The First Betti Number of the Smallest Closed Hyperbolic 3-Manifold

Marc Culler^{1∗}, Sa’ar Hersonsky^{2}, Peter B. Shalen^{3∗}

1Department of Mathematics, University of Illinois, Chicago, IL 60607, USA; culler@math.uic.edu 2Department of Mathematics, Cornell University, Ithaca, NY 14853, USA; saar@math.cornell.edu 3Department of Mathematics, University of Illinois, Chicago, IL 60607, USA; shalen@math.uic.edu

Introduction

It follows from work of Gromov, Jørgensen and Thurston (see [3]) that the real numbers which arise as volumes of hyperbolic 3-manifolds form a well-ordered set. It is not known at present which closed 3-manifold has the minimal volume (or whether such a manifold is unique). The techniques developed in the series of papers [6], [7], [8], [9], [1] bear on this question since they give volume estimates which depend on topological properties of the manifold. If a certain topological hypothesis can be shown to imply a volume bound that exceeds the volume of a known manifold, one obtains topological information about any minimal volume manifold. The first estimates to have interesting qualitative consequences of this sort appeared in the paper [1]. In the present paper we prove the following result.

Theorem A. If M is a closed orientable hyperbolic 3-manifold of minimal volume then the first Betti number of M is at most 2.

According to [15], the arithmetic 3-manifold obtained by (−5/1, −5/2) Dehn surgery on the Whitehead link has volume 0.94270 . . .. Thus Theorem A follows from:

∗Partially supported by a grant from the National Science Foundation.

Theorem B. Let M be a closed orientable hyperbolic 3-manifold with first Betti number at least 3. Then the volume of M is at least 0.94689.

In fact, we will prove a somewhat stronger result than Theorem B. Recall that if M is a closed orientable hyperbolic 3-manifold then a finitely generated subgroup of π_{1}(M) is said to be topologically tame if the corresponding covering space of M is homeomorphic to the interior of a compact 3-manifold. (The so-called “tameness conjecture” would imply that every finitely generated subgroup of π_{1}(M) is topologically tame.) As our main theorem we will prove:

Theorem 1.1. Let M be a closed orientable hyperbolic 3-manifold such that every 2generator subgroup of π_{1}(M) is topologically tame. Then either π_{1}(M) contains a 2generator subgroup of finite index, or the volume of M exceeds 0.94689.

Theorem B can be deduced from Theorem 1.1 by using [6; Prop. 10.2], which implies that if the first Betti number of M is at least 3 then every 2-generator subgroup of π_{1}(M) is of infinite index and topologically tame. The proof of the main theorem is contained in Section 1 of the paper. The proof depends on a number of subsidiary results. These results, or special cases of them, are stated as the need arises in the course of the proof. The rest of the paper is devoted to the proofs of the subsidiary results. The most interesting of these are Theorem 4.1, a refinement of Theorem 9.1 of [6], which is used via its Corollary 4.9; Proposition 5.1, a result restricting the set of distances among eight points on the 2-sphere; and Proposition 2.5, a hyperbolic geometry theorem which is essentially topological in nature, as its proof is based on an analysis of coverings of hyperbolic space by cylinders. In an attempt to clearly expose the logical structure of the proof we have included three appendices which contain proofs of certain technical results that are logically independent of the results in the body of the paper. The appendices are indexed by letters, so for example, Lemma A.1 is a reference to the first lemma in Appendix A. Throughout the paper there are times when we need approximate values of certain real constants. We will give these values as decimal numbers followed by ellipses, so 1.234 … denotes a real number in the interval [1.234, 1.235). We will be making computations in spherical geometry which will involve the following conventions and notation. We shall think of S^{2} as the unit sphere in ℝ^{3}. Any point P ∈ S^{2} may be written in the form (cos θ cos λ, sin θ cos λ, sin λ) with −π/2 ≤ λ ≤ π/2 and 0 ≤ θ < 2π. The latitude λ = λ(P) is uniquely determined by P; we have λ(P) = π/2 − ϕ(P), where ϕ(P) is the polar angle. The longitude θ = θ(P) is uniquely determined unless P is one of the poles N = (0, 0, 1) or S = (0, 0, −1).

We shall denote by ℓ : S^{2} \ {N, S} → S^{1} the projection map defined by

ℓ(P) = (cos θ(P), sin θ(P)).

For any two points P, Q ∈ S^{2} \ {N, S} we shall let Θ(P, Q) denote the circular distance between ℓ(P) and ℓ(Q). Thus Θ(P, Q) is the absolute value of the unique element of the interval (−π, π] which is congruent to θ(P) − θ(Q) modulo 2π. Note that ℓ : S^{2} \ {N, S} → S^{1} and Θ : (S^{2} \ {N, S}) × (S^{2} \ {N, S}) → [0, π] are continuous although θ : S^{2} \ {N, S} → [0, 2π) is not. A meridian in S^{2} is the closure of a fiber of the map ℓ. Two points P, P′ ∈ S^{2} lie on a common meridian if and only if either (i) P or P′ is a pole or (ii) P, P′ ∈ S^{2} − {N, S} and ℓ(P) = ℓ(P′). We will use the notation dist_{s}(P, Q) for the spherical distance between two points P and Q of S^{2}. To minimize confusion when we are working on the sphere at infinity of hyperbolic space, we will write dist_{h}(x, y) for the distance between two points x and y of the hyperbolic 3-space ℍ^{3}. We will use the notation Isom_{+}(ℍ^{3}) to denote the group of orientation preseving isometries of ℍ^{3}. If A, B and C are points in ℍ^{3} with A ≠ B ≠ C, we denote the angle between the lines BA and BC by ∠(A, B, C). If Γ is a group, we shall write H ≤ Γ to indicate that H is a subgroup of Γ. We shall use ⟨x_{1},…, x_{n}⟩ to denote the subgroup generated by elements x_{1},…, x_{n} of Γ. We thank John Smillie for suggesting the idea that led to the proof of Theorem 3.1.

1. Proof of the Main Theorem

In this section we give a derivation of the main estimate of the paper. The argument depends on results which are proved in later sections of the paper. We shall state these results as they are needed in the course of the argument.

Theorem 1.1. Let M be a closed orientable hyperbolic 3-manifold such that every 2generator subgroup of π_{1}(M) is topologically tame. Then either π_{1}(M) contains a 2generator subgroup of finite index, or the volume of M exceeds 0.94689.

Proof. We write M = ℍ^{3}/Γ where Γ < Isom_{+}(ℍ^{3}) is discrete, torsion-free, co-compact and, consequently, purely loxodromic.

The proof uses four carefully chosen constants. We set

\begin{aligned}β_{0} & = 0.51 π \\ ϵ_{0} & = 0.0065 \\ λ_{0} & = 1.00485 \\ δ_{0} & = 0.71497 π. \end{aligned}

It follows from [1; Cor. 7.3, Prop. 8.1 and Lemma 10.3] that if every 2-generator subgroup of π_{1}(M) = Γ has infinite index and is topologically tame, and if M contains a non-trivial closed geodesic whose length is less than some given positive number λ, then the volume of M is at least

V (λ) = \frac{πλ}{e^{λ} − 1}(\frac{e^{2λ} + 2e^{λ} + 5}{2(cosh \frac{λ}{2})(e^{λ} + 3)}) − \frac{πλ}{2}.

We have V (λ_{0}) = 0.94689 . . .. Thus the conclusion of Theorem 1.1 is certainly true if M contains a non-trivial closed geodesic of length ≤ λ_{0}. We shall now assume that every 2-generator subgroup of Γ is topologically tame and of infinite index, and that every non-trivial closed geodesic in M has length > λ_{0}. In particular, every non-trivial element of Γ has translation length > λ_{0}. We shall show that under these assumptions M contains a hyperbolic ball of radius \frac{1}{2}(log 3+ ϵ_{0}). As was observed by Meyerhoff [13], this implies the conclusion of Theorem 1.1 by an estimate due to Böröczky [4] for the density of a hyperbolic sphere-packing. Given an embedded ball in M, one obtains a sphere-packing by considering all of the lifts of the boundary sphere to hyperbolic space. Böröczky’s result gives an explicit estimate for the volume of the Dirichlet domain for the sphere-packing which, in this case, is also a Dirichlet domain for M. Applying this estimate, exactly as was done in [6; Corollary 10.4], one obtains that if M contains a ball of radius \frac{1}{2}(log 3 + ϵ_{0}) then the volume of M is at least 0.94689 . . .. Let us now assume that M contains no ball of radius \frac{1}{2}(log 3 + ϵ_{0}). We will show that this assumption leads to a contradiction. We begin by applying the following general result, which is proved in Section 2.

Proposition 2.5. Let Γ be a co-compact, torsion-free, discrete subgroup of Isom_{+}(ℍ^{3}). Let M denote the closed hyperbolic 3-manifold ℍ^{3}/Γ. Let Δ be a positive real number. Then either

(i) M contains a hyperbolic ball of radius Δ/2, or

(ii) there exist a point z ∈ ℍ^{3} and pairwise non-commuting elements ξ_{0}, ξ_{1}, ξ_{2}, ξ_{3} of Γ such that dist_{h}(z, ξ_{i} · z) < Δ for i = 0, 1, 2, 3.

We set Δ = log 3 + ϵ_{0} in Proposition 2.5. Since we have assumed that conclusion (i) does not hold, we find a point z of ℍ^{3} which is moved a distance less than \frac{1}{2}(log 3 + ϵ_{0}) by pairwise non-commuting elements ξ_{0}, ξ_{1}, ξ_{2}, ξ_{3} of Γ = π_{1}(M). Note that, since the ξ_{i} are non-trivial elements of Γ, they are loxodromic and have translation length > λ_{0}. Let us identify H^{3} conformally with the open unit ball in ℝ^{3} in such a way that z is the center of the ball. Then the unit sphere S^{2} is identified with the sphere at infinity, and every ray in ℍ^{3} emanating from z has a well-defined endpoint in S^{2}. For every (i, u) ∈ {0, 1, 2, 3} × {−1, 1}, let P_{(i,u)} ∈ S^{2} denote the endpoint of the ray that emanates from z and passes through the point ξ_{i}^{u}(z) (which is distinct from z since ξ_{i} is loxodromic). For any indices (i, u) and (j, v) in {0, 1, 2, 3}×{1, −1} we have dist_{s}(P_{(i,u)}, P_{(j,v)}) = ∠(ξ_{i}^{u}·z, z, ξ_{j}^{v}·z). In sections 3 and 4 we derive explicit lower bounds for the spherical distances between pairs of the points P_{(i,u)}. To state these results we must introduce certain auxiliary functions. First we define a function ϕ: ℝ^{+} × ℝ^{3} → ℝ by

ϕ(l, t, s, α) = \frac{2 cosh^{2} s − 2 sinh^{2} s cos α − cosh l − cos t}{cosh l − cos t}.

By direct calculation we find that

ϕ(λ_{0}, π, log 3 + ϵ_{0}, 0) = 0.56936 … < 1 < 3.40499 … = ϕ(λ_{0}, π, log 3 + ϵ_{0}, π).

Since ϕ is monotonically increasing with respect to the fourth variable, there is a unique α_{−∞} ∈ [0, π] such that ϕ(λ_{0}, π, log 3 + ϵ_{0}, α_{−∞}) = 1, and for any α ∈ (α_{−∞}, π] we have ϕ(λ_{0}, π, log 3 + ϵ_{0}, α) > 1. Solving numerically, one finds that α_{−∞} = 0.80060 . . .. We set

σ(α) = cosh^{−1} ϕ(λ_{0}, π, log 3 + ϵ_{0}, α)

for α ∈ (α_{−∞}, π]. We also need a constant K whose value is a slight perturbation of σ(β_{0}). By direct calculation we find that

ϕ(λ_{0}, π, log 3 − ϵ_{0}, β_{0}) = 1.98495 … > 1,

and

cosh^{−1} ϕ(λ_{0}, π, log 3 − ϵ_{0}, β_{0}) = 1.30822 ….

We set

K = 1.30822,

so that cosh^{−1} ϕ(λ_{0}, π, log 3 − ϵ_{0}, β_{0}) > K. The following result is included in the main result of Section 4. It is obtained by assigning particular values to the parameters appearing in the latter result.

Corollary 4.9. Let ξ and η be isometries which generate a subgroup of Isom_{+}(ℍ^{3}) which is discrete, free of rank 2, purely loxodromic and topologically tame. Suppose that ξ^{−1}η has translation length ≥ λ_{0}. Let z be a point of ℍ^{3} such that

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ_{0}.

Set α_{1} = ∠(ξ ·z, z, η ·z) and α_{−1} = ∠(ξ^{−1} ·z, z, η^{−1} ·z). Then α_{1} and α_{−1} lie in the interval [α_{−∞}, π] and

σ(α_{1}) + σ(α_{−1}) > 2K.

To first order, the conclusion of 4.9 consists of a lower bound on the sum of the angles α_{1} and α_{−1}. Corollary 4.9 is a refinement of Theorem 9.1 of [6], which gives a lower bound of log 3 for the quantity max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} without any restriction on α_{1} and α_{−1}. Suppose that (i, u) and (j, v) are indices in {0, 1, 2, 3} × {1, −1} with i ≠ j. We wish to apply Corollary 4.9 with ξ = ξ_{i}^{u}, η = ξ_{j}^{v}, (β, ϵ) = (β_{0}, ϵ_{0}), λ = λ_{0} and θ = π. Since the first Betti number of M is at least 3, and since ξ_{i} and ξ_{j} do not commute, it follows from [1; Cor. 7.2] and [5; Prop. 3.2] that ξ and η generate a free group F of rank 2 which is topologically tame. Since Γ is co-compact, F is purely loxodromic. Since ξ^{−1}η is a nontrivial element of Γ, it has translation length > λ_{0}. It follows from the defining properties of ξ_{0},…, ξ_{3} that dist_{h}(z, ξ_{i}^{u} · z) and dist_{h}(z, ξ_{j}^{v} · z) are less than log 3 + ϵ_{0}. Thus all the hypotheses of Corollary 4.9 hold. Setting α_{1} = ∠(ξ · z, z, η · z) = dist_{s}(P_{(i,u)}, P_{(j,v)}) and α_{−1} = ∠(ξ^{−1} · z, z, η^{−1} · z) = dist_{s}(P_{(i,−u)}, P_{(j,−v)}), we conclude that

σ(α_{1}) + σ(α_{−1}) > 2K.

This shows that for any two indices (i, u) and (j, v) in {0, 1, 2, 3} × {1, −1} such that i ≠ j, we have dist_{s}(P_{(i,u)}, P_{(j,v)}) > α_{−∞}, dist_{s}(P_{(i,−u)}, P_{(j,−v)}) > α_{−∞} and

σ(dist_{s}(P_{(i,u)}, P_{(j,v)})) + σ(dist_{s}(P_{(i,−u)}, P_{(j,−v)}) > 2K.

The next step is to apply the following Proposition about configurations of eight points on a sphere which is proved in Section 5.

Proposition 5.1. Suppose that we are given an indexed family

(P_{(i,u)})_{(i,u)∈{0,1,2,3}×{−1,1}}

of points in S^{2}. Assume that for any two indices (i, u) and (j, v) in {0, 1, 2, 3} × {−1, 1} with i ≠ j, we have dist_{s}(P_{(i,u)}, P_{(j,v)}) > α_{−∞}. Then either

(i) there is an element i of {0, 1, 2, 3} such that dist_{s}(P_{(i,1)}, P_{(i,−1)})) ≤ δ_{0}, or (ii) there exist indices (i, u) and (j, v) in {0, 1, 2, 3} × {−1, 1}, with i ≠ j, such that

σ(dist_{s}(P_{(i,u)}, P_{(j,v)})) + σ(dist_{s}(P_{(i,−u)}, P_{(j,−v)})) ≤ 2K.

Observe that the family (P_{(i,u)}) satisfies the hypothesis of Proposition 5.1 but, by 4.9, does not satisfy alternative (ii) of the conclusion. On the other hand, at the end of Section 4 we prove a result which shows that the family (P_{i,u}) does not satisfy alternative (i) either. The statement requires one more auxiliary function. For Δ > λ > 0 we define

ω(λ, Δ) = cos^{−1} (1 − \frac{2(cosh λ − 1)}{cosh Δ − 1}) .

Proposition 4.10. Let 0 < λ < Δ be real numbers. Let ξ be a loxodromic isometry of ℍ^{3} with translation length ≥ λ, and let z be a point of ℍ^{3} such that dist_{h}(z, ξ · z) < Δ. Then we have ∠(ξ^{−1} · z, z, ξ · z) > ω(λ, Δ).

Since each ξ_{i} has translation length > λ_{0} and since dist_{h}(z, ξ_{i} · z) < Δ, it follows from Proposition 4.10 that ∠(ξ_{i}^{−1} · z, z, ξ_{i} · z) > ω(λ_{0}, Δ). By direct computation we have

ω(λ_{0}, log 3 − ϵ_{0}) = (0.71497 …)π ≥ δ_{0}.

This means that dist_{s}(P_{(i,1)}, P_{(i,−1)}) > δ_{0} for i = 0, 1, 2, 3. Thus alternative (i) of Proposition 5.1 also fails to hold for the family (P_{(i,u)}). This gives a contradiction to Proposition 5.1, and the proof of Theorem 1.1 is therefore complete.

