Comptes Rendus
no. 10
Dynamical systems
On the Anosov character of the Pappus–Schwartz representations

Presented by: Claire Voisin

Viviane Pardini Valério1
1 Universidade Federal de Minas Gerais – ICEx – Departamento de Matemática, Av. Antônio Carlos, 6627, Belo Horizonte, Minas Gerais, CEP 31.270-901, Caixa Postal 702, Brazil
Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1037-1041.

In the paper Pappus's Theorem and The Modular Group (1993) [4], R.E. Schwartz observed that the classical Pappus theorem gives rise to an action of the modular group on the space of marked boxes. He inferred from this a 2-dimensional family of faithful representations of the modular group into the group of projective symmetries. These representations have a dynamical behavior very similar to the one of Anosov representations, even if they are never Anosov themselves. In this note, we announce the main result of V. Pardini Valério (2016) [3], which elucidates this Anosov character of the Schwartz representations by proving that their restrictions to the index-2 subgroup are limits of Anosov representations.

Dans l'article Pappus's Theorem and The Modular Group (1993) [4], R.E. Schwartz a mis en évidence le fait que le théorème classique de Pappus définit une action intéressante du groupe modulaire sur l'espace des boîtes marquées. Ceci lui a permis de construire une famille à deux paramètres de représentations fidèles du groupe modulaire dans le groupe de symétries projectives. Ces représentations ont un comportement dynamique très similaire à celui des représentations d'Anosov, bien que ne l'étant pas elles-mêmes. Dans cette note, nous annonçons le résultat principal de V. Pardini Valério (2016) [3], qui élucide ce caractère Anosov des représentations de Schwartz, en montrant que leurs restrictions au sous-groupe d'indice 2 sont chacune des limites des représentations d'Anosov.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.09.005

Viviane Pardini Valério 1

1 Universidade Federal de Minas Gerais – ICEx – Departamento de Matemática, Av. Antônio Carlos, 6627, Belo Horizonte, Minas Gerais, CEP 31.270-901, Caixa Postal 702, Brazil 
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Viviane Pardini Valério. On the Anosov character of the Pappus–Schwartz representations. Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1037-1041. doi : 10.1016/j.crma.2016.09.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.09.005/

[1] O. Guichard; A. Wienhard Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012) no. 2, pp. 357-438

[2] F. Labourie Anosov flows, surface groups and curves in projective space, Invent. Math., Volume 165 (2006) no. 1, pp. 51-114

[3] V. Pardini Valério Teorema de Pappus, Representções de Schwartz e Representações Anosov, UFMG, Brazil, January 2016 http://www.mat.ufmg.br/intranet-atual/pgmat/TesesDissertacoes/uploaded/Tese68.pdf (PhD thesis)

[4] R.E. Schwartz Pappus's theorem and the modular group, Publ. Math. Inst. Hautes Études Sci., Volume 78 (1993), pp. 187-206

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