Effective distance between nested Margulis tubes

We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.


Introduction
A key tool in the study of hyperbolic manifolds is the thick-thin decomposition. For any number > 0, a manifold M is decomposed into the -thin part, consisting of points on essential loops of length less than , and its complement the -thick part. Margulis proved the foundational result that there is a universal constant 3 > 0 such that for any hyperbolic 3-manifold M , the 3 -thin part is a disjoint union of cusps and tubes. Analogous statements hold in all dimensions, and for more general symmetric spaces. This result has had numerous important consequences in the study and classification of hyperbolic 3-manifolds and Kleinian groups. Thurston and Jorgensen used the Margulis lemma to describe the structure of the set of volumes of hyperbolic manifolds, with limit points occurring only via Dehn filling [27]. The Margulis lemma also plays a major role in the construction of model manifolds used in the proofs of the Ending Lamination Theorem [20,7] and the Density Theorem [22,23].
Since tubes and cusps are well-understood quotients of hyperbolic space by elementary groups, it seems that the thin parts of manifolds should be easy to analyze. However, in practice, the thin parts of a manifold are often very difficult to control. For example, the optimal value for the Margulis constant 3 is still unknown. The best known estimate is due to Meyerhoff [19]. Additionally, given an -thin tube, it is very difficult to analyze and bound simple quantities such as the radius of the tube in full generality. This is because the radius depends not only on , but also on the rotation and translation -the complex length -at the core of the tube. Although the radius is a continuous function of these parameters, it is non-differentiable in many places. See Proposition 3.10 for a formula, and Figure 2 for a graph.
We remark that the argument of arccosh in the lower bound of Theorem 1.1 may be less than 1, hence arccosh(·) is undefined. To remedy this, we employ the convention that an undefined value does not realize the maximum. The real point is that the lower bound is not very strong (less than /2) for any pair (δ, ) such that < √ 7.256 δ. On the other hand, the lower bound of Theorem 1.1 is sharp up to additive error for any pair (δ, ) where ≥ √ 7.256 δ. The upper bound is sharp for every pair (δ, ). See Section 1.3 for more details.
1.1. Motivation and applications. Ineffective bounds on the distances between thin tubes were previously observed by Minsky [20, Lemma 6.1] and Brock-Bromberg [5, Theorem 6.9], who credit the bound to Brooks and Matelski [8]. Universal bounds of this sort, depending only on δ and , are required in the proof of the Ending Lamination Theorem, both for punctured tori [20] and for general surfaces [21,7]. In particular, Minsky used such bounds in the proof of the "a priori bounds" theorem [21] that curves appearing in a hierarchy have universally bounded length. One consequence of "a priori bounds" is Brock's volume estimate for quasifuchsian manifolds and for mapping tori [3,4]. A second consequence is the result (due to Minsky, Bowditch, and Brock-Bromberg [2,6]) that distance in the curve complex of a surface S is coarsely comparable to electric distance in a 3-manifold of the form S × R.
In a slightly different direction, Brock and Bromberg applied the ineffective bounds on distances between tubes to cone-manifolds, establishing uniform bilipschitz estimates between the thick part of a cusped 3-manifold and the thick parts of its long Dehn fillings [5]. This application requires a version of Theorem 1.1 for solid tori with a cone-singularity at the core. In turn, the Brock-Bromberg result has been combined with the Ending Lamination Theorem to show that geometrically finite hyperbolic 3-manifolds are dense in the space of all (tame) hyperbolic 3-manifolds [22,23].
The past few years have witnessed an intense effort to make theorems in coarse geometry effective, that is, to make the constants explicit. Recent effective results include, for instance, [1,14,16]. Finding an effective version of the distance between thin tubes has been a major obstacle to extending those efforts. Theorem 1.1 provides such an effective result. Theorem 1.1 is already being applied to obtain effective versions of several results mentioned above. Futer and Taylor have outlined an effective "a priori bounds" theorem, combining Theorem 1.1 with sweepout arguments [14] and effective results about hierarchies [1]. Aougab, Patel, and Taylor have found an effective "electric distance" theorem, again using a combination of Theorem 1.1 and sweepout arguments. Finally, the authors of this paper have used Theorem 1.1 in combination with a number of cone-manifold estimates due to Hodgson and Kerckhoff [17,18] to effectivize the Brock-Bromberg bilipschitz theorem [13]. Our effective results on cone-manifolds require Theorem 1.1 to hold for singular solid tori.
