–Nielsen coordinates and Goldman brackets

L. O. Chekhov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: It is explicitly shown that the Poisson bracket on the set of shear coordinates defined by V. V. Fock in 1997 induces the Fenchel–Nielsen bracket on the set of gluing parameters (length and twist parameters) for pair-of-pants decompositions of Riemann surfaces $\Sigma_{g,s}$ with holes. These structures are generalized to the case of Riemann surfaces $\Sigma_{g,s,n}$ with holes and bordered cusps.
Bibliography: 49 titles.

Keywords: Darboux coordinates, Teichmüller spaces, Poisson structure, geodesics.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).


DOI: https://doi.org/10.4213/rm9972

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English version:
Russian Mathematical Surveys, 2020, 75:5, 929–964

Bibliographic databases:

UDC: 514.7+512.548
MSC: Primary 30F60, 57K20; Secondary 32G15
Received: 30.07.2020

Citation: L. O. Chekhov, “Fenchel–Nielsen coordinates and Goldman brackets”, Uspekhi Mat. Nauk, 75:5(455) (2020), 153–190; Russian Math. Surveys, 75:5 (2020), 929–964

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