2-frieze patterns and the cluster structure of the space of polygons

[Espace des 2-frises, espace de polygones et structure de variété amassée]

Morier-Genoud, Sophie ; Ovsienko, Valentin ; Tabachnikov, Serge

Annales de l'Institut Fourier, Tome 62 (2012), p. 937-987 / Harvested from Numdam

Résumé

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.

Abstract

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.

Détail

Publié le : 2012-01-01

MR 3013813 | Zbl pre06093169

DOI : https://doi.org/10.5802/aif.2713
Classification:  13F60,  14N05,  51M99
Mots clés: Frises, frises de Coxeter-Conway, espace de modules, algebre amassée, application pentagramme.

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