□

2. Intersections of cylinders in ℍ^{3}

In this section we prove Proposition 2.5, which was the starting point for the proof of Theorem 1.1. The proof is based on an analysis of coverings of hyperbolic space by cylinders, which is formulated in Proposition 2.1. We let ℍ̂^{3} denote the union of ℍ^{3} with the sphere at infinity, equipped with the standard topology that makes it homeomorphic to a closed 3-ball. In this section we will use the Beltrami-Klein model for ℍ̂^{3}, i.e. we will identify ℍ̂^{3} with the closed unit ball in ℝ^{3} in such a way that the lines in ℍ^{3} are open Euclidean line segments in ℝ^{3}. If we use this model, the Euclidean metric on the unit ball becomes a metric on ℍ̂^{3} which will be denoted dist_{e} to distinguish it from the hyperbolic metric dist_{h} on ℍ^{3}. A subset of ℍ^{3} is (strictly) convex in the hyperbolic sense if and only if it is identified with a set in the Beltrami-Klein model which is (strictly) convex in the Euclidean sense. For any closed set X ⊂ ℍ^{3}, we let X̂ denote the closure of X in ℍ̂^{3}, and we set Ẋ = X̂ −X. In particular, ℍ̇^{3} denotes the sphere at infinity. For any line l ⊂ ℍ^{3}, the set l̂ is identified with a closed line segment in the Beltrami-Klein model, and l̇ is identified with the set of endpoints of this segment. Recall that two lines l and l′ in ℍ^{3} are said to be parallel if l̇ ∩ l̇′ ≠ ∅. In this case, either l̇ ∩ l̇′ consists of a single point, or l = l′. If A is a line in ℍ^{3} and r is a positive real number, we shall denote by Z_{r}(A) the set of all points in ℍ^{3} whose hyperbolic distance from A is ≤ r. By a cylinder in ℍ^{3} we shall mean a set of the form Z = Z_{r}(A), where A ⊂ ℍ^{3} is a line and r is a positive number. The line A is uniquely determined by the set Z because Ȧ = ℍ̇^{3} ∩ Ẑ. It follows easily that r is also uniquely determined by Z. We shall refer to A and r respectively as the core and the radius of the cylinder Z. If Z is a cylinder with core A and radius r, then, by [6; Prop. 1.2], Z is a strictly convex subset of ℍ^{3}. It follows that Ẑ is a strictly convex, compact subset of ℍ̂^{3} and is therefore identified in the Beltrami-Klein model with a strictly convex, compact subset of the unit ball in ℝ^{3}. Since this set clearly has non-empty interior, it follows that Ẑ is a topological 3-ball, and that the boundary of this ball, ∂Ẑ, is identified with its frontier in ℝ^{3} in the Beltrami-Klein model. On the other hand, it is clear that Ż is equal to Ȧ, a two-point subset of Ḣ^{3}. Hence ∂Z = ∂Ẑ − Ż is a topological annulus, and coincides with the frontier of Z in ℍ^{3}. The first main result of this section is:

Proposition 2.1. Let Z be a locally finite collection of cylinders whose interiors cover ℍ^{3}. Suppose that the cores of any two distinct cylinders in Z are non-parallel (and in particular distinct). Let R ⊂ (0, ∞) denote the set of all radii of cylinders in Z. Suppose that R has a greatest element r_{0}, and let Z_{0} be any cylinder of radius r_{0} in Z. Then there are cylinders Z_{1}, Z_{2}, Z_{3} ∈ Z, distinct from one another and from Z_{0}, such that ∂Z_{0} ∩Int Z_{1} ∩Int Z_{2} ∩Int Z_{3} ≠ ∅. In particular we have Int Z_{0} ∩Int Z_{1} ∩Int Z_{2} ∩Int Z_{3} ≠ ∅.

Note that if the open set Int Z_{1} ∩ Int Z_{2} ∩ Int Z_{3} meets the frontier ∂Z_{0} of Int Z_{0} in ℍ^{3}, then Int Z_{1} ∩ Int Z_{2} ∩ Int Z_{3} must also meet the set Int Z_{0} itself. Thus the last assertion of the above proposition does follow from the first. The following two lemmas are needed for the proof of Proposition 2.1. An open subset U of a connected, non-simply-connected 2-manifold Σ will be termed inessential if (i) U has compact closure in Σ and (ii) if U is non-empty then, for every component V of U, the inclusion homomorphism π_{1}(V) → π_{1}(Σ) is trivial.

Lemma 2.2. Let Σ be a connected, non-compact, non-simply-connected 2-manifold without boundary. Let U be a locally finite collection of inessential open subsets of Σ that covers Σ. Then there are three distinct sets U_{1}, U_{2}, U_{3} ∈ U such that U_{1} ∩ U_{2} ∩ U_{3} ≠ ∅.

Proof. We may choose a compact set X_{U} ⊂ U for each U ∈ U, in such a way that the interiors of the sets X_{U} cover Σ. Let us fix a piecewise linear structure on Σ. After possibly enlarging the X_{U} we may assume that each X_{U} is a polyhedral subset of Σ. After replacing the X_{U} by their regular neighborhoods we may assume that they are compact polyhedral 2-manifolds with boundary. For each U there is a polyhedral disk D_{U} ⊂ Σ such that ∂D_{U} ⊂ X_{U} ⊂ D_{U}. Indeed, the hypothesis that each U ∈ U is inessential implies that each component C of ∂X_{U} bounds a disk Δ_{C} ⊂ Σ. We must have either X_{U} ⊂ Δ_{C} or Δ_{C} ∩X_{U} = C. But if Δ_{C} ∩X_{U} = C for every component C of ∂X_{U}, then the set X_{U} ∪⋃_{C} Δ_{C}, where C ranges over all components of ∂X_{U}, is a closed 2-manifold; this is impossible since Σ is connected and non-compact. Hence for some component C of ∂X_{U} we must have X_{U} ⊂ Δ_{C}. The disk D_{U} = Δ_{C} then has the asserted properties. We claim that there exists a set W ∈ U with the property that there is no U ∈ U for which D_{W} ⊂ Int D_{U}. Assume that this is false. Then there is a sequence (U_{i})_{i≥0} of sets in U such that if we set D_{i} = D_{U_{i}} for i = 0, 1,…, we have D_{i} ⊂ Int D_{i+1} for every i ≥ 1. Set D = ⋃_{i≥0} D_{i}. Then D is an open, simply-connected subset of Σ. Since Σ is connected and non-simply-connected, D must have a non-empty frontier. Let P be a point of the frontier of D, and let W be a connected neighborhood of P in Σ whose closure is compact. The set W meets D, and therefore meets D_{j} for some j ≥ 0; hence W meets D_{i} for every i ≥ j. On the other hand, there is no i for which W ⊂ D_{i}; for this would imply W ⊂ Int D_{i+1} ⊂ Int D, and P would not be on the frontier of D. Since W is connected it follows that W meets the boundary of D_{i} for each i ≥ j. But ∂D_{i} = ∂D_{U_{i}} ⊂ X_{U_{i}} ⊂ U_{i}; thus W ∩ U_{i} ≠ ∅ for each i ≥ j. Since it is clear from the choice of the sequence (U_{i})_{i≥0} that the U_{i} are all distinct, this contradicts the local finiteness of U. This proves the claim. Now if W ∈ U is the set given by the claim just proved, let us set S_{U} = ∂D_{W} ∩ X_{U} for each U ∈ U. Since Σ is covered by the sets Int X_{U} for U ∈ U, the simple closed curve ∂D_{W} is covered by the sets S_{U} for U ∈ U. Since X_{W} ⊂ D_{W}, we have S_{W} = ∅. Thus ∂D_{W} is covered by the sets S_{U} for W ≠ U ∈ U. Our choice of W guarantees that each S_{U} is a proper subset of ∂D_{W}. Since the S_{U} are open in ∂D_{W} and ∂D_{W} is connected, there are two sets U, U′ ∈ U, distinct from W and from each other, such that S_{U} ∩ S_{U′} ≠ ∅. We have

∅ ≠ S_{U} ∩ S_{U′} = ∂D_{W} ∩ Int X_{U} ∩ Int X_{U′} ⊂ X_{W} ∩ X_{U} ∩ X_{U′} ⊂ W ∩ U ∩ U′,

and the lemma is proved.

□

Lemma 2.3. Let Z_{0} and Z be two cylinders in ℍ^{3} with non-parallel cores, and let r_{0} and r denote their respective radii. Suppose that r_{0} ≥ r. Then Int Z ∩ ∂Z_{0} is inessential in ∂Z_{0}.

Proof. Let A and A_{0} denote the cores of Z and Z_{0} respectively. Since A and A_{0} are non-parallel, we have Ż ∩ Ż_{0} = Ȧ ∩ Ȧ_{0} = ∅. Hence Z ∩ Z_{0} ⊂ ℍ^{3} is compact. In particular Int Z ∩ ∂Z_{0} has compact closure in ∂Z_{0}. It remains to show that for every component V of Int Z ∩∂Z_{0}, the inclusion homomorphism π_{1}(V) → π_{1}(Σ) is trivial. We first prove this in the “generic” case in which A ∩ A_{0} = ∅. In this case, since A and A_{0} are not parallel, we have Â ∩ Â_{0} = ∅. To prove the assertion in this case, we begin by defining a map f : Â_{0} → ∂Ẑ_{0} as follows. Given any point z ∈ Â_{0}, we have z ∉ Â since Â ∩ Â_{0} = ∅. Hence there is a unique line L_{z} such that L̂_{z} contains z and L_{z} meets A perpendicularly. Let P_{z} denote the point of intersection of L_{z} with A. We have P_{z} ≠ z since z ∉ A. In the Beltrami-Klein model, L̂_{z} is a non-degenerate closed line segment. Let us write L̂_{z} = X_{z} ∪ Y_{z}, where X_{z} and Y_{z} are closed line segments such that P_{z} ∈ Y_{z} and X_{z} ∩ Y_{z} = {z}. Note that Y_{z} is always non-degenerate, but that X_{z} will be degenerate if z ∈ Ȧ_{0}. Since z ∈ Â_{0} ⊂ Ẑ_{0}, and since (according to the remarks at the beginning of this section) Ẑ_{0} is a strictly convex subset of the unit ball, there is a unique point of intersection of X_{z} with ∂Ẑ_{0}. We define f(z) to be this point of intersection. Note that we may characterize f(z) as the unique point of L̂_{z} ∩ ∂Ẑ_{0} such that dist_{e}(f(z), P_{z}) ≥ dist_{e}(f(z), z). Note also that f restricts to the identity on Ȧ_{0} and that f(A_{0}) ⊂ ∂Z_{0}. We claim that f : Â_{0} → ∂Ẑ_{0} is continuous. To prove this, it suffices to show that if (z_{i}) is a sequence of points of A_{0} converging to a point z ∈ Ȧ_{0}, then (f(z_{i})) converges to z. If this is false, then after passing to a subsequence we may assume that (f(z_{i})) converges to some point w ≠ f(z) in ˆ∂Z_{0}. Now since (z_{i}) converges to z, it is clear that the sequence (P_{i}) = (P_{z_{i}}) converges to P_{z} in the metric d_{e}, and hence that the sequence of line segments (L̂_{i}) = (L̂_{z_{i}}) converges to L_{z} in the Hausdorff metric defined by d_{e}. Since f(z_{i}) ∈ L_{i} for each i, it follows that w ∈ L_{z}. Similarly,

dist_{e}(w, P_{z}) = lim_{i→∞} dist_{e}(f(z_{i}), P_{i}) ≥ lim_{i→∞} dist_{e}(f(z_{i}), z_{i}) = dist_{e}(w, z).

From the above characterization of f(z) we conclude that w = f(z). This contradiction establishes the continuity of f. Now consider the subset S = ∂Z_{0} − f(A_{0}) of ∂Z_{0}. Since f restricts to the identity on Ȧ_{0} and f(A_{0}) ⊂ Z_{0}, we have S = ∂Ẑ_{0} − f(Â_{0}). The set f(Â_{0}) is compact and connected since f is continuous and Â_{0} is homeomorphic to a line segment. Since, according to the remarks at the beginning of this section, ∂Ẑ_{0} is a topological 2-sphere, it follows that each component of S is simply connected. We shall complete the proof that π_{1}(V) → π_{1}(Σ) is trivial for every component V of Int Z ∩ ∂Z_{0} by showing that Int Z ∩ ∂Z_{0} ⊂ S. For this purpose we consider an arbitrary point w ∈ Int Z ∩ ∂Z_{0}. Thus

dist_{h}(w, A) < r ≤ r_{0} = dist_{h}(w, A_{0}).  (2.3.1)

Assume that w ∉ S, so that w = f(z) for some z ∈ A_{0}. According to the definition of the map f, the point w lies on the line L_{z}, which meets A perpendicularly at P_{z}, and z lies on the segment of L_{z} with endpoints w and P_{z}. Hence

dist_{h}(w, A) = dist_{h}(w, P_{z}) ≥ (w, z) ≥ dist_{h}(w, A_{0}).

This contradicts (2.3.1), and the proof is thus complete in the case A ∩ A_{0} = ∅. Finally, we consider the case in which A ∩ A_{0} ≠ ∅. Assume that Int Z ∩ ∂Z_{0} has a component V for which π_{1}(V) → π_{1}(Σ) is non-trivial. Then there is a continuous map g : S^{1} → Int Z ∩∂Z_{0} which is homotopically non-trivial in Σ. Now let us choose a sequence of lines (A^{(i)}) in ℍ^{3} such that the sequence (Â^{(i)}) converges to (Â) in the Hausdorff metric defined by dist_{e}, and such that Â^{(i)} ∩ Â_{0} = ∅ for every i. Set Z^{(i)} = Z_{r}(A^{(i)}) for every i. By the case of the assertion already proved, π_{1}(V) → π_{1}(Σ) is trivial for every i and every component V of Int Z^{(i)} ∩ ∂Z_{0}. Hence there is no i for which g(S^{1}) ⊂ Z^{(i)}. This means that for each i there is a point w^{(i)} in the compact set g(S^{1}) such that dist_{h}(w^{(i)}, A^{(i)}) ≥ r. After passing to a subsequence we may assume that (w^{(i)}) converges to a point w ∈ g(S^{1}). In particular we have w ∈ Int A, so that dist_{h}(w, P) < r for some point P ∈ A. According to our choice of the A^{(i)}, there is a sequence of points (P^{(i)}) in ℍ̂^{3} such that P^{(i)} ∈ Â^{(i)} for every i and (P^{(i)}) converges to P in the metric dist_{e}. We must have P^{(i)} ∈ A^{(i)} for all large enough i; thus after again passing to a subsequence we may assume that P^{(i)} ∈ A^{(i)} for all i. It then follows that (P^{(i)}) converges to P in the metric dist_{h}. Hence for large enough i we have dist_{h}(w^{(i)}, P^{(i)}) < r. Since P^{(i)} ∈ A^{(i)} and dist_{h}(w^{(i)}, A^{(i)}) ≥ r, we have a contradiction.

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2.4. Proof of 2.1. Let U denote the collection of all subsets of ∂Z_{0} having the form ∂Z_{0} ∩ Int Z where Z is an element of Z distinct from Z_{0}. Since the interiors of the cylinders in Z cover ℍ^{3}, and since ∂Z_{0} is disjoint from Int Z_{0}, the collection U covers the topological open annulus ∂Z_{0}. Since Z is locally finite, so is U. By Lemma 2.3, each set in U is inessential in ∂Z_{0}. Hence by Lemma 2.2, applied with Σ = ∂Z_{0}, there are distinct sets U_{1}, U_{2}, U_{3} ∈ U with U_{1} ∩ U_{2} ∩ U_{3} ≠ ∅. Writing U_{i} = ∂Z_{0} ∩ Int Z_{i} with Z_{i} ∈ Z for i = 1, 2, 3, we obtain the conclusion of the proposition.

□

We are now ready to prove the result that was quoted in the proof of Theorem 1.1.

Proposition 2.5. Let Γ be a co-compact, torsion-free, discrete subgroup of Isom_{+}(ℍ^{3}). Let M denote the closed hyperbolic 3-manifold ℍ^{3}/Γ. Let Δ be a positive real number. Then either (i) M contains a hyperbolic ball of radius Δ/2 or (ii) there exist a point z ∈ ℍ^{3} and pairwise non-commuting elements ξ_{0}, ξ_{1}, ξ_{2}, ξ_{3} of Γ such that dist_{h}(z, ξ_{i} · z) < Δ for i = 0, 1, 2, 3.

Proof. As in [7], for every maximal cyclic subgroup of Γ we denote by Z(X) = Z_{Δ}(X) the set of all points z ∈ ℍ^{3} such that dist_{h}(z, ξ · z) < Δ for some non-trivial element ξ of X. The set Z(X) is the interior of a cylinder Z̄(X) if the maximal subgroup X is generated by an element of translation length < Δ, and otherwise Z(X) is empty. Let Z denote the collection consisting of all the cylinders Z̄(X), where X ranges over the maximal cyclic subgroups of Γ that are generated by elements of translation length < Δ. According to [7; Prop. 3.2], either M contains a hyperbolic ball of radius Δ/2 or Z covers ℍ^{3}. In the latter case we shall show that conclusion (ii) of the proposition holds.

The discreteness of Γ implies that for every point z ∈ ℍ^{3} there are only finitely many elements γ ∈ Γ such that dist(z, γ · z) ≤ Δ. Hence the collection Z is locally finite. Since Γ is discrete and co-compact, every element of Γ lies in a unique maximal cyclic subgroup, which is its centralizer. The stabilizer in Γ of any point of ℍ̇^{3} is either a maximal cyclic subgroup or the trivial group Thus if Z(X) is a cylinder in Z then X is the unique maximal cyclic subgroup fixing either point of Ż(X). It follows that any two distinct cylinders in Z have non-parallel cores. On the other hand, since Γ is co-compact, it contains only finitely many conjugacy classes of elements with translation length < Δ. It follows that the set R ⊂ (0, ∞), consisting of all radii of cylinders in Z, is finite and hence has a greatest element r_{0}. Let Z_{0} be any cylinder of radius r_{0} in Z. Then it follows from Proposition 2.1 that there are cylinders Z_{1}, Z_{2}, Z_{3} ∈ Z, distinct from one another and from R, such that Int Z_{0} ∩ Int Z_{1} ∩ Int Z_{2} ∩ Int Z_{3} ≠ ∅. We may write Z_{i} = Z(X_{i}) for i = 0, 1, 2, 3, where X_{i} is a maximal cyclic subgroup of Γ. For i = 0, 1, 2, 3 there is a non-trivial element ξ_{i} of X_{i} such that dist_{h}(ξ_{i} · z, z) < Δ. Since the ξ_{i} lie in distinct maximal cyclic subgroups of Γ, no two of them can commute.

