Pappus Theorem, Schwartz Representations and Anosov Representations
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, p. 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.
Received : 2016-10-24
Revised : 2017-11-01
Accepted : 2018-01-25
Published online : 2018-11-23
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
     author = {Barbot, Thierry and Lee, Gye-Seon and Val\'erio, Viviane Pardini},
     title = {Pappus Theorem, Schwartz Representations  and Anosov Representations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     pages = {2697-2741},
     doi = {10.5802/aif.3221},
     language = {en},
     url = {}
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.

[1] Acosta, Miguel Character varieties for real forms (2016) ( )

[2] Barbot, Thierry Three-dimensional Anosov flag manifolds, Geom. Topol., Tome 14 (2010) no. 1, pp. 153-191 | Article | MR 2578303

[3] Barbot, Thierry; Mérigot, Quentin Anosov AdS representations are quasi-Fuchsian, Groups Geom. Dyn., Tome 6 (2012) no. 3, pp. 441-483 | Article | MR 2961282

[4] Barrera, Waldemar; Cano Cordero, Angel; Navarrete, Juan Pablo Pappus’ Theorem and a construction of complex Kleinian groups with rich dynamics, Bull. Braz. Math. Soc. (N.S.), Tome 45 (2014) no. 1, pp. 25-52 | Article | MR 3194081 | Zbl 1344.37058

[5] Bochi, Jairo; Potrie, Rafael; Sambarino, Andrés Anosov Representations and dominated splittings (2017) ( )

[6] Bridgeman, Martin; Canary, Richard; Labourie, François; Sambarino, Andrés The pressure metric for Anosov representations, Geom. Funct. Anal., Tome 25 (2015) no. 4, pp. 1089-1179 | Article | MR 3385630

[7] Ghys, Étienne; De La Harpe, Pierre Sur les groupes hyperboliques d’après Mikhael Gromov, Birkhäuser, Progress in Mathematics, Tome 83 (1990), xii+285 pages (Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988) | Article | MR 1086648 | Zbl 0731.20025

[8] Gromov, Mikhael Hyperbolic groups, Essays in group theory, Springer (Mathematical Sciences Research Institute Publications) Tome 8 (1987), pp. 75-263 | Article | MR 919829

[9] Guichard, Olivier; Wienhard, Anna Anosov representations: domains of discontinuity and applications, Invent. Math., Tome 190 (2012) no. 2, pp. 357-438 | Article | MR 2981818

[10] Kapovich, Ilya; Benakli, Nadia Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, 2001), American Mathematical Society (Contemporary Mathematics) Tome 296 (2002), pp. 39-93 | Article | MR 1921706 | Zbl 1044.20028

[11] Katok, Svetlana; Ugarcovici, Ilie Symbolic dynamics for the modular surface and beyond, Bull. Am. Math. Soc., Tome 44 (2007) no. 1, pp. 87-132 | Article | MR 2265011

[12] Labourie, François Anosov flows, surface groups and curves in projective space, Invent. Math., Tome 165 (2006) no. 1, pp. 51-114 | Article | MR 2221137

[13] Lawton, Sean Generators, relations and symmetries in pairs of 3×3 unimodular matrices, J. Algebra, Tome 313 (2007) no. 2, pp. 782-801 | Article | MR 2329569

[14] Marquis, Ludovic Around groups in Hilbert geometry, Handbook of Hilbert geometry, European Mathematical Society (IRMA Lectures in Mathematics and Theoretical Physics) Tome 22 (2014), pp. 207-261 | MR 3329882

[15] Morier-Genoud, Sophie; Ovsienko, Valentin; Tabachnikov, Serge SL 2 ()-tilings of the torus, Coxeter-Conway friezes and Farey triangulations, Enseign. Math., Tome 61 (2015) no. 1-2, pp. 71-92 | Article | MR 3449283 | Zbl 1331.05022

[16] Orenstein, Paulo A métrica de Hilbert e aplicações (2009) (Scientic initiation work–Departamento de Matematica - PUC-Rio,

[17] Schwartz, Richard Pappus’s theorem and the modular group, Publ. Math., Inst. Hautes Étud. Sci., Tome 78 (1993), pp. 187-206 | MR 1259431 | Zbl 0806.57004

[18] Valério, Viviane Pardini On the Anosov character of the Pappus-Schwartz representations, C. R. Math. Acad. Sci. Paris, Tome 354 (2016) no. 10, pp. 1037-1041 | Zbl 1355.57019

[19] Valério, Viviane Pardini Teorema de Pappus, Representações de Schwartz e Representações Anosov, Federal University of Minas Gerais (Brazil) (2016) (Ph. D. Thesis)