Finally, Theorem 1.1 offers a useful step toward finding the Margulis constant 3 . The current state of knowledge is that 0.104 ≤ 3 ≤ 0.775, with the lower bound due to Meyerhoff [19] and the upper bound realized by the Weeks manifold. Furthermore, a theorem of Shalen [24,25], building on earlier work of Culler and Shalen [12], says that 0.29 is a Margulis number for all but finitely many hyperbolic 3-manifolds. That is, for all but finitely many choices of M , the 0.29-thin part of M is a disjoint union of cusps and tubes. Any manifold M failing this property must be closed and must have vol(M ) < 52.8. By combining Theorem 1.1 with our work on cone-manifolds, we produce an explicit lower bound on the injectivity radius of any exception to Shalen's theorem. This makes it theoretically feasible (although computationally impractical) to enumerate all manifolds with vol(M ) < 52.8 and injectivity radius bounded below, and to determine their Margulis numbers [13].
1.2. Distance between tubes, as a function. Let 0 < δ ≤ be injectivity radii, and consider a solid torus N = N α,λ,τ whose core geodesic has complex length λ + iτ and cone angle α ≤ 2π. The distance between the -and δ-tubes in N is defined carefully in Definition 3.1. We denote this distance by d α,λ,τ (δ, ). For our current discussion, it helps to note that d α,λ,τ (δ, ) is the difference of tube radii of the -tube and δ-tube, and that each tube radius is determined by taking a maximum of many smooth functions. See Proposition 3.10 for an exact formula. As a consequence, d α,λ,τ (δ, ) is a continuous but only piecewise smooth function of the quantities δ, , λ, and τ .  Figure 1. In each region, all complex lengths λ + iτ have the indicated power for = 0.2. That is: every based loop of length that realizes injectivity radius is freely homotopic to this power of the core. This figure was inspired by [11, Figure 2].
The failure of global smoothness can be explained as follows. For each value of > 0 and each complex length λ + iτ of the core of N , the radius of the -tube is determined by some power of the generator of π 1 (N ). This power can change as the data λ, τ, changes. For instance, Figure 1 shows the power of the core when = 0.2 is fixed but λ + iτ varies. Meanwhile, Example 3.9 shows how the power can depend on . Figure 2 shows the graph of d 2π,λ,τ (δ, ) when δ = 0.05 and = 0.2 are fixed but (λ, τ ) vary. Since the graph is the difference of a pair of wildly varying, piecewise-smooth functions, it is an extremely complicated terrain of deep valleys, narrow ridges, and sharp peaks. The sharp ridges are points of non-differentiabilty, and occur where the power of the core for δ changes. Other points of non-differentiability, where the power for changes, occur in the valleys. Even though δ and are fixed, the value of d α,λ,τ (δ, ) ranges a great deal: from approximately There are interesting examples illustrating the sharpness of both the upper and lower bounds of Theorem 1.1. As Proposition 5.7 will show, the upper bound of Theorem 1.1 is sharp for every pair (δ, ). It is attained if and only if N is a nonsingular tube whose core has complex length λ + iτ = δ + 0; that is, the core has length δ and trivial rotational part.
The lower bound of Theorem 1.1 is sharp up to additive error, which can be seen as follows. For every pair (δ, ) such that 0 < √ 7.256δ ≤ ≤ 0.3, Theorem 4.6 constructs a solid torus N = N 2π,λ,τ such that The core of N has complex length λ + iτ = 1/n 2 + 2πi/n, where n is the least natural number such that 1/n 2 ≤ δ. Meanwhile, Theorem 1.1 gives Since arccosh(x) ∼ log(2x) for large x, the expressions in (1.2) and (1.3) differ by an additive error. In fact, the additive difference is less than 2.2. One consequence of the above paragraphs is that Theorem 1.1 is sharpest when the solid torus N is nonsingular; that is, when N is the quotient of H 3 by a loxodromic isometry. Thus, while extending Theorem 1.1 to singular tubes introduces a few technical complications (for example see Propositions 3.10 and 5.6), this extension does not weaken the statement in any way. We emphasize that the extension to singular tubes is needed for our forthcoming applications to cone-manifolds and to bounding the Margulis numbers of (nonsingular) hyperbolic manifolds.
