Pappus Theorem, Schwartz Representations and Anosov Representations

Abstract

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,ℤ) into the group 𝒢 of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup PSL(2,ℤ) o of PSL(2,ℤ) under each representation ρ Θ is in the subgroup PGL(3,ℝ) 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,ℤ) o into PGL(3,ℝ) whose limit is the restriction of ρ Θ to PSL(2,ℤ) o . In this paper, we improve her work: for each ρ Θ , we build a 2-dimensional family of Anosov representations of PSL(2,ℤ) o into PGL(3,ℝ) containing ρ Θ ε and a 1-dimensional subfamily of which can extend to representations of PSL(2,ℤ) into 𝒢. Schwartz representations are therefore, in a sense, the limits of Anosov representations of PSL(2,ℤ) into 𝒢.

Locations

• arXiv (Cornell University) - View - PDF
• French digital mathematics library (Numdam) - View - PDF
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• Annals of the Fourier Institute (Institut Fourier) - View - PDF
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• Annales de l’institut Fourier - View - PDF