2-frieze patterns and the cluster structure of the space of polygons
Annales de l'Institut Fourier, Volume 62 (2012) no. 3, p. 937-987
We study 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n-gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.
Nous étudions une variante des frises de Coxeter-Conway appelée 2-frises. La réalisation géométrique de l’espace des 2-frises est l’espace des modules de polygones, dans le plan projectif ou dans l’espace vectoriel de dimension 3, qui est un analogue de l’espace des modules des courbes de genre 0 avec n points marqués. Nous montrons que l’espace des 2-frises admet une structure de variété amassée et nous en étudions les propriétés algébriques et arithmétiques.
DOI : https://doi.org/10.5802/aif.2713
Classification:  13F60,  14N05,  51M99
Keywords: Frieze patterns, Coxeter-Conway friezes, moduli space, cluster algebra, pentagram map.
@article{AIF_2012__62_3_937_0,
     author = {Morier-Genoud, Sophie and Ovsienko, Valentin and Tabachnikov, Serge},
     title = {2-frieze patterns and the cluster structure of the space of polygons},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {3},
     year = {2012},
     pages = {937-987},
     doi = {10.5802/aif.2713},
     mrnumber = {3013813},
     zbl = {pre06093169},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_3_937_0}
}
Morier-Genoud, Sophie; Ovsienko, Valentin; Tabachnikov, Serge. 2-frieze patterns and the cluster structure of the space of polygons. Annales de l'Institut Fourier, Volume 62 (2012) no. 3, pp. 937-987. doi : 10.5802/aif.2713. http://www.numdam.org/item/AIF_2012__62_3_937_0/

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