Pappus Theorem, Schwartz Representations and Anosov Representations

Abstract

In the paper Pappus's theorem and the modular group [13], R. Schwartz constructed a 2-dimensional family of faithful representations ρΘ of the modular group PSL(2, Z) into the group G of projective symmetries of the projective plane via Pappus Theorem. If PSL(2, Z)o denotes the unique index 2 subgroup of PSL(2, Z) and PGL(3, R) the subgroup of G consisting of projective transformations, then the image of PSL(2, Z)o under ρΘ is in PGL(3, R). The representations ρΘ share a very interesting property with Anosov representations of surface groups into PGL(3, R): It preserves a topological circle in the flag variety. However, the representation ρΘ itself cannot be Anosov since the Gromov boundary of PSL(2, Z) is a Cantor set and not a circle. In her PhD Thesis [15], V. P. Valerio elucidated the Anosov-like feature of the Schwartz representations by showing that for each representation ρΘ, there exists an 1-dimensional family of representations (ρ e Θ) e∈R of PSL(2, Z)o into PGL(3, R) such that ρ 0 Θ is the restriction of the Schwartz representation ρΘ to PSL(2, Z)o and ρ e Θ is Anosov for every e < 0. This result was announced and presented in her paper [14]. In the present paper, we extend and improve her work. For every representation ρΘ, we build a 2-dimensional family of representations (ρ λ Θ) λ∈R 2 of PSL(2, Z)o into PGL(3, R) such that ρ λ Θ = ρ e Θ for λ = (e, 0) and ρ λ Θ is Anosov for every λ ∈ R • , where R • is an open set of R 2 containing {(e, 0) | e < 0}. Moreover, among the 2-dimensional family of new Anosov representations, an 1-dimensional subfamily of representations can extend to representations of PSL(2, Z) into G, and therefore the Schwartz representations are, in a sense, on the boundary of the Anosov representations in the space of all representations of PSL(2, Z) into G.