Piecewise circular curves and Positivity
Burelle, Jean-Philippe1; Kirk, Ryan T.2
1Département de mathématiques, Université de Sherbrooke, Sherbrooke J1K 2R1 (Canada)
2Mathematics Department, East Carolina University, Greenville NC 27858 (USA)
Annales de l'Institut Fourier, Online first, 44 p.

We introduce the moduli space of generic circular $n$-gons in the Riemann sphere and relate it to a moduli space of Legendrian polygons. We prove that when $n=2k$, this moduli space contains a connected component homeomorphic to the Fock–Goncharov space of $k$-tuples of positive flags for $\mathsf {PSp}(4,\mathbb{R})$ and hence is a topological ball. We characterize this component geometrically as the space of simple circular $n$-gons with decreasing curvature.

Nous définissons l’espace de modules des $n$-gones circulaires génériques dans la sphère de Riemann et nous le relions à un espace de modules de polygones legendriens. Nous démontrons que lorsque $n$ est pair, cet espace de modules contient une composante homéomorphe à l’espace des $k$-uplets positifs de drapeaux dans $\mathsf {PSp}(4,\mathbb{R})$ défini par Fock et Goncharov, et est donc une boule topologique. Nous identifions cette composante de manière géométrique en tant que l’espace des polygones circulaires simples de courbure décroissante.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3764
Classification: 51M99, 32G15, 22F30
Keywords: Moduli spaces, higher Teichmüller theory, positivity, Legendrian knots, symplectic groups.
Mots-clés : Espaces de modules, théorie de Teichmüller en rang supérieur, positivité, noeuds Legendriens, groupes symplectiques.

Burelle, Jean-Philippe  1 ; Kirk, Ryan T.  2

1 Département de mathématiques, Université de Sherbrooke, Sherbrooke J1K 2R1 (Canada)
2 Mathematics Department, East Carolina University, Greenville NC 27858 (USA) 
Burelle, Jean-Philippe; Kirk, Ryan T. Piecewise circular curves and Positivity. Annales de l'Institut Fourier, Online first, 44 p.
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