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3. Angles and displacements, I

Corollary 4.9, which was used in the proof of Theorem 1.1, is a special case of the main result of Section 4, Theorem 4.1, which in turn may be regarded as a refined version of Theorem 9.1 of [6]. If two hyperbolic isometries ξ and η generate a free Kleinian group of rank 2 which is topologically tame and has no parabolic elements, then Theorem 9.1 of [6] asserts that an arbitrary point z of hyperbolic space is displaced at least a distance log 3 by either ξ or η. Theorem 4.1 gives an improved lower bound for this displacement involving the angles ∠(ξ · z, z, η · z) and ∠(ξ^{−1} · z, z, η^{−1} · z). To first order, the refined result says that the lower bound of log 3 can be increased slightly under the assumption that the sum of these two angles is small. Theorem 4.1 is proved in two stages. In this section we consider the case when z lies on the common perpendicular to the axes of ξ and η. Here we are able to take advantage of the existence of an involution that fixes z and conjugates ξ and η to their inverses. In particular, this symmetry implies that the two angles mentioned above are equal. The general case, where z need not lie on the common perpendicular, is handled in the next section. The proof of Theorem 9.1 of [6] makes use of a Patterson-Sullivan construction to produce a certain measure-theoretic decomposition of the limit set of the group Γ = ⟨ξ, η⟩ into four measures corresponding to the two generators and their inverses. As in [6; Lemma 5.3], the existence of an involution that fixes z and conjugates the generators to their inverses will be used to conclude that the measure associated to a generator has the same total mass as that associated to its inverse. Actually, as in [6; Prop. 5.2] these observations are used only in the special case in which every Γ-invariant positive super-harmonic function on ℍ^{3} is constant; the general case of Theorem 4.1 is reduced to this case. In this case the argument given in [6; Lemma 5.5 and Prop. 5.2] involves comparing the four measures with characteristic measures of four spherical caps whose areas sum to 1. The angle α = ∠(ξ · z, z, η · z) = ∠(ξ^{−1} · z, z, η^{−1} · z) is the spherical distance between the centers of two of these caps. In the crucial special case where the caps have areas close to 1/4, the assumption that α is small implies that there is a substantial overlap bvetween these two caps. This leads to a refinement of the estimate established in [6; Prop.5.2]. We will need some more notation regarding spherical geometry. If P is a point of S^{2} and r is a number in the interval (0, π), we shall denote by C(P, r) the spherical “cap” consisting of all points of S^{2} whose spherical distance from P is at most r. An easy computation shows that the area of C(P, r) is 2π(1 − cos r). For any three real numbers r_{1}, r_{2}, α in the interval (0, π) we shall denote by ι(α, r_{1}, r_{2}) the area of the intersection of two spherical caps C(P_{1}, r_{1}) and C(P_{2}, r_{2}), where P_{1} and P_{2} are two points of S^{2} such that dist_{s}(P_{1}, P_{2}) = α. A closed form expression for the function ι is derived in Appendix B. We define functions E : (0, ∞) → (0, \frac{1}{4}) by

E(ϵ) = \frac{1}{1 + 3e^{ϵ}}

and

r(ϵ) = cos^{−1}(1 − 2E(ϵ)).

We define functions I and f on (0, π) × (0, ∞) ⊂ ℝ^{2} by

I(α, ϵ) = \frac{1}{8π} ι(α, r(ϵ), r(ϵ))  (3.0.1)

and

f(α, ϵ) = \frac{1}{2} − E(ϵ) − I(α, ϵ).

Note that it follows from the definition of ι that it is non-negative and is monotonically decreasing as a function of the first variable. Hence I is also non-negative and monotonically decreasing as a function of the first variable. We denote by G the subset of ℝ^{3} consisting of all points (x, u, t) such that x > 1 and 0 ≤ t ≤ u. We define a real-valued function g on G by

g(x, u, t) = \frac{tx}{1 + t(x − 1)} + \frac{u − t}{(1 + u(x − 1))(1 + (2u − t)(x − 1))} + u.

We define a constant ϵ_{∞} = 0.05. We shall denote by D the open subset of (0, π)×(0, ϵ_{∞}) ⊂ ℝ^{2} consisting of all points (β, ϵ) satisfying the following conditions:

(1) f(β, ϵ) > 0;
(2) g(9e^{2ϵ}, \frac{1}{2} − E(ϵ), f(β, ϵ)) < 1; and
(3) I(β, ϵ) < \frac{1}{3}E(ϵ).

Note that if (1) holds then (9e^{2ϵ}, \frac{1}{2} − E(ϵ), f(β, ϵ)) ∈ G, so that condition (2) makes sense. The main result of this section is the following theorem, the proof of which will occupy the rest of the section.

Theorem 3.1. Let (β, ϵ) be any point in D. Let ξ and η be two loxodromic isometries of ℍ^{3} such that the group Γ generated by ξ and η is discrete, topologically tame, purely loxodromic and free on the generators ξ and η. Let z be a point on the common perpendicular to the axes of ξ and η. Suppose that ∠(ξ · z, z, η · z) < β. Then we have

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} ≥ log 3 + ϵ.

We follow the notation of Section 5 of [6]. We denote by λ_{γ,z} : S_{∞} → ℝ the conformal expansion factor of a hyperbolic isometry γ relative to a point z ∈ ℍ^{3}. It is shown in paragraph 2.4 of [6] that if we identify ℍ̂^{3} = ℍ^{3} ∪ S_{∞} conformally with the closed unit ball in ℝ^{3} in such a way that z is the origin and γ^{−1} · z is on the positive vertical axis, then the conformal expansion factor of γ is given by the formula

λ_{γ,z}(ζ) = (c − s cos ϕ)^{−1},  (3.1.1)

where c = cosh dist_{h}(z, γ · z), s = sinh dist_{h}(z, γ · z), and ϕ = ϕ(ζ) is the polar angle of ζ. In this paper we define the pole of γ (relative to z) to be the endpoint P ∈ S^{∞} of the ray emanating from z and passing through γ^{−1} · z. In the coordinate system just described, P is the north pole of S^{2} and the polar angle of a point of S^{2} is its spherical distance from P. Thus λ_{γ,z} is a positive-valued, decreasing function of the spherical distance of a point from the pole of γ. In particular, the pole is the unique maximum point of λ_{γ,z}. For any point z ∈ ℍ^{3} we denote by A_{z} the area measure on the sphere at infinity S_{∞} determined by the round metric centered at z, normalized so as to have total mass 1. Thus in the coordinate system described above, A is obtained from the ordinary area measure by dividing by 4π.

The following lemma is a key step in the proof of Theorem 3.1 which replaces one of the basic estimates in [6]. The original estimate contained in [6; Lemma 5.5], depends on the observation that if ν is a Borel measure on S_{∞} which is bounded above by A_{z}, then

∫_{S_{∞}}(λ_{γ,z})^{2} dν ≤ ∫_{C_{0}}(λ_{γ,z})^{2} dA_{z},

where C_{0} is a spherical cap centered at the pole of γ with area u = ν(S_{∞}). Here we observe that the cap C_{0} can be replaced by the union of a smaller cap of area t = ν(C_{0}) and an annulus of area u − t. Later we will use our condition on angles to get bounds on u and t. The conclusion that g(e^{2Δ}, u, t) ≥ 1 can, as we shall see, be regarded as giving a lower bound for the displacement Δ = dist_{h}(z, γ · z).

Lemma 3.2. Let γ be a loxodromic isometry of ℍ^{3} and let z be any point of ℍ^{3}. Set Δ = dist_{h}(z, γ · z). Let ν be a Borel measure on S_{∞} such that

(i) ν ≤ A_{z} and
(ii) ∫_{S_{∞}}(λ_{γ,z})^{2} dν = 1 − ν(S_{∞}).

Set u = ν(S_{∞}). Let C_{0} be the spherical cap with center at the pole of γ and with area u. Set t = ν(C_{0}). Then we have g(e^{2Δ}, u, t) ≥ 1 .

Proof. Let z ∈ ℍ^{3} be given. As in the discussion preceding the statement of the lemma, we identify ℍ̂^{3} = ℍ^{3} ∪ S_{∞} conformally with the closed unit ball in ℝ^{3} in such a way that z is the origin and γ^{−1} ·z is on the positive vertical axis. We set A = A_{z}, ϕ_{1} = arccos(1−2t), and C_{1} = C(P, ϕ_{1}). Since A is 1/4π times the area measure on S_{∞}, and since C_{1} has area 2π(1 − cos t), we have

A(C_{1}) = \frac{1}{2}(1 − cos ϕ_{1}) = t.

Now we set ϕ_{0} = cos^{−1}(1 − 2u) and C_{0} = C(P, ϕ_{0}). We also set ϕ_{R} = arccos(1 − 4u + 2t), and we let R ⊂ S_{∞} denote the annulus C(P, ϕ_{R}) \ Int C(P, ϕ_{0}). Then we have

A(R) = \frac{1}{2}(1 − cos ϕ_{R}) − \frac{1}{2}(1 − cos ϕ_{0}) = u − t.

Let us denote by ν_{1}, ν_{2} the restrictions of the measure ν to C_{0} and to S_{∞} \ C_{0} respectively. Then we have ν = ν_{1} + ν_{2}. Hence hypothesis (ii) may be rewritten in the form

∫_{C_{0}}(λ_{γ,z})^{2}dν_{1} + ∫_{S_{∞}−C_{0}}(λ_{γ,z})^{2}dν_{2} = 1 − u.  (3.2.1)

Since ν ≤ A we have ν_{1} ≤ A and ν_{2} ≤ A. Since, by the discussion preceding the statement of the lemma, λ_{γ,z}(ζ) is a positive-valued monotonically decreasing function of the polar angle ϕ, we apply Lemma 5.4 of [6] with C = C_{1}, μ_{0} = ν_{1}, μ = A, X = C_{0} and f = λ_{γ,z} to obtain the following inequality:

∫_{C_{0}}(λ_{γ,z})^{2}dν_{1} ≤ ∫_{C_{1}}(λ_{γ,z})^{2}dA.  (3.2.2)

Applying the same lemma from [6] with C = R, μ_{0} = ν_{2}, μ = A, X = S_{∞} \ C_{0} and f = λ_{γ,z} we obtain that

∫_{S_{∞}\C_{0}}(λ_{γ,z})^{2} dν_{2} ≤ ∫_{R}(λ_{γ,z})^{2} dA.  (3.2.3)

Now we evaluate the right hand sides of (3.2.2) and (3.2.3) using the formula (3.1.1) for λ_{γ,z}. We find that

∫_{C_{1}}(λ_{γ,z})^{2}dA = \frac{1}{4π} ∫_{0}^{2π} ∫_{0}^{ϕ_{1}} \frac{sin ϕ}{(c − s cos ϕ)^{2}} dϕdθ

= \frac{1}{2s} (\frac{1}{c − s} − \frac{1}{c − s cos ϕ_{1}} )

= \frac{te^{2Δ}}{1 + t(e^{2Δ} − 1)},

(3.2.4)

where in the last step we used that c = cosh Δ, s = sinh Δ and cos ϕ_{1} = 1 − 2t. Likewise, setting x = e^{2Δ}, we have

∫_{R}(λ_{γ,z})^{2}dA = ∫_{0}^{2ϕ} ∫_{ϕ_{0}}^{ϕ_{R}} \frac{sin ϕ}{(c − s cos ϕ)^{2}} dϕdθ)

= \frac{1}{2s}(\frac{1}{c − s cos ϕ_{0}} − \frac{1}{c − s cos ϕ_{R}})

= \frac{u − t}{(1 + u(x − 1))(1 + (2u − t)(x − 1))}.

(3.2.5)

Now adding (3.2.2) and (3.2.3), substituting for the right-hand sides the expressions given by (3.2.4) and (3.2.5), and rewriting the left-hand side of the resulting inequality as 1 − u by virtue of (3.2.1), we obtain

1 − u ≤ \frac{tx}{1 + t(x − 1)} + \frac{u − t}{(1 + u(x − 1))(1 + (2u − t)(x − 1))},

which is equivalent to the conclusion of the lemma.

□

The proof of Theorem 3.1, like that of Theorem 9.1 of [6], reduces to the case of a hyperbolic manifold which admits no non-constant positive super-harmonic functions. This case is contained in the next Proposition. We need two lemmas for the proof.

Lemma 3.3. Let ϵ be a given positive number. Let ξ and η be two loxodromic isometries of ℍ^{3}. Suppose that the group Γ generated by ξ and η is discrete and free on the generators ξ and η, and that every Γ-invariant, positive, superharmonic function on ℍ^{3} is constant. Let z be a point on the common perpendicular to the axes of ξ and η and set α = ∠(ξ · z, z, η · z). If we have

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ.

Then there exist real numbers Δ, t and u with Δ ∈ {dist_{h}(z, ξ · z), dist_{h}(z, η · z)}, u ∈ [E(ϵ), \frac{1}{2} − E(ϵ)] and t ∈ [0, u − I(α, ϵ)], such that

g(e^{2Δ}, t, u) ≥ 1.

Note that the inequalities Δ ≥ 0 and 0 ≤ t ≤ u imply that (e^{2Δ}, t, u) ∈ G, so that the last conclusion of the lemma makes sense if the others hold.

Proof. Let us identify ℍ̂^{3} = ℍ^{3} ∪ S_{∞} conformally with the closed unit ball in ℝ^{3} in such a way that z is the origin. Let P_{ξ} and P_{η} denote the poles of ξ and η relative to z. By definition P_{ξ} and P_{η} are the endpoints of the rays emanating from z and passing through ξ^{−1} · z and η^{−1} · z. Hence we have dist_{s}(P_{ξ}, P_{η}) = ∠(ξ^{−1} · z, z, η^{−1} · z). If we define τ ∈ Isom_{+}(ℍ^{3}) to be the rotation about L through an angle π, we have τξτ = ξ^{−1} and τητ = η^{−1}. Since τ fixes z it follows that

∠(ξ^{−1} · z, z, η^{−1} · z) = ∠(τξτ · z, z, τητ · z) = ∠(ξ · z, z, η · z) = α.

Hence

dist_{s}(P_{ξ}, P_{η}) = α.  (3.3.1)

We set Ψ = {ξ, ξ^{−1}, η, η^{−1}} ⊂ Γ. We also set A = A_{z}. According to [6; Lemma 5.3] and the hypotheses of the lemma there exist four Borel measures (ν_{ξ}, ν_{ξ^{−1}}, ν_{η}, ν_{η^{−1}}) on S_{∞} such that

( 3.3.2) A = ν_{ξ} + ν_{ξ^{−1}} + ν_{η} + ν_{η^{−1}},

( 3.3.3) for each ψ ∈ Ψ we have

∫_{S_{∞}}(λ_{ψ,z})^{2}dν_{ψ^{−1}} = 1 − ν_{ψ}(S_{∞}), and

( 3.3.4) ν_{ξ}(S_{∞}) = ν_{ξ^{−1}}(S_{∞}) and ν_{η}(S_{∞}) = ν_{η^{−1}}(S_{∞}).

Next we apply [6; Lemma 5.5] (after correcting a typographical error by interchanging the numerator and denominator of the fraction in the conclusion). Setting a = ν_{ψ}(S_{∞}) and b = 1 − a, we obtain that

log \frac{1 − ν_{ψ}(S_{∞})}{ν_{ψ}(S_{∞})} ≤ dist_{h}(z, ψ(z)) < log 3 + ϵ

for each ψ ∈ Ψ. We therefore obtain for each ψ ∈ Ψ the inequality

ν_{ψ}(S_{∞}) > \frac{1}{1 + 3e^{ϵ}} = E(ϵ).  (3.3.5)

Conditions (3.3.2) and (3.3.4) above imply that

ν_{ξ^{−1}}(S_{∞}) + ν_{η^{−1}}(S_{∞}) = \frac{1}{2}.  (3.3.6)

From (3.3.5) and (3.3.6) it follows that

ν_{ξ^{−1}}(S_{∞}), ν_{η^{−1}}(S_{∞}) < \frac{1}{2} − E(ϵ).  (3.3.7)

Let us choose two spherical caps C_{ξ}, C_{η}, centered at the P_{ξ} and P_{η}, such that A(C_{ξ}) = ν_{ξ^{−1}}(S_{∞}) and A(C_{η}) = ν_{η^{−1}}(S_{∞}). Since a spherical cap C(P, r) has area 2π(1−cos r), and since A is 1/4π times the area measure, we have C_{ξ} = C(P_{ξ}, r_{ξ}) and C_{η} = C(P_{ξ}, r_{η}), where r_{ξ} = cos^{−1}(1 − 2ν_{ξ^{−1}}(S_{∞})) and r_{η} = cos^{−1}(1 − 2ν_{η^{−1}}(S_{∞})). Since r(ϵ) = cos^{−1}(1 − 2E(ϵ)), it follows from (3.3.5) that

r_{ξ}, r_{η} ≥ r(ϵ).

If we set G = C_{ξ} ∩ C_{η}, then by (3.3.1) and (3.0.1) and the definition of the function ι, we have

A(G) = \frac{1}{4π} ι(α, r_{1}, r_{2}) ≥ \frac{1}{4π} ι(α, r(ϵ), r(ϵ)) = 2I(α, ϵ).  (3.3.8)

By property (3.3.2) of the ν_{ψ} we have

ν_{ξ^{−1}}(G) + ν_{η^{−1}}(G) ≤ A(G).

By symmetry we may assume that

ν_{ξ^{−1}}(G) ≤ \frac{A(G)}{2}.  (3.3.9)

Set t = ν_{ξ^{−1}}(C_{ξ}) and u = ν_{ξ^{−1}}(S_{∞}). By (3.3.5) and (3.3.7) we have

E(ϵ) < u < \frac{1}{2} − E(ϵ).  (3.3.10)

Using (3.3.2) and (3.3.9) and the definition of C_{ξ}, we find

\begin{aligned}t & = ν_{ξ^{−1}}((C_{ξ} \ G) ∪ G) = ν_{ξ^{−1}}(C_{ξ} \ G) + ν_{ξ^{−1}}(G) \\ & ≤ A(C_{ξ}) − A(G) + \frac{A(G)}{2} ≤ u − \frac{A(G)}{2}. \end{aligned}  (3.3.11)

Using (3.3.11) and (3.3.8), we get

t ≤ u − I(α, ϵ).  (3.3.12)

We may now apply Lemma 3.2 with γ = ξ and ν = ν_{ξ^{−1}}. Indeed, hypothesis (i) of Lemma 3.2 follows from (3.3.2), and hypothesis (ii) follows from (3.3.3) and (3.3.4). The above definitions of u, t now agree with those given in the statement of Lemma 3.2, while the cap C_{0} defined in the latter statement is C_{ξ}. It follows from Lemma 3.2 that

g(e^{2Δ}, u, t) ≥ 1,  (3.3.13)

where Δ = dist_{h}(z, ξ · z) = dist_{h}(z, ξ^{−1} · z). The conclusion of the theorem follows from (3.3.10), (3.3.12) and (3.3.13).

□

The second lemma is computational. To maintain the flow of the argument, we delay the proof until Appendix A.

Lemma A.1. Let g_{x}, g_{u}, g_{t} denote the partial derivatives of g with respect to the first, second and third variables respectively. The function g_{t} is positive everywhere on G. The functions g_{x} and g_{u} are positive at every point (x, u, t) ∈ G such that u < 1/2 and t > 2u/3.

Proposition 3.4. Let (β, ϵ) be any point in D. Let ξ and η be two loxodromic isometries of ℍ^{3} such that the group Γ generated by ξ and η is discrete and free on the generators ξ and η. Suppose that every Γ-invariant, positive, superharmonic function on ℍ^{3} is constant. Let z be a point on the common perpendicular to the axes of ξ and η. Suppose that ∠(ξ · z, z, η · z) < β. Then we have

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} ≥ log 3 + ϵ.

