Hyperbolic elements of a cofinite Fuchsian group move every point far

Let Γ ≤ PSL(2, ℝ) be cofinite. Then there is δ > 0 such that every hyperbolic element of Γ, the class of a matrix A with |tr A| > 2, moves every point of ℍ a hyperbolic distance at least δ. Equivalently, the translation lengths of the hyperbolic elements of Γ, the lengths of the closed geodesics of Γ \ ℍ, are bounded below by a positive constant: the systole of Γ \ ℍ is positive. Consequently an element of Γ moving some point less than δ fixes a point of ℍ or is parabolic: for small ε, the ε-thin part of Γ \ ℍ comes only from elliptic and parabolic elements.

The proof shows that a sufficiently short hyperbolic element moves some point exactly 1 / 4 and that this point, which lies far from the axis, is 1 / 4-thick. (The points on the axis itself are moved less than 1 / 4, so they are thin.) Conjugate a hyperbolic element to the dilation z ↦ exp s * z, and among the dilations of Γ along the same axis take the one with the least s₀ > 0. If s₀ is small, choose y with Im y = 1 so far from the imaginary axis that this dilation moves it exactly 1 / 4. Every other dilation in Γ along the axis moves y at least as far, by minimality, and so does every element exchanging the ends of the axis, because y is far from it. By the Margulis lemma Subgroup.conj_eq_or_conj_eq_inv_of_dist_le, an element of Γ moving y less than 1 / 4 preserves the axis, so it is one of these. Hence y lies in the 1 / 4-thick part of Γ.

The thick part of a Dirichlet domain of a cofinite group is bounded (Subgroup.IsCofinite.isBounded_dirichletDomain_inter_thickPart), so a translate of y lies in a fixed ball B. The corresponding conjugate of the minimal dilation moves a point of B by 1 / 4, so it is one of the finitely many elements of Γ moving the slightly larger ball to meet itself. Each of these that is hyperbolic moves every point by at least a fixed positive amount, and this bounds s₀ below.

Main results

• Subgroup.IsCofinite.exists_pos_le_dist_smul_of_two_lt_abs_trace: the hyperbolic elements of a cofinite group move every point at least a fixed positive distance.
• Subgroup.IsCofinite.exists_pos_exists_smul_eq_or_isParabolic_of_dist_smul_lt: an element of a cofinite group moving some point a small distance fixes a point of ℍ or is parabolic.

References

• Riccardo Benedetti and Carlo Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer, 1992, Chapter D (the thick-thin decomposition of a finite-volume hyperbolic manifold).
• Alan Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer, 1983, §5.4 (consequences of Jørgensen's inequality) and Chapter 7 (the displacement of a point by a Möbius transformation).

theorem Subgroup.IsCofinite.exists_pos_le_dist_smul_of_two_lt_abs_trace {Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)} (hΓ : Γ.IsCofinite) : ∃ δ > 0, ∀ (A : Matrix.SpecialLinearGroup (Fin 2) ℝ), ↑A ∈ Γ → 2 < |(↑A).trace| → ∀ (z : UpperHalfPlane), δ ≤ dist (A • z) z

Hyperbolic elements of a cofinite group move every point far. For a cofinite Γ ≤ PSL(2, ℝ) there is δ > 0 such that the class of every A ∈ SL(2, ℝ) with |tr A| > 2 lying in Γ moves every point of ℍ a hyperbolic distance at least δ. Equivalently, the translation lengths of the hyperbolic elements of Γ are bounded below by δ.

theorem Subgroup.IsCofinite.exists_pos_exists_smul_eq_or_isParabolic_of_dist_smul_lt {Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)} (hΓ : Γ.IsCofinite) : ∃ δ > 0, ∀ g ∈ Γ, ∀ (z : UpperHalfPlane), dist (g • z) z < δ → (∃ (w : UpperHalfPlane), g • w = w) ∨ g.IsParabolic

Short displacements in a cofinite group are elliptic or parabolic. For a cofinite Γ ≤ PSL(2, ℝ) there is δ > 0 such that every element of Γ moving some point of ℍ a hyperbolic distance less than δ fixes a point of ℍ or is parabolic.