∀ (𝒢 : Subgroup (GL (Fin 2) ℝ)) [inst : 𝒢.IsArithmetic] (a b : Matrix.SpecialLinearGroup (Fin 2) ℤ),
  a ≈ b →
    (fun g => ⟦⟨(Matrix.SpecialLinearGroup.mapGL ℝ) g⁻¹ • OnePoint.infty, ⋯⟩⟧) a =
      (fun g => ⟦⟨(Matrix.SpecialLinearGroup.mapGL ℝ) g⁻¹ • OnePoint.infty, ⋯⟩⟧) b