Proof. Assume that

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ.

Set α = ∠(ξ · z, z, η · z). According to Lemma 3.3, there exist real numbers Δ, u_{0} and t_{0}, where Δ ∈ \{dist_{h}(z, ξ · z), dist_{h}(z, η · z)\}, u_{0} ∈ [E(ϵ), \frac{1}{2} − E(ϵ)] and t_{0} ∈ [0, u_{0} − I(α, ϵ)], such that g(e^{2Δ}, t_{0}, u_{0}) ≥ 1. We write x_{0} = e^{2Δ}. Let us set t_{1} = u_{0} − I(β, ϵ). Since I is monotonically decreasing as a function of the first variable, we have t_{1} ≥ u_{0} − I(α, ϵ) ≥ t_{0}. We also have t_{1} ≤ u_{0} since I(β, ϵ) is non-negative. It follows that the line segment σ_{1} = {x_{0}} × [t_{0}, t_{1}] × {u_{0}} is contained in G. According to Lemma A.1, the partial derivative g_{t} is positive on G and hence

g(x_{0}, t_{1}, u_{0}) ≥ g(x_{0}, t_{0}, u_{0}) ≥ 1.

Now let us set x_{2} = 9e^{2ϵ}, u_{2} = \frac{1}{2} − E(ϵ) and t_{2} = f(β, ϵ) = \frac{1}{2} − E(ϵ) − I(β, ϵ). Since Δ ∈ {dist_{h}(z, ξ · z), dist_{h}(z, η · z)}, our assumption implies that Δ < log 3 + ϵ and hence that x_{0} < x_{2}. Since u_{0} was taken to be ≤ \frac{1}{2} − E(ϵ), we have u_{2} ≥ u_{0} and t_{2} ≥ t_{1}. Now consider the line segment σ_{2} from (x_{0}, u_{0}, t_{1}) to (x_{2}, u_{2}, t_{2}). Since u_{2} − t_{2} = I(β, ϵ) = u_{0} − t_{1}, we have u − t = I(β, ϵ) for every (x, u, t) ∈ σ_{2}. Since I(β, ϵ) ≥ 0 it follows that σ_{2} ⊂ G. On the other hand, by hypothesis we have (β, ϵ) ∈ D; and applying condition (3) of the definition of D we find that for every (x, u, t) ∈ D we have

u − t = I(β, ϵ) < \frac{1}{3}E(ϵ) ≤ \frac{1}{3}u_{0} ≤ \frac{1}{3}u,

so that t > \frac{2}{3}u. Hence by Lemma A.1, the partial derivatives g_{x}, g_{u} and g_{t} are all positive on σ_{2}. Since x_{2} ≥ x_{0}, u_{2} ≥ u_{0} and t_{2} ≥ t_{1}, it follows that

g(x_{2}, u_{2}, t_{2}) ≥ g(x_{0}, u_{0}, t_{1}) ≥ 1.

However, according to condition (2) of the definition of D, we have

g(x_{2}, u_{2}, t_{2}) = g(9e^{2ϵ}, \frac{1}{2} − E(ϵ), f(β, ϵ)) < 1.

This contradiction completes the proof.

□

The proof of Theorem 3.1 requires one more proposition, the proof of which is deferred to Appendix C.

Proposition C.1. Let ξ and η be two loxodromic isometries of ℍ^{3} without any common fixed point. Denote by L the common perpendicular to the axes A_{ξ} and A_{η} of ξ and η respectively. Let z_{0} be any point of L. Then there exist continuous one-parameter families (ξ_{t})_{0≤t≤1} and (η_{t})_{0≤t≤1} of loxodromic isometries of ℍ^{3} with the following properties:

(i) ξ_{0} = ξ and η_{0} = η;

(ii) for every t the axes of ξ_{t} and η_{t} are perpendicular to L;

(iii) the functions t ↦ dist(z_{0}, ξ_{t} · z_{0}) and t ↦ dist(z_{0}, η_{t} · z_{0}) are monotonically

decreasing on [0, 1]; (iv) the function t ↦ ∠(ξ_{t} · z_{0}, z_{0}, η_{t} · z_{0}) is (weakly) monotonically decreasing on

[0, 1]; and

(v) the isometries ξ_{1} and η_{1} have the same axis.

3.5. Proof of Theorem 3.1. We argue by contradiction. Assume that

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ.

Suppose first that Γ is topologically tame and is not geometrically finite. Then it follows from [5; Theorem 7.2] that ℍ^{3} admits no non-constant positive Γ-invariant superharmonic functions. The assertion of the theorem now follows from Proposition 3.4. Now suppose that Γ is geometrically finite. Let L denote the common perpendicular to the axes A_{ξ} and A_{η} of ξ and η respectively. Let (ξ_{t})_{0≤t≤1} and (η_{t})_{0≤t≤1} be oneparameter families having the properties stated in Proposition C.1. Let V denote the complex affine variety P SL_{2}(ℂ) × P SL_{2}(ℂ) endowed with the classical topology, and consider the path ρ: [0, 1] → V defined by ρ(t) = (ξ_{t}, η_{t}). By property (i) of C.1 and the hypotheses of the theorem, we have that (ξ_{0}, η_{0}) is a point in the Schottky space CC ⊂ V, i.e. the group Γ = ⟨ξ, η⟩ is a geometrically finite Kleinian group which is free of rank 2 and has no parabolics. By property (v) of C.1 we have that ⟨ξ_{1}, η_{1}⟩ is not free of rank 2 and therefore (ξ_{1}, η_{1}) does not lie in CC. By [12], CC is an open subset of V. Set t_{0} = inf{t ∈ [0, 1]|(ξ_{t}, η_{t}) ∉ CC}. It follows that (ξ_{t_{0}}, η_{t_{0}}) is in the frontier of CC. By property (ii) of C.1, the point z is on the common perpendicular to the axes of ξ_{t_{0}} and η_{t_{0}}. By the hypothesis of the theorem and property (iv) of C.1 we have

∠(ξ_{t_{0}} · z, z, η_{t_{0}} · z) < β.

Similarly, our assumption and property (iii) of C.1 imply that

max{dist_{h}(z, ξ_{t_{0}} · z), dist_{h}(z, η_{t_{0}} · z)} < log 3 + ϵ.

By [6; Theorem 8.2] there exists a sequence (ξ_{i}, η_{i}) in the frontier of CC in V which converges to (ξ_{t_{0}}, η_{t_{0}}), and such that for every i the group Γ_{i} =< ξ_{i}, η_{i} > is purely loxodromic and free on the generators ξ_{i} and η_{i}, and ℍ^{3} admits no non-constant positive Γ_{i}-invariant superharmonic functions. For large enough i we have

∠(ξ_{i} · z, z, η_{i} · z) < β  (3.5.1)

and

max{dist_{h}(z, ξ_{i} · z), dist_{h}(z, η_{i} · z)} < log 3 + ϵ.  (3.5.2)

Denote by L_{i} the common perpendicular to the axes of ξ_{i} and η_{i}. Let z_{i} denote the foot of the perpendicular from z to L_{i}. The sequence of lines (L_{i}) converges to the common perpendicular to the axes of ξ_{t_{0}} and η_{t_{0}}, which is L by property (ii) of C.1. Since z ∈ L we have dist_{h}(z, L_{i}) → 0, and hence z_{i} → z. It follows from (3.5.1) and (3.5.2) that

∠(ξ_{i} · z_{i}, z_{i}, η_{i} · z_{i}) < β  (3.5.3)

and

max{dist_{h}(z_{i}, ξ_{i} · z_{i}), dist_{h}(z_{i}, η_{i} · z_{i})} < log 3 + ϵ  (3.5.4)

for large i. Since (β, ϵ) ∈ D and z_{i} ∈ L_{i}, and since ℍ^{3} admits no non-constant positive Γ_{i}-invariant superharmonic functions, it follows from (3.5.3) and Proposition 3.4 that

max{dist_{h}(z_{i}, ξ_{i} · z_{i}), dist_{h}(z_{i}, η_{i} · z_{i}) ≥ log 3 + ϵ

for large i. This contradicts (3.5.4), and the proof of the theorem is thus complete.

□

4. Angles and displacements, II

In this section we complete the proof of Theorem 4.1, of which a special case, Corollary 4.9, was used in the proof of Theorem 1.1. Theorem 4.1 is an extension of the result of the previous section to the case where the point z does not lie on the common perpendicular of the axes of ξ and η. Since the point z is not invariant under the involution that conjugates the generators to their inverses, we have two angles to consider. We define a function ϕ: ℝ^{+} × ℝ^{3} → ℝ by

ϕ(l, t, s, α) = \frac{2 cosh^{2} s − 2 sinh^{2} s cos α − cosh l − cos t}{cosh l − cos t}.

We denote by V the open subset of ℝ^{+} × ℝ^{3} consisting of all points (l, t, s, α) such that ϕ(l, t, s, α) > 1. We define a function ρ: V → ℝ by

ρ(l, t, s, α) = cosh^{−1} ϕ(l, t, s, α).

Recall that the region D ⊂ ℝ^{2} was defined in the discussion preceding Theorem 3.1. The main result of the section is:

Theorem 4.1. Let λ and θ be real numbers with λ > 0 and 0 ≤ θ ≤ π. Let (β, ϵ) be a point of the region D ⊂ ℝ^{2}. Let ξ and η be isometries which generate a subgroup of Isom_{+}(ℍ^{3}) which is discrete, free of rank 2, purely loxodromic and topologically tame. Suppose that ξ^{−1}η has translation length ≥ λ and twist angle ≤ θ. Let z be a point of ℍ^{3} such that

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ

Set α_{1} = ∠(ξ · z, z, η · z) and α_{−1} = ∠(ξ^{−1} · z, z, η^{−1} · z). Then the points (λ, θ, log 3 + ϵ, α_{1}) and (λ, θ, log 3 + ϵ, α_{−1}) are contained in V. If in addition we have (λ, θ, log 3 − ϵ, β) ∈ V, then

ρ(λ, θ, log 3 + ϵ, α_{1}) + ρ(λ, θ, log 3 + ϵ, , α_{−1}) ≥ 2ρ(λ, θ, log 3 − ϵ, β).

The proof of Theorem 4.1 requires several lemmas. We begin with two geometric lemmas. The first of these was used in the proof of [6; Prop. 5.2] but was inadvertently omitted from the published paper. The reference to “Proposition 1.15” in the latter paper should have been a reference to the lemma below.

Lemma 4.2. Let γ be a loxodromic isometry of ℍ^{3} with axis A_{γ}. Let L be a line which meets A_{γ} orthogonally at a point w_{0}. Suppose that z is any point of ℍ^{3} and that z_{0} ∈ L is the orthogonal projection of z to the line L. Then

dist_{h}(z, γ · z) ≥ dist_{h}(z_{0}, γ · z_{0}).

Proof. Let r: ℍ^{3} → L denote the map which sends each point of ℍ^{3} to it orthogonal projection in the line L, i.e., to the nearest point of L. It follows from [11; Lemma 1.3.4] that r is a distance decreasing retraction from ℍ^{3} to L. It follows from Lemma C.2 that the displacement of a point under the loxodromic isometry γ is a monotonically increasing function of its distance from A_{γ}. Thus it suffices to show that the distance from z_{0} to A_{γ} is not larger than the distance from z to A_{γ}. Let w denote the point of A_{γ} which is nearest to z. Then we must show that dist_{h}(z, w) ≥ dist_{h}(z_{0}, w_{0}). But we have z_{0} = r(z) and w_{0} = r(w); this completes the proof since r is distance decreasing.

□

Lemma 4.3. Let L be a line in ℍ^{3} and let R: ℍ^{3} → L be the function that assigns to each point of ℍ^{3} its distance from the line L. Then R is a convex function.

Proof. Let r: ℍ^{3} → L be the nearest point retraction. Let z and w be two points of ℍ^{3} and let m be the midpoint of the segment from z to w and let m′ ∈ L denote the midpoint of the segment from r(z) to r(w). We must show that R(m) ≤ \frac{1}{2}(R(z) + R(w)). It follows from [10; Lemme 2] that

dist_{h}(m′, m)) ≤ \frac{1}{2}(dist_{h}(r(z), z) + dist_{h}(r(w), w)) = \frac{1}{2}(R(z) + R(w)).

(The quoted result is stated in [10] only for ℍ^{2}, but the proof works equally well in three dimensions.) But R(m) = dist_{h}(r(m), m)) ≤ dist_{h}(m′, m) by the definition of r. The lemma follows.

□

Lemma 4.4. Let l, t, s, λ, θ, s′, α_{1}, α_{−1} and α_{0} be real numbers with l ≥ λ > 0, s > s′ > 0 and 0 ≤ t ≤ θ ≤ π. Suppose that (l, t, s, α_{1}), (l, t, s, α_{−1}) and (l, t, s′, α_{0}) lie in V and that

ρ(l, t, s, α_{1}) + ρ(l, t, s, α_{−1}) ≥ 2ρ(l, t, s′, α_{0}).

Then the points (λ, θ, s, α_{1}), (λ, θ, s, α_{−1}), and (λ, θ, s′, α_{0}) are contained in V and we have

ρ(λ, θ, s, α_{1}) + ρ(λ, θ, s, α_{−1}) ≥ 2ρ(λ, θ, s′, α_{0}).

Proof. First note that the function ϕ is monotone decreasing in the variable l for l ≥ 0, and monotone increasing in the variable t for 0 ≤ t ≤ π. Since the set V is the region in which ϕ > 1, and since λ ≤ l and θ ≥ t, it follows immediately that the points (λ, θ, s, α_{1}), (λ, θ, s, α_{−1}), and (λ, θ, s′, α_{0}) lie in V. For the rest of the argument we introduce functions ρ_{i}(x, y) for i ∈ {−1, 0, 1} defined as follows:

ρ_{1}(x, y) = cosh^{−1} (\frac{2 cosh^{2} s − 2 sinh^{2} s cos α_{1} − cosh x − cos y}{cosh x − cos y} ) ,
ρ_{−1}(x, y) = cosh^{−1} (\frac{2 cosh^{2} s − 2 sinh^{2} s cos α_{−1} − cosh x − cos y}{cosh x − cos y} ), and
ρ_{0}(x, y) = cosh^{−1} (\frac{2 cosh^{2} s′ − 2 sinh^{2} s′ cos α_{0} − cosh x − cos y}{cosh x − cos y} ) .

By hypothesis we have that the inequality

ρ_{1}(x, y) + ρ_{−1}(x, y) − 2ρ_{0}(x, y) ≥ 0

holds with x = l, y = t. We must show that the same inequality holds with x = λ ≤ l, and y = θ ≥ t. We will consider a linear path (x(s), y(s)) for 0 ≤ s ≤ 1 with (x(0), y(0)) = (l, t) and (x(1), y(1)) = (λ, θ). We claim that the quantity

w(s) = ρ_{1}(x(s), y(s)) + ρ_{−1}(x(s), y(s)) − 2ρ_{0}(x(s), y(s))

is non-negative on the entire path. To establish the claim we will show that w′(s) > 0 whenever w(s) = 0. Since w(0) ≥ 0, this implies w′(0) > 0. Thus if s_{0} is the first positive number for which w(s_{0}) = 0 then we must have w′(s_{0}) ≤ 0. Since this is impossible we must have w(s) > 0 for all s ∈ (0, 1]. Since x(s) decreases monotonically while y(s) increases monotonically we must show that the condition ρ_{1}(x, y) + ρ_{−1}(x, y) − 2ρ_{0}(x, y) = 0 implies the inequalities

\frac{∂ρ_{1}}{∂u} + \frac{∂ρ_{−1}}{∂u} − \frac{∂ρ_{0}}{∂u} < 0

and

\frac{∂ρ_{1}}{∂v} + \frac{∂ρ_{−1}}{∂v} − \frac{∂ρ_{0}}{∂v} > 0.

Computing partial derivatives with respect to x and y we find

\frac{∂ρ_{i}}{∂x} = \frac{− sinh x}{cosh x − cos y} (\frac{cosh ρ_{i} + 1}{sinh ρ_{i}})

and

\frac{∂ρ_{i}}{∂y} = \frac{− sin y}{cosh x − cos y} (\frac{cosh ρ_{i} − 1}{sinh ρ_{i}}) .

If we set

p(z) = \frac{cosh z + 1}{sinh z}  and  q(z) = \frac{cosh z − 1}{sinh z}

then to complete the proof of the claim we must prove that ρ_{1} + ρ_{−1} = 2ρ_{0} implies p(ρ_{1}) + p(ρ_{−1}) > 2p(ρ_{0}) and q(ρ_{1}) + q(ρ_{−1}) < 2q(ρ_{0}). This is shown by computing second derivatives of p and q:

p′′(z) = \frac{(cosh z + 1)^{2}}{sinh^{3} z} > 0

and

q′′(z) = \frac{−(cosh z − 1)^{2}}{sinh^{3} z} < 0.

□

Lemma 4.5. Let s, s_{0}, λ and θ be real numbers with s, s_{0}, λ > 0 and 0 ≤ θ ≤ π. Let ξ and η be loxodromic isometries of ℍ^{3} having no common fixed point on the sphere at infinity. Suppose that ξ^{−1}η is also loxodromic, and that it has translation length ≥ λ and twist angle ≤ θ. Let L denote the common perpendicular to the axes of ξ and η. Let z be any point of ℍ^{3}, and let z_{0} denote the foot of the perpendicular from z to L. Suppose that dist_{h}(z, ξ · z) < s, dist_{h}(z, η · z) < s and dist_{h}(z_{0}, ξ · z_{0}) > s_{0}. Set α_{1} = ∠(ξ · z, z, η · z), α_{−1} = ∠(ξ^{−1} · z, z, η^{−1} · z), and α_{0} = ∠(ξ · z_{0}, z_{0}, η · z_{0}). Suppose that cos α_{i} < (tanh s_{0})/(tanh s) for i = 1, 0, −1. Then we have (λ, θ, s, α_{1}), (λ, θ, s, α_{−1}) ∈ V. If in addition we have (λ, θ, s_{0}, α_{0}) ∈ V, then

ρ(λ, θ, s, α_{1}) + ρ(λ, θ, s, α_{−1}) ≥ 2ρ(λ, θ, s_{0}, α_{0}).

