Jørgensen's inequality #
Let Γ ≤ PSL(2, ℝ) be a discrete subgroup and let A, B ∈ SL(2, ℝ) lift elements of Γ.
Jørgensen's inequality says that unless B normalises ⟨A⟩ in a degenerate way,
|tr² A - 4| + |tr [A, B] - 2| ≥ 1.
For hyperbolic A, that is |tr A| > 2, the exceptional cases are B A B⁻¹ = A and
B A B⁻¹ = A⁻¹ in PSL(2, ℝ): the elements B preserving the pair of fixed points of A on
the ideal boundary, which they fix or exchange. For elliptic A, that is |tr A| < 2, the only
exceptional case is B A B⁻¹ = A: the elements B fixing the fixed point of A in ℍ. Thus in
a discrete group a hyperbolic element of small translation length, or an elliptic element of
small rotation angle, forces every element outside these exceptional cases to have commutator
trace far from 2. This is a standard input for collar and Margulis-type estimates in Fuchsian
groups. For parabolic A, where tr² A = 4, the only exceptional case is again B A B⁻¹ = A:
the elements B fixing the fixed point of A on the ideal boundary. This case is Shimizu's lemma,
Subgroup.inv_le_abs_apply_one_zero_of_upperRightHom_mem: after conjugating A to the
translation z ↦ z + w, the trace tr [A, B] - 2 is w² c², where c is the lower-left entry
of the conjugated B. Since A = ±1 commutes with every B, the three cases together give the
inequality for every pair of elements outside the exceptional cases.
The hyperbolic and elliptic proofs are Jørgensen's, and use his sequence B₀ = B,
Bₙ₊₁ = Bₙ A Bₙ⁻¹ (TauCeti.jorgensenSeq), which stays in Γ.
For hyperbolic A, conjugate A to the dilation D = diag(u, u⁻¹), u = exp (s / 2)
(TauCeti.UpperHalfPlane.exists_eq_conj_dilation_of_two_lt_abs_trace). For
B = !![a, b; c, d] one has tr² D - 4 = (u - u⁻¹)² and tr [D, B] - 2 = -(u - u⁻¹)² b c, so
the inequality reads (u - u⁻¹)² (1 + |b c|) ≥ 1. If it fails, the products βₙ = bₙ cₙ of the
iterates satisfy βₙ₊₁ = -(1 + βₙ) βₙ (u - u⁻¹)², so they tend to 0 geometrically.
Conjugating Bₙ₊₁ by a suitable power of D balances its two off-diagonal entries, and the
resulting elements of Γ converge to D. Discreteness makes one of them equal to D, so some
Bₙ₊₁ is diagonal. Running the iteration backwards, B is diagonal or antidiagonal, which is to
say B D B⁻¹ = D^{±1}.
For elliptic A fixing z ∈ ℍ, put κ = (4 - tr² A) / 2 > 0 and cₙ = cosh (dist (Bₙ • z) z).
The point z is moved by Bₙ₊₁ as far as Bₙ⁻¹ • z is moved by A, so the displacement
formula Matrix.SpecialLinearGroup.cosh_dist_smul_of_smul_eq_self gives
cₙ₊₁ - 1 = κ (cₙ + 1) (cₙ - 1), and by
Matrix.SpecialLinearGroup.trace_commutatorElement_of_smul_eq_self the inequality reads
κ (1 + c₀) ≥ 1. If it fails and B does not fix z, the numbers cₙ - 1 > 0 decrease
strictly, so the Bₙ are infinitely many distinct elements of Γ moving z a bounded distance,
contradicting proper discontinuity. No convergence argument is needed.
Main results #
Subgroup.one_le_four_mul_sinh_sq_mul_of_dilation_mem: Jørgensen's inequality for the dilationz ↦ exp s * z, in coordinates.one_le_abs_trace_sq_sub_four_add_abs_trace_commutatorElement_sub_two_of_two_lt_abs_trace, in the namespaceSubgroup: Jørgensen's inequality for a hyperbolic element, in trace form.Subgroup.one_le_four_sub_trace_sq_mul_cosh_sq_of_smul_eq_self: Jørgensen's inequality for an elliptic element fixingz, in terms of the distanceBmovesz.one_le_abs_trace_sq_sub_four_add_abs_trace_commutatorElement_sub_two_of_abs_trace_lt_two, in the namespaceSubgroup: Jørgensen's inequality for an elliptic element, in trace form.Subgroup.one_le_abs_trace_commutatorElement_sub_two_of_isParabolic: Jørgensen's inequality for a parabolic element, which is Shimizu's lemma in trace form.Subgroup.one_le_abs_trace_sq_sub_four_add_abs_trace_commutatorElement_sub_two: Jørgensen's inequality for every element, in trace form.
References #
- Troels Jørgensen, On discrete groups of Möbius transformations, American Journal of Mathematics 98 (1976), 739–749.
- Alan Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer, 1983, §5.4.
Jørgensen's sequence for a dilation #
Jørgensen's inequality #
Jørgensen's inequality for a dilation. Let Γ ≤ PSL(2, ℝ) be discrete and contain the
dilation z ↦ exp s * z and the class of B = !![a, b; c, d] ∈ SL(2, ℝ). Unless B conjugates
this dilation to itself or to its inverse, 4 sinh² (s / 2) (1 + |b c|) ≥ 1.
The trace form #
Jørgensen's inequality for hyperbolic elements. Let Γ ≤ PSL(2, ℝ) be a discrete subgroup
and let A, B ∈ SL(2, ℝ) lift elements of Γ, with A hyperbolic: |tr A| > 2. Unless B
conjugates the class of A to itself or to its inverse, that is, unless B fixes or exchanges
the two fixed points of A on the ideal boundary,
|tr² A - 4| + |tr [A, B] - 2| ≥ 1.
Elliptic elements #
Jørgensen's inequality for an element fixing a point. Let Γ ≤ PSL(2, ℝ) be discrete and
let A ∈ SL(2, ℝ) lift a nontrivial element of Γ fixing z ∈ ℍ. Every B ∈ Γ not fixing z
satisfies (4 - tr² A) cosh² (dist (B • z) z / 2) ≥ 1. When A acts as the rotation through
angle 2θ about z this reads 2 |sin θ| cosh (dist (B • z) z / 2) ≥ 1.
Jørgensen's inequality for elliptic elements. Let Γ ≤ PSL(2, ℝ) be a discrete subgroup
and let A, B ∈ SL(2, ℝ) lift elements of Γ, with A elliptic: |tr A| < 2. Unless B
commutes with A in PSL(2, ℝ), that is, unless B fixes the fixed point of A in ℍ,
|tr² A - 4| + |tr [A, B] - 2| ≥ 1.
Parabolic elements #
Jørgensen's inequality for parabolic elements. Let Γ ≤ PSL(2, ℝ) be a discrete subgroup
and let A, B ∈ SL(2, ℝ) lift elements of Γ, with A parabolic, so that tr² A = 4. Unless B
commutes with A in PSL(2, ℝ), that is, unless B fixes the fixed point of A on the ideal
boundary, |tr [A, B] - 2| ≥ 1. This is Shimizu's lemma in trace form.
Every element #
Jørgensen's inequality. Let Γ ≤ PSL(2, ℝ) be a discrete subgroup and let
A, B ∈ SL(2, ℝ) lift elements of Γ. Unless B conjugates the class of A to itself or to
its inverse, |tr² A - 4| + |tr [A, B] - 2| ≥ 1.