Pappus Theorem, Schwartz Representations and Anosov Representations
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2697-2741.

In the paper Pappus’s theorem and the modular group, R. Schwartz constructed a 2-dimensional family of faithful representations ρ Θ of the modular group PSL(2,) into the group 𝒢 of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup PSL(2,) o of PSL(2,) under each representation ρ Θ is in the subgroup PGL(3,) of 𝒢 and preserves a topological circle in the flag variety, but ρ Θ is not Anosov. In her PhD Thesis [18, 19], V. P. Valério elucidated the Anosov-like feature of Schwartz representations: for every ρ Θ , there exists a 1-dimensional family of Anosov representations ρ Θ ε of PSL(2,) o into PGL(3,) whose limit is the restriction of ρ Θ to PSL(2,) o . In this paper, we improve her work: for each ρ Θ , we build a 2-dimensional family of Anosov representations of PSL(2,) o into PGL(3,) containing ρ Θ ε and a 1-dimensional subfamily of which can extend to representations of PSL(2,) into 𝒢. Schwartz representations are therefore, in a sense, the limits of Anosov representations of PSL(2,) into 𝒢.

Dans l’article Pappus’s theorem and the modular group R. Schwartz a montré que le Théorème de Pappus fournissait une famille à deux paramètres de représentations ρ Θ du groupe modulaire PSL(2,) dans le groupe 𝒢 de symétries projectives du plan projectif. L’image de l’unique sous-groupe PSL(2,) o d’indice 2 de PSL(2,) par chaque ρ Θ de PSL(2,) o est contenue dans le sous-groupe PGL(3,) de 𝒢 formé des transformations projectives, et préserve un cercle topologique dans la variété des drapeaux. Cependant, elles ne sont pas Anosov. Dans sa thèse [18, 19], V. P. Valério a élucidé ce comportement de type Anosov des représentations de Schwartz. Pour chaque représentation ρ Θ , il existe une famille à un paramètre (ρ Θ ε ) ε de représentations Anosov de PSL(2,) o dans PGL(3,) telles que ρ Θ 0 soit la restriction de ρ Θ à PSL(2,) o et de sorte que ρ Θ ε soit Anosov pour ε<0. Dans le présent article, nous améliorons son travail. Pour chaque représentation ρ Θ , nous construisons une famille à deux paramètres de représentations Anosov (ρ Θ λ ) λ 2 de PSL(2,) o vers PGL(3,) contenant les ρ Θ ε , ainsi qu’une sous-famille à un paramètre de représentations qui s’étendent en des représentations de PSL(2,) vers 𝒢. Ceci montre qu’en un certain sens, les représentations de Schwartz sont dans le bord de l’ensemble des représentations Anosov dans l’espace de toutes les représentations de PSL(2,) vers 𝒢.

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DOI: 10.5802/aif.3221
Classification: 37D20,  37D40,  20M30,  22E40,  53A20
Keywords: Pappus Theorem, modular group, group of projective symmetries, Farey triangulation, Schwartz representation, Gromov-hyperbolic group, Anosov representation, Hilbert metric
Barbot, Thierry 1; Lee, Gye-Seon 2; Valério, Viviane Pardini 3

1 Université d’Avignon et des Pays de Vaucluse, Laboratoire de Mathématiques Campus Jean-Henri Fabre 301 rue Baruch de Spinoza BP 21239 84916 Avignon Cedex 9 (France)
2 Mathematisches Institut, Universität Heidelberg Im Neuenheimer Feld 205 69120 Heidelberg (Germany)
3 Universidade Federal de São João del Rei Departamento de Matemática e Estatística Praça Frei Orlando, 170, Centro, CEP: 36307-352 São João del Rei, Minas Gerais (Brazil)
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Barbot, Thierry; Lee, Gye-Seon; Valério, Viviane Pardini. Pappus Theorem, Schwartz Representations  and Anosov Representations. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2697-2741. doi : 10.5802/aif.3221. https://aif.centre-mersenne.org/articles/10.5802/aif.3221/

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