Proof. According to Lemma 4.2, we have

s_{0} ≤ dist_{h}(z_{0}, ξ · z_{0}) ≤ dist_{h}(z, ξ · z) ≤ s  (4.5.1)

and

s_{0} ≤ dist_{h}(z_{0}, η · z_{0}) ≤ dist_{h}(z, η · z) ≤ s.  (4.5.2)

In particular s_{0} ≤ s. For i = 1, 0, −1 we consider the function

f_{i}(x, y) = cosh x cosh y − sinh x sinh y cos α_{i}.

We have

\frac{∂}{∂x}f_{i}(x, y) = cosh x cosh y (tanh x − tanh y cos α_{i})

and

\frac{∂}{∂y} f_{i}(x, y) = cosh x cosh y (tanh y − tanh x cos α_{i}).

In view of the hypothesis, it follows that f_{i}(x, y) is monotone both in x and in y for x, y ∈ [s_{0}, s] and for i = 1, 0, −1. In particular we have f_{i}(s_{0}, s_{0}) ≤ f_{i}(x, y) ≤ f_{i}(s, s) for x, y ∈ [s_{0}, s]; that is,

1 + (1 − cos α_{i}) sinh^{2} s_{0} ≤ cosh x cosh y − sinh x sinh y cos α_{i}
≤ 1 + (1 − cos α_{i}) sinh^{2} s

(4.5.3)

for i = 1, 0, −1 and for all x, y ∈ [s_{0}, s]. Now define τ ∈ Isom_{+}(ℍ^{3}) to be the rotation about L through an angle π. We have τξτ = ξ^{−1} and τητ = η^{−1}. Hence

α_{−1} = ∠(τξτ · z, z, τητ · z) = ∠(ξτ · z, τ · z, ητ · z).

If we set z_{1} = z and z_{−1} = τ · z, it follows from the equation above and the definitions of α_{0} and α_{1} that

∠(ξ · z_{i}, z_{i}, η · z_{i}) = α_{i}  (4.5.4)

for i = 1, 0, −1. We also have dist_{h}(z_{−1}, ξ · z_{−1}) = dist_{h}(τ · z, ξτ · z) = dist_{h}(z, τξτ · z) = dist_{h}(z, ξ^{−1} · z) = dist_{h}(z_{1}, ξ · z_{1}). Combining this with (4.5.1), and setting X_{i} = dist_{h}(z_{i}, ξ · z_{i}), we get

s_{0} ≤ X_{i} ≤ s

for i = 1, 0, −1. Similarly, setting Y_{i} = dist_{h}(z_{i}, η · z_{i}) and using (4.5.2), we find that

s_{0} ≤ Y_{i} ≤ s

for i = 1, 0, −1. We set D_{i} = dist_{h}(ξ · z_{i}, η · z_{i}) for i = 1, 0, −1, and apply the first hyperbolic law of cosines to the the hyperbolic triangle with vertices z_{i}, ξ · z_{i} and η · z_{i}; by (4.5.4), the angle at the vertex z_{i} is α_{i}. This gives

cosh D_{i} = cosh X_{i} cosh Y_{i} − sinh X_{i} sinh Y_{i} cos α_{i}.  (4.5.5)

Since X_{i}, Y_{i} ∈ [s_{0}, s], we can combine (4.5.3) and (4.5.5) to conclude that

1 + (1 − cos α_{i}) sinh^{2} s_{0} ≤ cosh D_{i} ≤ 1 + (1 − cos α_{i}) sinh^{2} s  (4.5.6)

for i = 1, 0, −1. It follows from the definition of D_{i} that

D_{i} = dist_{h}(z_{i}, ξ^{−1}η · z_{i}).

Let l and t denote the translation length and twist angle of ξ^{−1}η. By hypothesis we have 0 ≤ λ ≤ l and 0 ≤ t ≤ θ ≤ π. If we define R_{i} to be the orthogonal distance from z_{i} to the axis of ξ^{−1}η, then it follows from the formula for the displacement of point under a loxodromic isometry (see Lemma C.2) that

sinh^{2} R_{i} = \frac{cosh D_{i} − cosh l}{cosh l − cos t} .

Combining this with (4.5.6) we obtain

\frac{1 + (1 − cos α_{i}) sinh^{2} s_{0} − cosh l}{cosh l − cos t} ≤ sinh^{2} R_{i} ≤ \frac{1 + (1 − cos α_{i}) sinh^{2} s − cosh l}{cosh l − cos t} .

Using the identity cosh 2R = 1 + 2 sinh^{2} R and the definition of ϕ, we obtain

ϕ(l, t, s, α_{i}) ≤ cosh 2R_{i} ≤ ϕ(l, t, s, α_{i}).  (4.5.7)

Since ϕ is monotone decreasing with respect to the first variable and monotone increasing with respect to the second variable, we have

ϕ(λ, θ, s, α_{i}) ≥ ϕ(l, t, s, α_{i}) ≥ cosh 2R_{i} ≥ 1.

It follows that (l, t, s, α_{i}) and (λ, θ, s, α_{i}) lie in V for i = 1, 0, −1. This includes the first assertion of the lemma. Since (l, t, s, α_{i}) ∈ V, we may conclude from (4.5.7) and the monotonicity of the hyperbolic cosine that

ρ(l, t, s, α_{i}) ≥ 2R_{i}  (4.5.8)

for i = 1, 0, −1. It also follows from (4.5.7) that cosh 2R_{0} = ϕ(l, t, s′, α_{0}) for some s′ ∈ [s_{0}, s]. This implies that (l, t, s′, α_{0}) ∈ V and that

ρ(l, t, s′, α_{0}) = 2R_{0}.  (4.5.9)

Now let R : ℍ^{3} → ℝ denote the function that assigns to each point of ℍ^{3} its minimum hyperbolic distance from the axis of ξ^{−1}η; thus R(z_{i}) = R_{i} for i = 1, 0, −1. According to Lemma 4.3, R is a convex function on ℍ^{3}. Since z_{0} is the midpoint of the segment joining z_{1} to z_{−1}, we have

R_{1} + R_{−1} ≥ 2R_{0}.  (4.5.10)

Combining (4.5.8), (4.5.9), and (4.5.10), we find that

ρ(l, t, s, α_{1}) + ρ(l, t, s, α_{−1}) ≥ 2R_{1} + 2R_{−1} ≥ 4R_{0} = 2ρ(l, t, s′, α_{0}).

Since we have s′ ≤ s, λ ≤ l and t ≤ θ ≤ π, it follows from Lemma 4.4 that (λ, θ, s, α_{1}), (λ, θ, s, α_{−1}), and (λ, θ, s′, α_{0}) lie in V and that

ρ(λ, θ, s, α_{1}) + ρ(λ, θ, s, α_{−1}) ≥ 2ρ(λ, θ, s′, α_{0}).  (4.5.11)

Let us now assume that (λ, θ, s_{0}, α_{0}) ∈ V. Then since s_{0} ≤ s′, and since ϕ is monotonically increasing with respect to the third variable and the hyperbolic cosine is monotonically increasing on (0, ∞), we have ρ(λ, θ, s_{0}, α_{0}) ≤ ρ(λ, θ, s′, α_{0}). Combining this with (4.5.11) we conclude that

ρ(λ, θ, s, α_{1}) + ρ(λ, θ, s, α_{−1}) ≥ 2ρ(λ, θ, s_{0}, α_{0}).
□

Lemma 4.6. Let ϵ be a given positive number. Let ξ and η be two loxodromic isometries of ℍ^{3}. Suppose that the group Γ generated by ξ and η is discrete, free on the generators ξ and η, and topologically tame. Let z be any point of ℍ^{3}. Suppose that

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ.

Then we have

min{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} > log 3 − ϵ.

Proof. In [1; Theorem 6.1(a)] it was shown that under the hypotheses of the lemma we have

\frac{1}{1 + e^{dist_{h}(z,ξ·z)}} + \frac{1}{1 + e^{dist_{h}(z,η·z)}} ≤ \frac{1}{2}

for every z ∈ ℍ^{3}. If the conclusion of the lemma is false then one of the two quantities dist_{h}(z, ξ · z) and dist_{h}(z, η · z) is less than log 3 − ϵ while the other is greater than log 3 + ϵ. Hence

\frac{1}{1 + 3e^{ϵ}} + \frac{1}{1 + 3e^{−ϵ}} ≤ \frac{1}{2}.  (4.6.1)

But the left-hand side of (4.6.1) can be written in the form

\frac{1 + 3 cosh ϵ}{5 + 3 cosh ϵ},

which is monotonically increasing function of ϵ and takes the value \frac{1}{2} at 0. This contradicts (4.6.1) since the given value of ϵ is strictly positive.

□

Lemma 4.7. Let ϵ be a positive number less than log 3. Let ξ and η be isometries which generate a subgroup of Isom_{+}(ℍ^{3}) which is discrete, free of rank 2, purely loxodromic and topologically tame. Let z be a point of ℍ^{3} such that

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ.

Then we have

cos ∠(ξ · z, z, η · z) < \frac{cosh^{2}(log 3 + ϵ) − cosh(log 3 − ϵ)}{sinh^{2}(log 3 − ϵ)}.

Proof. Since max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ it follows from Lemma 4.6 that dist_{h}(z, η · z) > log 3 − ϵ)and dist_{h}(z, ξ · z) > log 3 − ϵ). On the other hand, since ξ and ξ^{−1}η generate the same group as ξ and η, we may apply Lemma 4.6 with ξ^{−1}η in place of η to conclude that dist_{h}(z, ξη^{−1} · z) > log 3 − ϵ. Now consider the triangle with vertices z, ξ · z and η · z. The sides adjacent to the vertex z have lengths X = dist_{h}(z, ξ · z) and Y = dist_{h}(z, η · z). The third side has length D = dist_{h}(ξ · z, η · z) = dist_{h}(z, ξ^{−1}η · z). Since X and Y lie in (log 3 − ϵ, log 3 + ϵ) and since D > log 3 − ϵ, the first hyperbolic law of cosines gives

cos α = \frac{cosh X cosh Y − cosh D}{sinh X sinh Y} ≤ \frac{cosh^{2}(log 3 + ϵ) − cosh(log 3 − ϵ)}{sinh^{2}(log 3 − ϵ)}.
□

4.8. Proof of Theorem 4.1. Let L denote the common perpendicular to the axes of ξ and η. Let z_{0} denote the foot of the perpendicular from z to L. Let us set α_{0} = ∠(ξ·z_{0}, z_{0}, η·z_{0}). We must have α_{0} ≥ β; for if α_{0} were < β, then since (β, ϵ) ∈ D, it would follow from Theorem 3.1 that max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} ≥ log 3 + ϵ, a contradiction to the hypothesis of Theorem 4.1. We shall apply Lemma 4.5, taking s = log 3 + ϵ and s_{0} = log 3 − ϵ. By hypothesis we have dist_{h}(z, ξ · z) < log 3 + ϵ. According to Lemma 4.2 we have

dist_{h}(z_{0}, ξ · z_{0}) ≤ dist_{h}(z, ξ · z) < log 3 + ϵ.

Hence by Lemma 4.6 we have dist_{h}(z_{0}, η · z_{0}) > log 3 − ϵ. In order to apply Lemma 4.5 we must still check that for i = 1, 0, −1 we have cos α_{i} < (tanh(log 3 − ϵ))/(tanh(log 3 + ϵ). To this end we observe that by Lemma 4.7 we have α_{1} = cos ∠(ξ · z, z, η · z) < a, where

a = \frac{cosh^{2}(log 3 + ϵ) − cosh(log 3 − ϵ)}{sinh^{2}(log 3 − ϵ)} .

Applying Lemma 4.7 with ξ^{−1} and η^{−1} in place of ξ and η gives α_{−1} < a; and the same lemma, with z_{0} in place of z, gives α_{0} < a. Now recall that from the definition of D, we have ϵ < ϵ_{∞}. Hence a < a_{∞}, where

a_{∞} = \frac{cosh^{2}(log 3 + ϵ_{∞}) − cosh(log 3 − ϵ_{∞})}{sinh^{2}(log 3 − ϵ_{∞})} = 0.80060 ….

Thus we have α_{i} < a for i = 1, 0, −1. But

\frac{(tanh(log 3 − ϵ))}{(tanh(log 3 + ϵ)} > \frac{(tanh(log 3 − ϵ_{∞}))}{(tanh(log 3 + ϵ_{∞})} = 0.95591 … > a.

Hence we indeed have cos α_{i} < (tanh(log 3 − ϵ))/(tanh(log 3 + ϵ) for i = 1, 0, −1. It now follows from Lemma 4.5 that (λ, θ, log 3 + ϵ, α_{1}) and ρ(λ, θ, log 3 + ϵ, α_{−1}) lie in V. Now suppose that (λ, θ, log 3 − ϵ, β) ∈ V. Since α_{0} ≥ β, and since the function ϕ is monotone increasing in the fourth variable, it follows that (λ, θ, log 3 − ϵ, α_{0}) ∈ V and that

ρ(λ, θ, log 3 − ϵ, α_{0}) ≥ ρ(λ, θ, log 3 − ϵ, β).

Since (λ, θ, log 3 − ϵ, α_{0}) ∈ V, Lemma 4.5 gives

ρ(λ, θ, log 3 + ϵ, α_{1}) + ρ(λ, θ, log 3 + ϵ, α_{−1}) ≥ 2ρ(λ, θ, log 3 − ϵ, α_{0}).

Combining the two inequalities above, we conclude that

ρ(λ, θ, log 3 + ϵ, α_{1}) + ρ(λ, θ, log 3 + ϵ, α_{−1}) ≥ 2ρ(λ, θ, log 3 − ϵ, β),

and the theorem is proved.

□

We now specialize Theorem 4.1 to the case which was needed to prove Theorem 1.1. Recall, from Section 1, that ϵ_{0} = 0.0065, and that

{α ∈ [0, π] | ρ(λ_{0}, π, log 3 + ϵ_{0}, α) ≥ 1} = [α_{−∞}, π].

By comparing the definition of σ in Section 1 with the definition of ρ we find that

σ(α) = ρ(λ_{0}, π, log 3 + ϵ_{0}, α)

for α ∈ (α_{−∞}, π]. Also recall from Section 1 that the constant

K = 1.30822

is defined so that ρ(λ_{0}, π, log 3 − ϵ_{0}, β_{0}) = cosh^{−1} ϕ(λ_{0}, π, log 3 − ϵ_{0}, β_{0}) > K. Moreover, the point (β_{0}, ϵ_{0}) lies in the region D since we have:

(1) f(β_{0}, ϵ_{0}) = 0.23139 … > 0;
(2) g(9e^{2ϵ_{0}}, \frac{1}{2} − E(ϵ_{0}), f(β_{0}, ϵ_{0})) = 0.98623 … < 1; and
(3) I(β_{0}, ϵ_{0}) = 0.03964 … < 0.08292 … = \frac{1}{3}E(ϵ_{0}).

Here we have computed I(β_{0}, ϵ_{0}) = \frac{1}{8π} ι(β_{0}, ϵ_{0}) using Proposition B.1. Thus by specializing Theorem 4.1 we obtain:

Corollary 4.9. Let ξ and η be isometries which generate a subgroup of Isom_{+}(ℍ^{3}) which is discrete, free of rank 2, purely loxodromic and topologically tame. Suppose that ξ^{−1}η has translation length ≥ λ_{0}. Let z be a point of ℍ^{3} such that

max{dist_{h}(z, ξ · z), dist_{h}(z, η · z)} < log 3 + ϵ_{0}.

Set α_{1} = ∠(ξ ·z, z, η ·z) and α_{−1} = ∠(ξ^{−1} ·z, z, η^{−1} ·z). Then α_{1} and α_{−1} lie in the interval [α_{−∞}, π] and

σ(α_{1}) + σ(α_{−1}) > 2K.
□

We conclude this section by proving another result that was used in the proof of Theorem 1.1. It is similar in flavor to the above results, although independent of them. Recall that we define the function ω(λ, Δ) for Δ > λ > 0 by

ω(λ, Δ) = cos^{−1}(1 − \frac{2(cosh λ − 1)}{cosh Δ − 1}).

Proposition 4.10. Let λ and Δ be positive real numbers. Let ξ be a loxodromic isometry of ℍ^{3} with translation length ≥ λ, and let z be a point of ℍ^{3} such that dist_{h}(z, ξ · z) < Δ. Then we have ∠(ξ^{−1} · z, z, ξ · z) > ω(λ, Δ).

Proof. We let l and θ denote the translation length and twist angle of ξ, so that ξ^{2} has length 2l and twist angle 2θ. We set D = dist_{h}(z, ξ · z) and D′ = dist_{h}(z, ξ^{2} · z), and we denote by R the distance from z to the axis of ξ (which is also the axis of ξ^{2}). In Appendix C we derive a formula for the displacement of a point of ℍ^{3} under the action of a loxodromic isometry. According to Lemma C.2, we have

sinh^{2} R = \frac{cosh D − cosh l}{cosh l − cos θ}

and likewise

sinh^{2} R = \frac{cosh D′ − cosh 2l}{cosh 2l − cos 2θ} .

Hence

cosh D′ = \frac{cosh D − cosh l}{cosh l − cos θ} (cosh 2l − cos 2θ) + cosh 2l.  (4.10.1)

Now consider the triangle with vertices z, ξ · z and ξ^{−1} · z. The sides adjoining the vertex z have length D, and the third side has length dist_{h}(ξ^{−1} · z, ξ · z) = dist_{h}(z, ξ^{2} · z) = D′. Setting w = ∠(ξ^{−1} · z, z, ξ · z) and applying the first hyperbolic law of cosines we find that

cosh D′ = cosh^{2} D − sinh^{2} D cos w = 1 + sinh^{2} D(1 − cos w).

Combining this with (4.10.1) we obtain

1 − cos w = \frac{1}{sinh^{2} D} (\frac{cosh D − cosh l}{cosh l − cos θ} (cosh 2l − cos 2θ) + cosh 2l − 1)
= \frac{2}{sinh^{2} D} ((cosh l + cos θ)(cosh D − cosh l) + cosh^{2} l − 1)
≥ \frac{2}{sinh^{2} D} ((cosh l − 1)(cosh D − cosh l) + cosh^{2} l − 1)
= \frac{2}{sinh^{2} D} (cosh D + 1)(cosh l − 1) = \frac{2(cosh l − 1)}{cosh D − 1}.

Since 0 < D < Δ and l ≥ λ, it follows that

1 − cos w > \frac{2(cosh λ − 1)}{cosh Δ − 1},

which is equivalent to the conclusion of the proposition.

