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 ρΘ 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 ρΘ is in the subgroup PGL(3, R) of G and preserves a topological circle in the flag variety, but ρΘ is not Anosov. In her PhD Thesis, V. P. Valério elucidated the Anosov-like feature of Schwartz representations: For every ρΘ, there exists a 1-dimensional family of Anosov representations ρ ε Θ of PSL(2, Z)o into PGL(3, R) whose limit is the restriction of ρΘ to PSL(2, Z)o. In this paper, we improve her work: For each ρΘ, we build a 2-dimensional family of Anosov representations of PSL(2, Z)o into PGL(3, R) containing ρ ε Θ 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 .

Relatively Anosov representations via flows II: Examples

“…Given a marked box 𝔅, let 𝜌 𝔅 ∶ 𝖯𝖲𝖫(2, ℤ) → 𝖯𝖦𝖫(3, ℝ) be a Pappus-Schwartz representation as defined in [36,Th. 2.4] (see also [7]). This representation is defined as follows.…”

“…By [36,Sec. 3.2,3.3] (see also [7,Sec. 5.3]), there is a continuous 𝜌 𝔅 -equivariant map 𝜉 𝔅 = (𝜉 1 𝔅 , 𝜉 2 𝔅 )∶ 𝜕 ∞ ℍ 2 ℝ → 𝐏(ℝ 3 ) × Gr 2 (ℝ 3 ).…”

“…By [36, Sec. 3.2, 3.3] (see also [7, Sec. 5.3]), there is a continuous $$\rho _\mathfrak {B}$$‐equivariant map $$$$\begin{equation*} \xi _\mathfrak {B}= (\xi _\mathfrak {B}^1,\xi ^2_\mathfrak {B}) \colon \partial _\infty \operatorname{\mathbb {H}}^2_{\operatorname{\mathbb {R}}} \rightarrow \mathbf {P}(\operatorname{\mathbb {R}}^3) \times \operatorname{Gr}_2(\operatorname{\mathbb {R}}^3).$$…”

“…Schwartz' beautiful construction comes from iterating Pappus's theorem [36], and he also showed that these representations have many of the properties that relatively Anosov representations (not yet defined at the time) have. We should also note that Barbot–Lee–Valério proved that these representations are limits of families of Anosov representations of word hyperbolic groups [7].…”

“…If 𝑠 2 − 𝑠 1 ⩾ 𝑇 0 , then applying Lemma 6.2(2) to the intervals [0, 𝑠 1 ], [𝑠 2 , 𝑡] and Lemma 6.3 to the interval [𝑠 1 , 𝑠 2 ] yields 𝑣 for all 𝑌 ∈ Θ𝑘 (𝑣) and nonzero 𝑍 ∈ Ξ𝑑−𝑘 (𝑣). Otherwise, if 𝑠 2 − 𝑠 1 ⩽ 𝑇 0 , then applying Lemma 6.2(2) to the intervals [0, 𝑠 1 ], [𝑠 2 , 𝑡] and Equation(7) to the interval [𝑠 1 , 𝑠 2 ] yields…”