Pappus Theorem, Schwartz Representations and Anosov Representations

In the paper \textitPappus’s theorem and the modular group, R. Schwartz constructed a 2-dimensional family of faithful representations \rho_Θof the modular group \mathrmPSL(2,\mathbbZ) into the group \mathscrG of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup \mathrmPSL(2,\mathbbZ)_o of \mathrmPSL(2,\mathbbZ) under each representation \rho_Θis in the subgroup \mathrmPGL(3,\mathbbR) of \mathscrG and preserves a topological circle in the flag variety, but \rho_Θis \emphnot Anosov. In her PhD Thesis, V. P. Valério elucidated the Anosov-like feature of Schwartz representations: for every \rho_Θ, there exists a 1-dimensional family of Anosov representations ρ^\varepsilon_Θ of \mathrmPSL(2,\mathbbZ)_o into \mathrmPGL(3,\mathbbR) whose limit is the restriction of \rho_Θto \mathrmPSL(2,\mathbbZ)_o. In this paper, we improve her work: for each \rho_Θ, we build a 2-dimensional family of Anosov representations of \mathrmPSL(2,\mathbbZ)_o into \mathrmPGL(3,\mathbbR) containing ρ^\varepsilon_Θ and a 1-dimensional subfamily of which can extend to representations of \mathrmPSL(2,\mathbbZ) into \mathscrG. Schwartz representations are therefore, in a sense, the limits of Anosov representations of \mathrmPSL(2,\mathbbZ) into \mathscrG.