□

5. Geography

This section is devoted to the proof of the proposition about configurations of eight points on the sphere which was used in the proof of Theorem 1.1. Recall that the constants δ_{0} = 0.71497 π, K = 1.30822 and α_{−∞} = 0.80060 … were defined in Section 1. Recall also that the function σ has domain [α_{−∞}, π]. By inspection of the definition of σ in Section 1 one sees that for every x > α_{−∞} we have

σ(x) = cosh^{−1}(A − B cos(x))  (5.0.2)

where the constants A and B are defined by

A = \frac{2 cosh^{2}(log 3 + ϵ_{0}) − cosh λ_{0} + 1}{cosh λ_{0} + 1} = 1.98717 …
B = \frac{2 sinh^{2}(log 3 + ϵ_{0})}{cosh λ_{0} + 1} = 1.41781 …

The main result of this section, which was quoted in Section 1, is:

Proposition 5.1. Suppose that we are given an indexed family

(P_{(i,u)})_{(i,u)∈{0,1,2,3}×{−1,1}}

of points in S^{2}. Assume that for any two indices (i, u) and (j, v) in {0, 1, 2, 3} × {−1, 1} with i ≠ j, we have dist_{s}(P_{(i,u)}, P_{(j,v)}) > α_{−∞}. Then either

(i) there is an element i of {0, 1, 2, 3} such that dist_{s}(P_{(i,1)}, P_{(i,−1)}) ≤ δ_{0}, or (ii) there exist indices (i, u) and (j, v) in {0, 1, 2, 3} × {−1, 1}, with i ≠ j, such that

σ(dist_{s}(P_{(i,u)}, P_{(j,v)})) + σ(dist_{s}(P_{(i,−u)}, P_{(j,−v)})) ≤ 2K.

We will need a formula for the distance between two points on the unit sphere. We use the conventions established in the introduction. If P and P′ are points of S^{2}, the spherical distance dist_{s}(P, P′) is equal to the angle between the (unit) position vectors of P and P′ in ℝ^{3}. If P, P′ ∉ {N, S}, so that Θ(P, P′) is defined, we find by writing down the inner product and using the identity cos(θ − θ′) = cos Θ(P, P′) that

cos dist_{s}(P, P′) = cos λ(P) cos λ(P′) cos Θ(P, P′) + sin λ(P) sin λ(P′).

Proposition 5.1 is deduced by a combinatorial argument from three estimates which are stated below as Lemmas 5.2, 5.7 and 5.8.

Lemma 5.2. Let P, P′, Q and Q′ be points of S^{2} . Suppose that

(a) Q and Q′ lie on a common meridian and 0 < λ(Q) = −λ(Q′),

(b) dist_{s}(P, P′) > δ_{0} and dist_{s}(Q, Q′) = 2λ(Q) > δ_{0},

(c) dist_{s}(P, Q), dist_{s}(P′, Q′), dist_{s}(P′, Q)) and dist_{s}(P, Q′) are all ≥ α_{−∞}, (c′) σ(dist_{s}(P, Q)) + σ(dist_{s}(P′, Q′)) > 2K, and (c′′) σ(dist_{s}(P′, Q)) + σ(dist_{s}(P, Q′)) > 2K.

Then |λ(P)| and |λ(P′)| are less than π/6.

Lemma 5.2 will follow from the next three lemmas.

Lemma 5.3. Let a and b be positive real numbers. If |a − b| < 1 then the function cosh^{−1}(a−b cos x) has negative second derivative at each point in the interior of its domain.

Proof. We may assume that a + b > 1 since otherwise the domain has empty interior. One checks that the second derivative is given by

\frac{−q(cos x)}{((a − b cos x)^{2} − 1)^{3/2}}

where q is the quadratic polynomial function given by

q(t) = ab^{2}t^{2} − (a^{2}b + b^{3} − b)t + ab^{2}.

Thus it suffices to check that the polynomial q is everywhere positive. Clearly q(0) > 0, and the discriminant of q is

b^{2}(1 − (a + b)^{2})(1 − (a − b)^{2}),

which is negative since a + b > 1 and |a − b| < 1.

□

We let A′ denote the function given by A′(x, y) = A − 2B sin x sin y.

Lemma 5.4. Let P, Q, and Q′ be points on S^{2}. Assume that Q and Q′ lie on the same meridian and that 0 ≤ λ(Q) = −λ(Q′). Assume also that dist_{s}(P, Q) > α_{−∞}. Then

σ(dist_{s}(P, Q)) = cosh^{−1}(A′(λ(P), λ(Q)) − B cos dist_{s}(P, Q′)).

Proof. This follows directly from (5.0.2) and the identity

cos(dist_{s}(P, Q)) = cos(dist_{s}(P, Q′)) + 2 sin λ(P) sin λ(Q).

The identity is an immediate consequence of the spherical distance formula, given that λ(Q′) = −λ(Q).

□

Lemma 5.5. Let P, Q, and Q′ be points on S^{2}. Assume that Q and Q′ lie on the same meridian and that δ_{0}/2 ≤ λ(Q) = −λ(Q′). Assume also that dist_{s}(P, Q) ≥ α_{−∞} and that λ(P) ≥ π/6. Then 2π − δ_{0} − dist_{s}(P, Q′) ≥ α_{−∞} and

σ(dist_{s}(P, Q)) + σ(2π − δ_{0} − dist_{s}(P, Q′)) < 2K.

Proof. Our hypothesis on the latitudes of Q and Q′ implies that dist_{s}(Q, Q′) > δ_{0}. Since the perimeter of a spherical triangle is at most 2π we have

2π − δ_{0} − dist_{s}(P, Q′) ≥ 2π − dist_{s}(Q, Q′) − dist_{s}(P, Q′) ≥ dist_{s}(P, Q) ≥ α_{−∞}.

In particular, the left hand side of the inequality in the conclusion is defined. It follows from the spherical distance formula that if the configuration of the three points P, Q and Q′ is modified by moving these points along their meridians toward the equator while preserving the condition λ(Q) = −λ(Q′), then dist_{s}(P, Q) increases while dist_{s}(P, Q′) decreases. Hence it suffices to consider the case where λ(P) = π/6 and λ(Q) = δ_{0}/2. Then the expression σ(dist_{s}(P, Q)) + σ(2π − δ_{0} − dist_{s}(P, Q′)) is defined and, by (5.0.2) and Lemma 5.4, is equal to

cosh^{−1}(A′(λ(P), λ(Q)) − B cos dist_{s}(P, Q′)) + cosh^{−1}(A − B cos(2π − δ_{0} − dist_{s}(P, Q′))).

We will use A′ to denote the quantity A′(π/6, δ_{0}/2) = 0.70911 . . .. It follows from Lemma 5.4 that the left hand side of the inequality in the conclusion of the lemma is given by f(dist_{s}(P, Q′)), where

f(x) = cosh^{−1}(A′ − B cos(x)) + cosh^{−1}(A − B cos(2π − δ_{0} − x)).

Thus it suffices to show that f(x) < 2K for all x in the intersection of the domain of f with the interval [0, π]. (This intersection is easily seen to be an interval, which we will denote J.) Note that f′′(x) < 0 for all x in J, since Lemma 5.3 implies that each of the summands of f has negative second derivative on the interior of its domain. To complete the proof we give a numerical estimate of the maximum value of f on J. To make the estimate we will exhibit two points x_{1} and x_{2} with x_{1} < x_{2} such that f′(x_{1}) > 0, and f′(x_{2}) < 0. By the concavity of f we then know that f(x) < f(x_{1}) + f′(x_{1})(x_{2} − x_{1}) for all x in J. We set

\begin{aligned}x_{1} & = 2.3; \\ x_{2} & = 2.4. \end{aligned}

We then have

\begin{aligned}f(x_{1}) & = 2.52535 …; \\ f′(x_{1}) & = 0.09792 …; \\ f′(x_{2}) & = −0.11091 …; \\ f(x_{1}) + f′(x_{1})(x_{2} − x_{1}) & = 2.53514 … ≤ 2K = 2.61644. \end{aligned}

This completes the proof of the lemma.

□

5.6. Proof of Lemma 5.2. Because of the symmetry of the hypotheses, it suffices to show that |λ(P)| < π/6. Assume to the contrary that |λ(P)| ≥ π/6. Then in view of hypotheses (a), (b) and (c), we may apply Lemma 5.5 to deduce that 2π − δ_{0} − dist_{s}(P, Q′) ≥ α_{−∞} and that

σ(dist_{s}(P, Q)) + σ(2π − δ_{0} − dist_{s}(P, Q′)) < 2K.

On the other hand, using hypothesis (b) and the fact that a spherical triangle has perimeter at most 2π, we find that

2π − δ_{0} − dist_{s}(P, Q′) > 2π − dist(P, P′) − dist_{s}(P, Q′) ≥ dist(P′, Q′).

Hence

σ(dist_{s}(P, Q)) + σ(dist_{s}(P′, Q′)) < 2K.

But this contradicts hypothesis (c′).

□

We need two more lemmas for the proof of Proposition 5.1.

lemma 5.7. Let 0 < Δ < π and 0 < Λ < π/2 be constants. Suppose that R and R′ are points of S^{2} such that |λ(R)| and |λ(R′)| are less than Λ, and dist_{s}(R, R′) ≥ Δ. Then

cos Θ(R, R′) < \frac{cos(Δ) + sin^{2} Λ}{cos^{2} Λ}.

In particular, if |λ(R)| and |λ(R′)| are less than π/6 and dist_{s}(R, R′) > δ_{0} then Θ(R, R′) > 2π/3.

Proof. We have

cos Θ(R, R′) = \frac{cos dist_{s}(R, R′) − sin λ(R) sin λ(R′)}{cos λ(R) cos λ(R′)} ≤ \frac{cos Δ − sin λ(R) sin λ(R′)}{cos λ(R) cos λ(R′)}.

We consider the function

f(x, y) = \frac{cos Δ − sin x sin y}{cos x cos y} = cos Δ sec x sec y − tan x tan y.

One checks by elementary calculus that f is differentiable in a neighborhood of the rectangle [−Λ, Λ] × [−Λ, Λ], that f has only one critical point in the interior of the rectangle, which is a saddle located at the origin, and that the maxima occur at the two points ±(−Λ, Λ). This gives the first conclusion. Direct computation shows that

\frac{cos δ_{0} + sin^{2} π/6}{cos^{2} π/6} = −0.50022 … < −\frac{1}{2}.

The second conclusion follows.

□

Lemma 5.8. Let P, P′, Q and Q′ be points of S^{2}\{N, P }. Suppose that both dist_{s}(P, Q) and dist_{s}(P′Q′) are greater than α_{−∞}, and that

σ(dist_{s}(P, Q)) + σ(dist_{s}(P′, Q′)) > 2K.

Suppose further that |λ(P)|, |λ(Q)|, |λ(P′)| and |λ(Q′)| are all less than π/6. Then

Θ(P, Q) + Θ(P′, Q′) > \frac{2π}{3} .

Proof. Applying Lemma 5.7 with P and Q playing the roles of P and P′ in the latter result, and taking Λ = π/6 and Δ = dist_{s}(P, Q), we obtain

cos dist_{s}(P, Q) ≥ \frac{3}{4} cos Θ(P, Q) − \frac{1}{4}.

Similarly,

cos dist_{s}(P′, Q′) ≥ \frac{3}{4} cos Θ(P′, Q′) − \frac{1}{4}.

Recall that σ(x) = cosh^{−1}(A−B cos x). Since B > 0 we have that σ is monotone increasing on the intersection of its domain with the interval [0, π]. Thus by the inequalities above,

σ(dist_{s}(P, Q)) ≤ cosh^{−1}((A + \frac{1}{4}B) − \frac{3}{4}B cos Θ(P, Q))

and

σ(dist_{s}(P′, Q′)) ≤ cosh^{−1}((A + \frac{1}{4}B) − \frac{3}{4}B cos Θ(P′, Q′)).

We define a function on the interval [0, π] by f(x) = cosh^{−1}((A + \frac{1}{4}B) − \frac{3}{4}B cos x). Note that A − \frac{1}{2}B > 1, so this definition makes sense. The inequalities above show that

f(Θ(P, Q)) + f(Θ(P′, Q′)) ≥ σ(dist_{s}(P, Q)) + σ(dist_{s}(P′, Q′)) ≥ 2K  (5.8.1)

The function f is monotonically increasing, and we have f(0) = 0.72972 … and f(π) = 1.89609 … while 2K − f(0) = 1.88671 … and 2K − f(2π/3) = 0.89958 . . .. It follows that f^{−1}(2K − f(x)) is defined for all x in [0, \frac{2π}{3} ], so we may define a monotonically increasing function g on the interval [0, \frac{2π}{3} ] by g(x) = \frac{2π}{3} − f^{−1}(2K − f(x)). We now assume, contrary to the conclusion of the lemma, that

Θ(P, Q) + Θ(P′, Q′) ≤ \frac{2π}{3} .

By symmetry we may also assume that Θ(P, Q) ≤ \frac{π}{3} . From the inequality (5.8.1) and the monotonicity of f we conclude that

2K − f(Θ(P, Q)) ≤ f(Θ(P′, Q′)) ≤ f(\frac{2π}{3} − Θ(P, Q)).

Since 2K − f(Θ(P, Q)) is in the domain of the increasing function f^{−1} we have

f^{−1}(2K − f(Θ(P, Q))) ≤ \frac{2π}{3} − Θ(P, Q)

and hence that

g(Θ(P, Q)) = \frac{2π}{3} − f^{−1}(2K − f(Θ(P, Q))) ≥ Θ(P, Q)).

Since the function g is increasing and

g(\frac{2π}{3} ) = 1.54878 … ∈ [\frac{π}{3} , \frac{2π}{3} ] ⊂ [Θ(P, Q), \frac{2π}{3} ],

we have shown (under our assumption) that g maps the interval [Θ(P, Q), 2π/3] into itself. In particular for every positive integer k the k^{th} iterate g^{k}(π/3) is positive. But this leads to a contradiction because direct computation shows that g^{5}(π/3) = −0.30973 … < 0.

□

5.9. Proof of Proposition 5.1. Suppose that we are given an indexed family

(P_{(i,u)})_{(i,u)∈{0,1,2,3}×{−1,1}}

of points in S^{2} for which conclusions (i) and (ii) of 5.1 both fail to hold. Thus the (P_{(i,u)}) satisfy the following conditions:

(-i) for every i ∈ {0, 1, 2, 3} we have dist_{s}(P_{(i,1)}, P_{(i,−1)})) > δ_{0}; and

(-ii) for every two indices (i, u) and (j, v) in {0, 1, 2, 3} × {−1, 1} such that i ≠ j, we

have

σ(dist_{s}(P_{(i,u)}, P_{(j,v)})) + σ(dist_{s}(P_{(i,−u)}, P_{(j,−v)})) > 2K.

We shall derive a contradiction. After a rotation of the sphere we may assume that P_{(0,1)} and P_{(0,−1)} lie on a common meridian, that λ(P_{(0,1)}) ≥ 0 and that λ(P_{(0,−1)}) = −λ(P_{(0,−1)}). Since conditions (-i) and (-ii) are open, we may assume after perturbing the remaining (P_{(i,u)}) that none of the six points P_{(1,±1)}, P_{(2,±1)}, P_{(3,±1)} has latitude 0 and that no two of these six points lie on a common meridian. In particular these six points are distinct. Let us consider any index i ∈ {1, 2, 3}. We wish to apply Lemma 5.2, taking P = P_{(i,1)}, P′ = P_{(i,−1)}, Q = P_{(0,1)} and Q′ = P_{(0,−1)}. We have arranged that P_{(0,1)} and P_{(0,−1)} lie on a common meridian, that λ(P_{(0,1)}) ≥ 0 and that λ(P_{(0,−1)}) = −λ(P_{(0,−1)}). Thus hypothesis (a) of 5.2 holds. Hypothesis (b) follows from condition (-i) above, while hypothesis (c) follows directly from the hypotheses of the proposition. Hypothesis (c′) of 5.2 is simply condition (-ii) above with the given i ∈ {1, 2, 3} and with u = 1, j = 0 and v = 1. Likewise, hypothesis (c′′) is condition (-ii) with the given i and with u = 1, j = 0 and v = −1. Thus, for each (i, u) ∈ {1, 2, 3} × {−1, 1}, we have |λ(P_{i,u})| < π/6. Now consider any two indices (i, u), (j, v) ∈ {1, 2, 3} × {−1, 1} such that i ≠ j. We shall apply Lemma 5.8 taking P = P_{(i,u)}, Q = P_{(j,v)}, P′ = P_{(i,−u)}, Q′ = P_{(j,−v)}. According to (-ii) we have

σ(dist_{s}(P_{(i,u)}, P_{(j,v)}) + σ(dist_{s}(P_{(i,−u)}, P_{(j,−v)}) > 2K

and

σ(dist_{s}(P_{(i,u)}, P_{(j,−v)}) + σ(dist_{s}(P_{(i,−u)}, P_{(j,v)}) > 2K.

On the other hand, we have that λ(P_{(i,u)}), λ(P_{(j,v)}), λ(P_{(i,−u)}), and λ(P_{(j,−v)}) are all less than π/6. Thus the hypotheses of 5.8 all hold, and it follows that

Θ(P_{(i,u)}, P_{(j,v)}) + Θ(P_{(i,−u)}, P_{(j,−v)}) > \frac{2π}{3} .  (5.9.1)

For each (i, u) ∈ {1, 2, 3} × {−1, 1}, the hypothesis of Lemma 5.7 holds if we set P = P_{(i,u)} and Q = Q_{(i,−u)}. Hence we have

Θ(P_{(i,u)}, P_{(i,−u)}) > 2π/3.  (5.9.2)

Recall that ℓ is the retraction with meridian fibers from S^{2} \{N, S} to the equator. We set ζ_{(i,u)} = ℓ(P_{(i,u)} for each (i, u) ∈ {1, 2, 3}×{−1, 1}. The six points ζ_{(1,±1)}, ζ_{(2,±1)}, ζ_{(3,±1)} are distinct because no two of the six points P_{(1,±1)}, P_{(2,±1)}, P_{(3,±1)} lie on a common meridian. The six-element set Z = {ζ_{(i,u)} : (i, u) ∈ {1, 2, 3} × {−1, 1}} has two natural fixed-point-free involutions. The first, which we denote τ_{1}, is defined by τ_{1}(ζ_{(i,u)}) = ζ_{(i,−u)}. To define the second involution, which we denote by τ_{2}, we observe that since Z is a six-element subset of S^{1}, there exists for each ζ ∈ Z a unique ζ′ ∈ Z such that each component of S^{1} \ {ζ, ζ′} contains exactly two elements of Z \ {ζ, ζ′}. We set τ_{2}(ζ) = ζ′. We consider two cases. Case 1. The involutions τ_{1} and τ_{2} coincide. In this case we may assume, after relabeling some of the P_{(i,u)} if necessary, that as one circumnavigates S^{1} in the counterclockwise sense beginning with ζ_{(1,1)}, the ζ_{i,u} appear in the order

ζ_{(1,1)}, ζ_{(2,1)}, ζ_{(3,1)}, ζ_{(1,−1)}, ζ_{(2,−1)}, ζ_{(3,−1)}.

