A cofinite Fuchsian group has finitely many elliptic orbits

Let Γ ≤ PSL(2, ℝ) be cofinite. An elliptic point of Γ is a point w ∈ ℍ fixed by a nonidentity element of Γ. This file shows that the thin part of Γ \ ℍ near the elliptic points is controlled uniformly, and deduces that Γ has only finitely many orbits of elliptic points.

• Separation. There is c > 0 such that every element of Γ either fixes a given elliptic point w or moves it at least c. If g ≠ 1 fixes w and k • w ≠ w, then the commutator ⁅g, k⁆ has trace 2 + (4 - tr² g) / 2 · (cosh (dist (k • w) w) - 1) > 2, so it is hyperbolic. It moves k • w to g • k • w, at most twice as far as k moves w. Since the hyperbolic elements of a cofinite group move every point at least a fixed positive distance (Subgroup.IsCofinite.exists_pos_le_dist_smul_of_two_lt_abs_trace), this bounds the displacement of w below.
• Elliptic thin parts. There is τ > 0 such that a point moved less than τ by a nonidentity element g fixing w is moved at least τ by every element of Γ not fixing w. If g rotates through a small angle, so that 4 - tr² g is small, this is the Margulis lemma in trace form (Subgroup.conj_eq_or_conj_eq_inv_of_cosh_dist): an element k moving the point little conjugates g to g or g⁻¹, so k fixes the fixed point w of g. Otherwise the point is close to w, and an element moving it little moves w less than the separation constant.
• Finiteness. At each elliptic point w choose a nonidentity g fixing w whose rotation angle is least, and a point y that g moves exactly τ / 2. Every other element fixing w moves y at least as far, and every element not fixing w moves it at least τ, so y is τ / 2-thick. The ball of radius τ / 4 about y lies in the τ-thin part of w, so by the previous point the balls chosen at representatives of distinct elliptic orbits have pairwise disjoint translates. They all have the same positive area, and a fundamental domain has finite area, so there are only finitely many elliptic orbits.

Compare Subgroup.finite_ellipticOrbits, which deduces the same finiteness from compactness of the cusp compactification. Here finite covolume suffices, as is needed to bound the part of a Dirichlet domain lying in the thin part near elliptic points.

Main results

• Subgroup.IsCofinite.exists_pos_le_dist_smul_of_smul_eq_self: the elliptic points of a cofinite group are moved at least a fixed positive distance by every element not fixing them.
• Subgroup.IsCofinite.exists_pos_le_dist_smul_of_dist_smul_lt: a point moved little by an element fixing an elliptic point w is moved far by every element not fixing w.
• Subgroup.IsCofinite.finite_ellipticOrbits: a cofinite Fuchsian group has finitely many elliptic orbits.

References

• Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992, §4.2 (finitely many elliptic cycles, deduced there from Siegel's theorem).
• 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 (Jørgensen's inequality and its consequences).

theorem Subgroup.IsCofinite.exists_pos_le_dist_smul_of_smul_eq_self {Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)} (hΓ : Γ.IsCofinite) : ∃ c > 0, ∀ g ∈ Γ, g ≠ 1 → ∀ (w : UpperHalfPlane), g • w = w → ∀ k ∈ Γ, k • w ≠ w → c ≤ dist (k • w) w

Elliptic points of a cofinite group are not moved a short distance. For a cofinite Γ ≤ PSL(2, ℝ) there is c > 0 such that if a nonidentity element of Γ fixes w ∈ ℍ, then every element of Γ not fixing w moves it a hyperbolic distance at least c.

theorem Subgroup.IsCofinite.exists_pos_le_dist_smul_of_dist_smul_lt {Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)} (hΓ : Γ.IsCofinite) : ∃ τ > 0, ∀ g ∈ Γ, g ≠ 1 → ∀ (w : UpperHalfPlane), g • w = w → ∀ k ∈ Γ, k • w ≠ w → ∀ (y : UpperHalfPlane), dist (g • y) y < τ → τ ≤ dist (k • y) y

The thin part at an elliptic point. For a cofinite Γ ≤ PSL(2, ℝ) there is τ > 0 with the following property. If a nonidentity element g ∈ Γ fixes w ∈ ℍ and moves y ∈ ℍ less than τ, then every element of Γ not fixing w moves y a hyperbolic distance at least τ. In particular the τ-thin parts of distinct elliptic points are disjoint.

theorem Subgroup.IsCofinite.finite_ellipticOrbits {Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)} (hΓ : Γ.IsCofinite) : Γ.ellipticOrbits.Finite

A cofinite Fuchsian group has finitely many elliptic orbits. For a cofinite Γ ≤ PSL(2, ℝ), only finitely many Γ-orbits in ℍ consist of points with nontrivial stabilizer in Γ.