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.

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: 10.5802/aif.2713
Classification: 13F60, 14N05, 51M99
Keywords: Frieze patterns, Coxeter-Conway friezes, moduli space, cluster algebra, pentagram map.
Mot clés : Frises, frises de Coxeter-Conway, espace de modules, algebre amassée, application pentagramme.
Morier-Genoud, Sophie 1; Ovsienko, Valentin 2; Tabachnikov, Serge 3

1 Institut de Mathématiques de Jussieu UMR 7586 Université Pierre et Marie Curie 4, place Jussieu, case 247 75252 Paris Cedex 05
2 CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
3 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2713/

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