(The relabeling may involve permuting the indices i = 1, 2, 3 and, for certain values of i, interchanging the labels of P_{(i,−1)} and P_{(i,1)}.) To unify the notation we set P_{(4,u)} = P_{(1,−u)} and ζ_{4,u} = ζ_{(1,−u)} for u = ±1. The points of the set Z divide S^{1} into six arcs A_{(1,±1)}, A_{(2,±1)}, A_{(3,±1)}, where A_{(i,u)} has endpoints ζ_{(i,u)} and ζ_{(i+1,u)} for i = 1, 2, 3. The length of the arc A_{(i,u)} is greater than or equal to the circular distance between its endpoints, which by definition is Θ(P_{(i,u)}, P_{(i+1,u)}). Hence

2π = ∑_{i=1}^{3} length A_{(i,1)} + ∑_{i=1}^{3} length A_{(i,−1)}

≥ ∑_{i=1}^{3} Θ(P_{(i,1)}, P_{(i+1,1)}) + ∑_{i=1}^{3} Θ(P_{(i,−1)}, P_{(i+1,−1)})

= ∑_{i=1}^{3}(Θ(P_{(i,1)}, P_{(i+1,1)}) + Θ(P_{(i,−1)}, P_{(i+1,−1)})).

But it follows from (5.9.1) that

Θ(P_{(i,1)}, P_{(i+1,1)}) + Θ(P_{(i,−1)}, P_{(i+1,−1)}) > 2π/3.

for i = 1, 2, 3. Thus we have a contradiction in this case. Case 2. The involutions τ_{1} and τ_{2} are distinct. In this case we may assume, after relabeling the P_{(i,u)} if necessary, that τ_{1}(ζ_{(1,1)}) = (ζ_{(1,−1)}) is distinct from τ_{2}(ζ_{(1,1)}); that is, some component A of S^{1} \ {ζ_{(1,1)}, ζ_{(1,−1)}} contains at least three points of Z \ {ζ_{(1,1)}, ζ_{(1,−1)}}. Hence for some p ∈ {2, 3}, the arc A contains ζ_{(p,1)} and ζ_{(p,−1)}. After further relabeling the P_{(i,u)} if necessary we may assume that A contains ζ_{(2,1)} and ζ_{(2,−1)}. Then the four element set Z′ = {ζ_{(i,u)} : i ∈ {1, 2}, u ∈ {−1, 1}} divides S^{1} into four arcs B_{1}, B_{2}, C_{1}, C_{−1}, where B_{i} has endpoints P_{(i,1)} and P_{(i,−1)}, and C_{u} has endpoints P_{(1,u)} and P_{(2,u)}. Reasoning as in Case 1, we find that

2π = length B_{1} + length B_{2} + length C_{1} + length C_{−1}
≥ Θ(P_{(1,1)}, P_{(1,−1)}) + Θ(P_{(2,1)}, P_{(2,−1)}) + Θ(P_{(1,1)}, P_{(2,1)}) + Θ(P_{(1,−1)}, P_{(2,−1)}).

But we have

Θ(P_{(1,1)}, P_{(2,1)}) + Θ(P_{(1,−1)}, P_{(2,−1)}) > \frac{2π}{3}

by (5.9.1), and by (5.9.2) each of the terms Θ(P_{(1,1)}, P_{(1,−1)}) and Θ(P_{(2,1)}, P_{(2,−1)}) is > \frac{2π}{3} . Thus we have a contradiction in this case as well.

□

Appendix A. Monotonicity of g

In this appendix we prove the following monotonicity statement, which was needed in the proof of Theorem 3.1. The definition of the function g and the domain G appear immediately before the statement of 3.1.

Lemma A.1. Let g_{x}, g_{u}, g_{t} denote the partial derivatives of g with respect to the first, second and third variables respectively. The function g_{t} is positive everywhere on G. The functions g_{x} and g_{u} are positive at every point (x, u, t) ∈ G such that u < 1/2 and t > 2u/3.

proof. On the set G we define functions A = A(x, t), B = B(x, u) and C = C(x, u, t) by A = 1 + t(x − 1), B = 1 + u(x − 1) and C = 1 + (2u − t)(x − 1). Since x > 1 and 0 < t < u for (x, u, t) ∈ G, we have 0 < A < B < C on G. The definition of g may be rewritten in the form

g(x, u, t) = \frac{tx}{A} + \frac{(u − t)x}{BC} + u.

Differentiating with respect to t, we find that

g_{t}(x, u, t) = \frac{x}{A^{2}} − \frac{x}{B^{2}} ,

and the right-hand side is positive on G since 0 < A < B. This proves the first assertion. Retaining the assumptions (x, u, t) ∈ G, u < 1/2 and t > 2u/3, we now differentiate with respect to u. We obtain

g_{u}(x, u, t) = 1 + \frac{x}{B^{2}C^{2}} (BC − (u − t)(x − 1)(2B + C)).

In order to show that g_{u}(x, u, t) > 0, it certainly suffices to show that BC − (u − t)(x − 1)(2B + C) > 0. Again using that 0 < B < C, that x > 1 and that t < u, we find that

BC − (u − t)(x − 1)(2B + C) > C(B − 3(u − t)(x − 1)).

Now, again using the definition of B and the inequalities x > 1 and 3t/2 < u, we find that

B − 3(u − t)(x − 1) = 1 + (x − 1)(3t − 2u) > 0.

Thus g_{u}(x, u, t) is indeed positive. Differentiating with respect to x we find that g_{x} is a rational function in x, u and t with denominator (ABC)^{2}. We introduce a new variable v = t/u and define

h(x, u, v) = (A(x, u, uv)B(x, u, uv)C(x, u, uv))^{2}g_{x}(x, u, uv).

It suffices to show that h(x, u, v) > 0 for all x > 0 and for all (u, v) in the rectangle R = {(u, v)|0 < u < 1/2, 2/3 < v < 1}. We find, with the assistance of a symbolic computation program, that

h(x, u, v) = ∑_{i=0}^{4} p_{i}(u, v)x^{i}

where

p_{0} = u(1 − u)(vu + 1 − 2u)(1 − vu)(1 − (2v − v^{2})(2u − u^{2}))
p_{1} = 4vu^{2}(2 − v)(u − 1)^{2}(1 − vu)(vu + 1 − 2u)
p_{2} = u^{3}((−6v^{4} + 24v^{3} − 24v^{2})u^{3} + (6v^{4} − 24v^{3} + 12v^{2} + 24v)u^{2}+
(−2v^{4} + 8v^{3} + 8v^{2} − 32v)u + (−8v^{2} + 16v − 2))
p_{3} = 4vu^{4}(2 − v)(1 − vu)(vu + 1 − 2u)  and
p_{4} = vu^{5}(2 − v)(2 − (2v − v^{2})(1 + u)).

Using the inequalities 2v − v^{2} < 1 and 2u − u^{2} < 3/4 one checks easily that p_{0}, p_{1}, p_{3} and p_{4} are positive for (u, v) ∈ R. It remains to show that p_{2} is positive for (u, v) ∈ R. For this it suffices to show that

\begin{aligned}q(u, v) = & (−6v^{4} + 24v^{3} − 24v^{2})u^{3} + (6v^{4} − 24v^{3} + 12v^{2} + 24v)u^{2}+ \\ & (−2v^{4} + 8v^{3} + 8v^{2} − 32v)u + (−8v^{2} + 16v − 2) \end{aligned}

is positive. We will show that q_{u}(u, v) < 0 and q_{v}(u, v) > 0 for (u, v) ∈ R and thus that q(u, v) > q(1/2, 2/3) = 19/324 for (u, v) ∈ R.

Differentiating, we find that

q_{u}(u, v) = −2v(2 − v)(2 + v − 3vu)(3vu + (4 − v − 6u)).

Thus by inspection q_{u} < 0 for (u, v) ∈ R. Finally we have

\begin{aligned}q_{v} = & (−3v^{3} + 9v^{2} − 16v)u^{3} + (3v^{2} − 9v^{2} + 3v + 24)u^{2}+ \\ & (−v^{3} + 3v^{2} + 2v − 2)u + (2 − 2v). \end{aligned}

One checks that the coefficient of u^{3} is negative for 2/3 < v < 1 and hence, since u^{3} < u^{2}, we have

q_{v}(u, v) ≥ (24 − 13v)u^{2} + (3v^{2} − v^{3})u + (2 − 2v)(1 − u).

The right hand side of this inequality is positive for v ≤ 1 and 0 < u < 1/2. This completes the proof of the lemma.

□

Appendix B. Intersections of spherical caps

In this appendix we use the Gauss-Bonnet theorem to derive a formula for the area of the intersection of two spherical caps. This formula was needed in the proof of Corollary 4.9. Recall that ι(α, r_{1}, r_{2}) denotes the area of the intersection of two spherical caps C(P_{1}, r_{1}) and C(P_{2}, r_{2}), where P_{1} and P_{2} are two points of S^{2} such that dist_{s}(P_{1}, P_{2}) = α. We shall denote by U the region in ℝ^{3} consisting of all points (x_{1}, x_{2}, x_{3}) such that 0 < x_{i} < π for i = 1, 2, 3 and such that x_{1} + x_{2} > x_{3}, x_{2} + x_{3} > x_{1} and x_{3} + x_{1} > x_{2}. For any (x_{1}, x_{2}, x_{3}) ∈ U we have

−1 < \frac{cos x_{3} − cos x_{1} cos x_{2}}{sin x_{1} sin x_{2}} < 1.

Hence we may define a function J: U → (0, π) by

J(x_{1}, x_{2}, x_{3}) = arccos(\frac{cos x_{3} − cos x_{1} cos x_{2}}{sin x_{1} sin x_{2}} ).

Note that the region U is by definition invariant under permutations of the coordinates of ℝ^{3}; hence for any point (x_{1}, x_{2}, x_{3}) ∈ U and any permutation π ∈ S_{3}, we have (x_{π(1)}, x_{π(2)}, x_{π(3)}) ∈ U, so that J(x_{π(1)}, x_{π(2)}, x_{π(3)}) is defined. We shall prove:

Proposition B.1. Let α, r_{1}, r_{2} be numbers in the interval (0, π). If r_{1} + r_{2} ≤ α then ι(α, r_{1}, r_{2}) = 0. If r_{1} + α ≤ r_{2} or r_{2} + α ≤ r_{1} then ι(α, r_{1}, r_{2}) = | cos r_{1} − cos r_{2}|. Finally, if (r_{1}, r_{2}, α) ∈ U then

ι(r_{1}, r_{2}, α) = 2π − 2(J(r_{1}, α, r_{2}) cos r_{1} + J(r_{2}, α, r_{1}) cos r_{2} + J(r_{1}, r_{2}, α)).

Suppose that P_{1}, P_{2} are points in S^{2} and that r_{1}, r_{2} are numbers in (0, π). Let us set C_{i} = C(P_{i}, r_{i}) and α = dist_{s}(P_{1}, P_{2}). It is clear that Int C_{2} ∩ Int C_{1} = ∅ if and only if r_{1} + r_{2} ≤ α. Furthermore, we have C_{1} ⊂ C_{2} if and only if r_{1} + α ≤ r_{2}; in this case, since C_{i} has area 2π(1 − cos r), we have ι(r_{1}, r_{2}, α) = 2π(cos r_{2} − cos r_{1}). Similarly, we have C_{2} ⊂ C_{1} if and only if r_{2} + α ≤ r_{1}, and in this case ι(r_{1}, r_{2}, α) = 2π(cos r_{2} − cos r_{1}). and likewise C_{2} ⊂ C_{1} if and only if r_{2} + α ≤ r_{1}. This establishes the first two assertions of Proposition B.1, and also shows that the boundary circles of C_{1} and C_{2} cross if and only if (r_{1}, r_{2}, α) ∈ U. In this case it is clear that A_{1} = ∂C_{1} ∩ C_{2} and A_{2} = ∂C_{2} ∩ C_{1} are arcs and that C_{1} ∩ C_{2} is a topological disk whose boundary A_{1} ∪ A_{2} consists of two smooth arcs. We shall refer to such a configuration as a digon. The rest of the section is devoted to the proof of the final assertion of Proposition B.1. The proof depends on Lemmas B.2 and B.3 below. At certain points in the proof it will be necessary to keep track of orientations. We shall always give S^{2} the orientation induced from the restriction to the unit ball of the standard orientation of ℝ^{3}. When we consider a spherical cap C(P, r) we shall always give it the orientation obtained by restricting the orientation of S^{2}. This orientation of C(P, r) induces an orientation of ∂C(P, r).

Lemma B.2. Let P_{1}, P_{2} be points in S^{2}, let r_{1} and r_{2} be numbers, set α = dist_{s}(P_{1}, P_{2}), and suppose that (r_{1}, r_{2}, α) ∈ U. Then length of the arc ∂C(P_{1}, r_{1}) ∩ C(P_{2}, r_{2}) is equal to 2J(r_{1}, α, r_{2}) sin r_{1}. Furthermore, the exterior angles of the digon C(P_{1}, r_{1}) are both equal to J(r_{1}, r_{2}, α).

Proof. Set C_{i} = C(P_{i}, r_{i}). Let Q and Q′ denote the endpoints of A. It is clear that the exterior angles of C_{1} ∩ C_{2} at Q and Q′ have the same value ϵ. Let p_{i}, q and q′ denote the position vectors of P_{i} and Q in ℝ^{3}. We have

cos r_{2} = cos dist_{s}(Q, P_{2}) = q · P_{i},  (B.2.1)

where · denotes the scalar product. Let ∂C_{i} be given the orientation induced from the orientation of C_{i}, and let v_{i} denote the unit tangent vector to C_{i} at Q which is positive with respect to this orientation. We have

v_{1} · v_{2} = cos ϵ.  (B.2.2)

Note that q is orthogonal to S^{2} and hence to the v_{i}, and that p_{i} is orthogonal to C_{i} and hence to v_{i}. The angle between p_{i} and q is dist_{s}(P_{i}, Q) = r_{i}. In view of our orientation conventions, it follows that

v_{i} = (csc r_{i})p_{i} × q,  (B.2.3)

where × denotes the vector product. After a rotation of the sphere, we may assume that P_{1} is the north pole and that P_{2} has longitude 0 and polar angle α. Thus p_{1} = (0, 0, 1) and p_{2} = (sin α, 0, cos α). The endpoints of the arc A = ∂C_{1} ∩ C_{2} have polar angle r_{1}, and their longitudes are opposite in sign and equal in absolute value. Thus we have q = (sin r_{1} cos θ, sin r_{1} sin θ, cos r_{1}), Q′ = (sin r_{1} cos θ, − sin r_{1} sin θ, cos r_{1}). After a reflection of the sphere we may assume that θ > 0. From (B.2.1) we obtain

cos r_{2} = (sin r_{1} cos θ, sin r_{1} sin θ, cos r_{1}) · (sin α, 0, cos α),

which implies that

θ = J(r_{1}, α, r_{2}).  (B.2.4)

Since the circle ∂C(P_{1}, r_{1}) has Euclidean radius sin r_{1}, and since the arc A subtends an angle of 2θ = 2J(r_{1}, α, r_{2}) in this circle, the length of A is 2J(r_{1}, α, r_{2}) sin r_{1}. This is the first assertion of the lemma. From (B.2.3) we have

v_{1} = \frac{1}{sin r_{1}} (a_{2}, −a_{1}, 0)

and

v_{2} = \frac{1}{sin r_{2}} (a_{2} cos α, −a_{1} cos α + a_{3} sin α, −a_{2} sin α).

Computing the scalar product of the v_{i} from these expressions and substituting the value of θ given by (B.2.4) we find that

v_{1} · v_{2} = \frac{cos α − cos r_{1} cos r_{2}}{sin r_{1} sin r_{2}} ,

which by (B.2.2) implies that ϵ = J(r_{1}, r_{2}, α).

□

The next lemma gives the geodesic curvature of the boundary of a spherical cap. Recall that geodesic curvature is a signed quantity and that its sign depends on both an orientation of the curve and an orientation of the ambient surface. We have already fixed an orientation of S^{2}.

Lemma B.3. Let P be any point P ∈ S^{2} and let any r be any number in the interval (0, π). Let us regard the circle ∂C(P, r), as an oriented curve in S^{2} with the orientation induced from that of C(P, r). Then ∂C(P, r) has constant geodesic curvature cot r.

Proof. Since the stabilizer of C = C(P, r) in the group of orientation-preserving isometries of S^{2} acts transitively on the points of ∂C, the geodesic curvature of ∂C is a constant κ. Since ∂C is a Euclidean circle of radius sin r, its length is 2π sin r. On the other hand, C(P, r) has area 2π(1 − cos r) and constant Gaussian curvature K = 1. According to the Gauss-Bonnet theorem we have

2π = 2χ(C(P, r)) = ∫_{C} K + ∫_{∂C} κ = 2π(1 − cos r) + 2πκ sin r,

from which the conclusion follows.

□

B.4. Proof of Proposition B.1. By the remarks following the statement of the Proposition, we need only consider the case (α, r_{1}, r_{2}) ∈ U. Let P_{1} and P_{2} denote points of S^{2} with dist_{s}(P_{1}, P_{2}) = α. Again by the remarks following the statement, the boundaries of C_{1} = C(P_{1}, r_{1}) and C_{2} = C(P_{2}, r_{2}) cross, and G = C_{1} ∩ C_{2} is a digon. Its boundary consists of two circular arcs A_{1} = ∂C_{1} ∩ C_{2} and A_{2} = ∂C_{2} ∩ C_{1}, which by Lemma B.2 have lengths l_{1} = 2J(r_{1}, α, r_{2}) sin r_{1} and l_{2} = 2J(r_{2}, α, r_{1}) sin r_{2} respectively. Let us fix an orientation on S^{2}, so that G and the C_{i} inherit orientations. We give A_{i} the orientation induced from G, which is the same as the one induced from C_{i}. By Lemma B.3, A_{i} ⊂ ∂C_{i} has constant geodesic curvature κ_{i} = cot r_{i}. The digon G has constant Gaussian curvature K = 1, and its exterior angles are equal to ϵ = J(r_{1}, r_{2}, α) by Lemma B.2. By the Gauss-Bonnet theorem we have

2π = 2πχ(G) = K area(G) + l_{1}κ_{1} + l_{2}κ_{2} + 2ϵ
= ι(α, r_{1}, r_{2}) + 2J(r_{1}, α, r_{2}) cos r_{1} + 2J(r_{2}, α, r_{1}) cos r_{2} + 2J(r_{1}, r_{2}, α),

which implies the conclusion of the proposition.

□

Appendix C. A path of 2-generator groups

In this appendix we prove the remaining result that was required for the proof of Theorem 3.1. We give a construction of certain 1-parameter families of 2-generator subgroups of Isom_{+}(ℍ^{3}); these paths were used in the proof of Theorem 3.1 for the reduction to the case of groups which admit no non-constant invariant super-harmonic functions. In this appendix we also derive a formula, which is included in Lemma C.2, for the displacement of a point of ℍ^{3} under a loxodromic isometry. This formula was used in Lemma 4.2 and Proposition 4.10 as well as in [9].

Proposition C.1. Let ξ and η be two loxodromic isometries of ℍ^{3} without any common fixed point. Denote by L the common perpendicular to the axes A_{ξ} and A_{η} of ξ and η respectively. Let z_{0} be any point of L. Then there exist continuous one-parameter families (ξ_{t})_{0≤t≤1} and (η_{t})_{0≤t≤1} of loxodromic isometries of ℍ^{3} with the following properties:

(i) ξ_{0} = ξ and η_{0} = η;

(ii) for every t the axes of ξ_{t} and η_{t} are perpendicular to L;

(iii) the functions t ↦ dist(z_{0}, ξ_{t} · z_{0}) and t ↦ dist(z_{0}, η_{t} · z_{0}) are monotonically

decreasing on [0, 1]; (iv) the function t ↦ ∠(ξ_{t} · z_{0}, z_{0}, η_{t} · z_{0}) is monotonically decreasing on [0, 1]; and (v) the isometries ξ_{1} and η_{1} have the same axis.

The proof of Proposition C.1 depends on the following three lemmas, C.2, C.3 and C.4. The first assertion of C.2 was used, but not proved, in [9].

Lemma C.2. Let γ be a loxodromic isometry of ℍ^{3} with axis A, translation length l and twist angle θ. Let z be any point of ℍ^{3}, let z′ denote the foot of the perpendicular from z to A, and let R denote the perpendicular distance from z to A. Then we have

cosh dist_{h}(z, γ · z) = Δ(R),

where Δ = Δ_{l,θ} is the function defined on [0, ∞) by

Δ(R) = cosh^{−1}(cosh l + sinh^{2} R(cosh l − cos θ)).

We also have

∠(z′, z, γ · z) = α(R),

where α = α_{l,θ} is the function defined on [0, ∞) by

α(R) = cos^{−1} \frac{(sinh R)(cosh R)(cosh l − cos θ)}{\sqrt{(cosh l + sinh^{2} R(cosh l − cos θ))^{2} − 1}}.

Furthermore, Δ(R) is strictly monotone increasing on [0, ∞) and tends to ∞ as R → ∞, while α(R) is strictly monotone decreasing on [0, ∞) and tends to 0 as R → ∞.

Proof. We identify ℍ^{3} conformally with the upper half-space ℝ^{2} × ℝ^{+} in such a way that A = {0} × ℝ^{+}, and so that γ is given by γ · (w, t) = (e^{l+iθ}w, e^{l}t). We may also suppose the identification to have been made in such a way that z_{0} = (s, 1) for some positive real number s. The distance from any point (w, t) ∈ ℍ^{3} to A is given by the formula

sinh dist_{h}((w, t), A) = \frac{|z|}{t}.  (C.2.1)

Here the right-hand side is the tangent of the angle between the ray \overline{A} and the ray which has origin 0 and passes through (w, t). Formula (C.2.1) is the 3-dimensional analogue of formula (7.20.3) of [2], and follows from applying the latter formula to the hyperbolic plane spanned by A and (w, t). In the same way, formula (7.2.1(ii)) of [2] implies that the distance between any two points (w, t) and (w′, t′) of ℍ^{3} is given by

cosh dist_{h}((w, t), (w′, t′)) = 1 + \frac{|w − w′|^{2} + (t − t′)^{2}}{2tt′} .  (C.2.2)

It follows from (C.2.2) that sinh R = sinh dist_{h}((s, 1), A) = s. Now let us apply (C.2.2) to the points z_{0} = (s, 1) and γ · z_{0} = (e^{l+iθ}s, e^{l}). This gives

cosh dist(z_{0}, γ · z_{0}) = 1 + \frac{s^{2}|e^{l+iθ} − 1|^{2} + (e^{l} − 1)^{2}}{2e^{l}}
= (cosh l − cos θ)s^{2} + cosh l = cosh Δ(R),

and the first assertion of the lemma follows. To prove the second assertion, we first apply the hyperbolic Pythagorean theorem to the right triangle with vertices z′, γ · z′ and γ · z. The lengths of the sides adjacent to the right angle are dist_{h}(z′, γ · z′) = l and dist_{h}(γ · z′, γ · z) = R. Hence, setting c = dist_{h}(z′, γ · z), we have

cosh c = cosh l cosh R.

Now we apply the first hyperbolic law of cosines to the triangle with vertices z′, z and γ · z, whose side lengths are dist_{h}(z, γ · z) = Δ(R), dist_{h}(γ · z, z′) = c and dist_{h}(z′, z) = R. This gives

cos ∠(z′, z, γ · z) = \frac{cosh Δ(R) cosh R − cosh c}{sinh Δ(R) sinh R}
= \frac{cosh Δ(R)(cosh R − cosh l)}{sinh Δ(R) sinh R}
= cos α(R),

and the second assertion follows. It is clear from the definition of the function Δ(R) that it is strictly monotone increasing and tends to ∞ with R. In order to establish the properties of α(R), we set A = cosh l−cos θ and B = cosh l. Note that B ≥ A > 0 and B > 1. For each R ∈ [0, ∞) we have α(R) = cosh^{−1} \sqrt{g(sinh R)}, where g is defined on [0, ∞) by

g(x) = \frac{A(x + 1)x}{(Ax + B)^{2} − 1}.

It is clear that g(x) tends to 1 as x → ∞, and hence that α(R) tends to 0 as R → ∞. To complete the proof, we need only show that g is strictly monotone increasing on [0, ∞). Differentiating g we find that

((Ax + B)^{2} − 1)^{2}A^{−1}g′(x) = 2A(B − A)x^{2} + 2(B^{2} − 1)x + B^{2} − 1,

where the right hand side is strictly positive for x > 0 since B ≥ A > 0 and B > 1. Hence g′(x) > 0 for every x > 0.

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In the statement of the next lemma, we fix a line L ⊂ ℍ^{3} and a point z_{0} ∈ L. We identify ℍ̂^{3} = ℍ^{3} ∪ S_{∞} conformally with the closed unit ball in ℝ^{3} in such a way that z_{0} is the origin and L is the vertical axis. In particular, S_{∞} is identified with S^{2} in such a way that the end points of L are the north and south poles N and S. We denote by G_{L} the subgroup of Isom_{+}(ℍ^{3}) consisting of all isometries that stabilize the line L and preserve orientation on L. Then G_{L} is a Lie subgroup of Isom_{+}(ℍ^{3}) and in particular it has the structure of a manifold. The non-trivial elements of G_{L} are precisely the loxodromic and elliptic isometries of ℍ^{3} having L as axis.

Lemma C.3. Let ξ be a loxodromic isometry of ℍ^{3} whose axis meets L perpendicularly at a point z_{0}. Let us define a map f_{ξ}: G_{L} → S_{∞} = S^{2} as follows: for each γ ∈ G_{L} we define f_{ξ}(γ) ∈ S_{∞} to be the endpoint of the ray from z_{0} to γξγ^{−1} · z_{0}. Then f_{ξ} maps G_{L} homeomorphically onto S^{2} \ {N, S}. Furthermore, there is a strictly monotone increasing function h_{ξ}: [0, ∞) → [0, π/2) such that for every γ ∈ G_{L} we have

|λ(f_{ξ}(γ))| = h_{ξ}(dist_{h}(z_{0}, γξγ^{−1} · z_{0})).

Finally, for each point ζ on the equator of S^{2} = S_{∞}, the isometry f_{ξ}^{−1}(ζ) ∈ G_{L} maps the axis of ξ onto the line which passes through z_{0} and has ζ as an end point.

Proof. Let us fix a transverse orientation for the line L. For each w = e^{2πiθ} ∈ S^{1}, let τ_{w} denote the isometry which fixes L pointwise and whose restriction to each plane orthogonal to L is a rotation through an angle θ which is counterclockwise in terms of the chosen transverse orientation. For each r ∈ ℝ \ {0} let h_{r} denote the hyperbolic isometry with axis L having translation length |r|, and having S or N as its attracting fixed point according as r is positive or negative. (To say that h_{r} is hyperbolic means that it leaves each plane through L invariant.) Let h_{r} denote the identity map on ℍ^{3}. Then (r, w) ↦ τ_{w} ∘ h_{r} is a homeomorphism of ℝ × S^{1} onto G_{L}. In particular G_{L} is a topological 2-manifold homeomorphic to ℝ × S^{1}. Let l and θ denote the translation length and twist angle of ξ, and let A_{ξ} denote its axis. Let Δ = Δ_{l,θ} and α = α_{l,θ} be defined as in the statement of Lemma C.2. Consider any r ≥ 0 and any w ∈ S^{1}, and set γ = τ_{w} ∘ h_{r}. The isometry γξγ^{−1} has axis γ · A_{ξ}, and its translation length and twist angle are l and θ. The point z_{0} lies at a distance r from γ · A_{ξ}. The foot of the perpendicular from z_{0} to γ · A_{ξ} is γ · z_{0}. Hence by Lemma C.2 we have ∠(γ · z_{0}, z_{0}, γξγ^{−1} · z_{0}) = α(r). On the other hand, since r ≥ 0, the point γ · z_{0} lies on the ray L_{+} ⊂ L which has origin z_{0} and endpoint N. Hence

λ(f_{ξ}(γ)) = \frac{π}{2} − ∠(N, z_{0}, γξγ^{−1} · z_{0}) = \frac{π}{2} − ∠(γ · z_{0}, z_{0}, γξγ^{−1} · z_{0}) = \frac{π}{2} − α(r).

Similarly, for r ≤ 0 we find that λ(f_{ξ}(γ)) = α(|r|) − \frac{π}{2} . Thus for all w ∈ S^{1} and r ∈ ℝ we have

λ(f_{ξ}(τ_{w} ∘ h_{r})) = β(r),  (C.3.1)

where β(r) = |\frac{π}{2} − α(|r|)|. By Lemma C.2 the function α(R) decreases monotonically from π/2 to 0 as R varies from 0 to ∞. Hence β(r) increases montonically from 0 to π/2 as r increases from −∞ to ∞. It therefore follows from (C.3.1) that f_{ξ} : G_{L} → S^{2} \ {N, S} is a proper map.

We claim that f_{ξ} is also one-to-one. Indeed, suppose that for some γ = (τ_{w} ∘ h_{r}) and γ′ = (τ_{w′} ∘ h_{r′}) we have f_{ξ}(γ) = f_{ξ}(γ′). By (C.3.1) it follows that β(r) = β(r′). By the monotonicity of β it follows that r = r′. Hence (γ′) = τ_{w′/w} ∘ γ. Since τ_{w′/w} fixes z_{0} ∈ L we have γ′αγ′^{−1} · z_{0} = τ_{w′/w} · (γαγ^{−1} · z_{0}). In view of the definition of f_{α}, this implies that f_{α}(γ′) = τ_{w′/w} · f_{α}(γ). Since we have assumed that f_{α}(γ′) = ·f_{α}(γ), we conclude that w = w′. This shows that f is one-to-one. Since f is a proper one-to-one map between the 2-manifolds G_{L} and S^{2} \ {N, S}, it is a homeomorphism of G_{L} onto S^{2} −{N, S}. This is the first assertion of the lemma. To prove the second assertion, we first note that for any γ = τ_{w} ∘ h_{r} we have |λ(f_{ξ}(γ))| = \frac{π}{2} − α(r) by virtue of (C.3.1). On the other hand, since z_{0} is at a distance r from the axis of γξγ^{−1}, which has twist angle θ and translation length l, it follows from Lemma C.2 that dist_{h}(z_{0}, γξγ^{−1} · z_{0}) = Δ(|r|). Since, by C.2, the function Δ(R) increases monotonically from l to ∞ as R increases from 0 to ∞, and \frac{π}{2} −α(R) increases from 0 to \frac{π}{2} as R increases from 0 to ∞, the second assertion of C.3 follows if we set h_{ξ} = α ∘ Δ^{−1}. Finally, suppose that ζ is a point on the equator of S^{2} = S^{∞}, and let B denote the line which passes through z_{0} and has ζ as an end point. Since ζ is on the equator, B is perpendicular to L at z_{0}. As A is also perpendicular to L at z_{0}, we have τ_{w} · A = B for some w ∈ S^{1}. We may suppose w to be chosen so that the ray A_{+} ⊂ A which has origin z_{0} and passes through ξ · z_{0} is transformed by τ_{w} onto the ray B_{+} ⊂ B which has origin z_{0} and end point ζ. Then τ_{w}ξτ_{w}^{−1} · z_{0} = τ_{w}ξ · z_{0} ∈ B_{+}, and hence f_{ξ}(τ_{w}) = ζ. In view of the first assertion of the lemma we may therefore write f_{ξ}^{−1}(ζ) = τ_{w}. In particular we have f_{ξ}^{−1}(ζ)(A) = B. This is the final assertion of the lemma.

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Lemma C.4. For any two points P and Q of S^{2} \ {N, S}, there exist two continuous paths p, q: [0, 1] → S^{2} \ {N, S} with the following properties:

(i) p(0) = P and q(0) = Q;

(ii) the functions λ ∘ p and λ ∘ q are (weakly) monotonically decreasing on [0, 1]; (iii) the function t ↦ dist_{s}(p(t), q(t)) is (weakly) monotonically decreasing on [0, 1];

and (iv) the points p(1) and q(1) coincide and lie on the equator.

Proof. We may assume without loss of generality that θ(Q) = 0 and that 0 ≤ θ(P) ≤ π. We define p(t) for 0 ≤ t ≤ \frac{1}{2} by setting θ(p(t)) = (1 − 2t)θ(P) and λ(p(t)) = λ(P). For \frac{1}{2} ≤ t ≤ 1 we define p(t) by θ(p(t)) = 0 and λ(p(t)) = (2 − 2t)λ(P). We define q(t) = Q for 0 ≤ t ≤ \frac{1}{2}, and for \frac{1}{2} ≤ t ≤ 1 we define q(t) by θ(q(t)) = 0 and λ(q(t)) = (2 − 2t)λ(Q).

Conclusions (i), (ii) and (iv) of the lemma are now clear. It is also clear that the function t ↦ Θ(p(t), q(t)) is monotone decreasing on [0, 1]. Since by the spherical distance formula we have

cos dist_{s}(p(t), q(t)) = cos λ(p(t)) cos λ(q(t)) cos Θ(p(t), q(t)) + sin λ(p(t)) sin λ(q(t)),

it follows readily that t ↦ cos dist_{s}(p(t), q(t)) is monotone increasing on [0, 1], so that conclusion (iii) holds as well.

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C.5. Proof of C.1. According to Lemma C.3 we have homeomorphisms f_{ξ} and f_{η} from G_{L} to S^{2} \ {N, S}. We set P = f_{ξ}(I) and Q = f_{η}(I), where I denotes the identity element of G_{L}. With these choices of P and Q, let p and q be paths satisfying conclusions (i)—(iv) of Lemma C.4. For each t ∈ [0, 1] we set γ_{t} = f_{ξ}^{−1}(p(t)) and δ_{t} = f_{η}^{−1}(q(t)). We set ξ_{t} = γ_{t}ξγ_{t}^{−1} and η_{t} = γ_{t}ηγ_{t}^{−1}. It is clear that (ξ_{t})_{0≤t≤1} and (η_{t})_{0≤t≤1} are continuous one-parameter families of isometries of ℍ^{3}. As ξ and η are loxodromic with axes A_{ξ} and A_{η}, the isometries ξ_{t} and η_{t} are also loxodromic and have axes A_{ξ_{t}} = γ_{t} · A_{ξ} and A_{η_{t}} = δ_{t} · A_{η}. Since A_{ξ} and A_{η} are perpendicular to L and since γ_{t}, δ_{t} ∈ G_{L}, it follows that A_{ξ_{t}} and A_{η_{t}} are also perpendicular to L. This is conclusion (ii) of Proposition C.1. We have γ_{0} = f_{ξ}^{−1}(P) = I and hence ξ_{0} = ξ; similarly η_{0} = η. This is conclusion (i) of C.1. For any t ∈ [0, 1], using the function Λ_{ξ} given by Lemma C.3, we have

Λ_{ξ}(dist(z_{0}, ξ_{t} · z_{0})) = Λ_{ξ}(dist(z_{0}, γ_{t}ξγ_{t}^{−1} · z_{0})) = |λ(f_{ξ}(γ_{t}))| = |λ(p(t))|.

Since Λ_{ξ} is strictly monotone increasing by C.3 and λ ∘ p is monotonically decreasing by conclusion (iii) of C.4, it follows that t ↦ dist(z_{0}, ξ_{t} · z_{0}) is monotonically decreasing on [0, 1]; the same argument shows that and t ↦ dist(z_{0}, η_{t} · z_{0}) is monotonically decreasing on [0, 1]. This is conclusion (iii) of C.1. For any t ∈ [0, 1] we have

∠(ξ_{t} · z_{0}, z_{0}, η_{t} · z_{0}) = ∠(γ_{t}ξγ_{t}^{−1} · z_{0}, z_{0}, γ_{t}ηγ_{t}^{−1} · z_{0}).

In view of the definition of f_{ξ} and f_{η} the right-hand side of the equation above is equal to the spherical distance between f_{ξ}(γ_{t}) = p(t) and f_{η}(γ_{t}) = q(t). Hence conclusion (iv) of C.1 follows from conclusion (iii) of C.4. Finally, according to conclusion (iv) of C.4, we have p(1) = q(1) = ζ, where ζ is some point of the equator of S^{2}. Thus we have γ_{1} = f_{ξ}^{−1}(ζ), and the last sentence of Lemma C.3 implies that A_{ξ_{1}} = γ_{1} · A_{ξ} is the line passing through z_{0} and having ζ as an end point. The same argument shows that A_{η_{1}} is the very same line, and thus conclusion (v) of C.1 holds.

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