The results of this paper have an interesting relation to the discussion of Margulis tubes in the work of Minsky (see [20,Section 6 Proposition 1.4. Let 0 < δ < . Let N be a horocusp whose -thick part, N ≥ , is non-empty. Then the δ-thin and -thick parts of N are separated by distance Proof. Let x ∈ N be a point such that injrad(x) = δ/2. Then there must be a lift x ∈ N and a parabolic covering transformation ϕ, such that d( x, ϕ x) = δ. By [10, Lemma A.2], a horocyclic segment from x to ϕ x has length 2 sinh(δ/2). Similarly, if y ∈ N is a point such that injrad(y) = /2, then there is a horocyclic segment from y to ϕ y of length 2 sinh( /2). Now, a standard calculation shows that the horospheres containing x and y are separated by distance We observe that the distance d N (δ, ) satisfies both the upper and lower bounds of Theorem 1.1. Thus Theorem 1.1 also applies to cusps.
1.5. Organization. Section 2 lays out definitions and sets up notation that will be used for the remainder of the paper. Section 3 proves Proposition 3.10, which gives an exact formula for the radius of an -thin tube.
Section 4 describes a family of examples showing the sharpness of the lower bound of Theorem 1.1. Section 5 proves the upper bound of Theorem 1.1 and shows that it is sharp.
The lower bound of Theorem 1.1 requires a delicate case analysis, treating shallow and deep tubes separately. For a maximally sharp estimate, we rely on a result of Zagier [19], later improved by Cao, Gehring, and Martin [9]. We obtain a bound on the Euclidean metric on the tube boundary in Section 6, and prove a lower bound on the depth of an -tube in Section 7. These ingredients are combined to give the final proof in Section 8.
Appendix A contains several elementary lemmas in hyperbolic trigonometry that are useful elsewhere in the paper.
1.6. Acknowledgements. Futer was supported in part by NSF grant DMS-1408682. Purcell was supported in part by the Australian Research Council. All three authors acknowledge support from NSF grants DMS-1107452, 1107263, 1107367, "RNMS: Geometric Structures and Representation Varieties" (the GEAR Network), which funded an international trip to collaborate on this paper.
We thank Ian Biringer and Yair Minsky for a number of enlightening conversations. We also thank Tarik Aougab, Marc Culler, Priyam Patel, and Sam Taylor for their helpful suggestions.

Tubes and equidistant tori
In this section, we set notation and give definitions for solid tori, tubes, and injectivity radii that will be used for the remainder of the paper.
To form a nonsingular hyperbolic tube, one starts with a neighborhood of a geodesic in H 3 and takes a quotient under a loxodromic isometry fixing that geodesic. In order for our results to hold for both nonsingular and singular tubes, we will take a quotient of a more complicated space, as in the following definition. Fix 0 ∈ H 3 to be an arbitrary basepoint.
Definition 2.1. Let σ ⊂ H 3 be a bi-infinite geodesic. LetĤ 3 denote the metric completion of the universal cover of (H 3 − σ). Letσ be the set of points added in the completion.
The spaceĤ 3 can be regarded as an infinite cyclic branched cover of H 3 , branched over σ. The branch setσ ⊂Ĥ 3 is a singular geodesic with infinite cone angle.
There is a natural action of C (as an additive group) onĤ 3 , where z ∈ C translatesσ by distance Re(z) and rotates by angle Im(z). Sinceσ has infinite cone angle, we have that angles of rotation are real-valued. Conversely, every isometry ϕ ofĤ 3 that preserves orientation on bothĤ 3 andσ comes from this action, and has a well-defined complex length z = ζ + iθ. We can therefore write ϕ = ϕ ζ+iθ .
Definition 2.3. Consider a group G = Z × Z of isometries ofĤ 3 , generated by an elliptic ψ iα and a loxodromic ϕ = ϕ λ+iτ , where α > 0 and λ > 0. The quotient space N α,λ,τ is an open solid torus whose core curve Σ is a closed geodesic of complex length λ + iτ , and with a cone singularity of angle α at the core. We call N = N α,λ,τ a model solid torus.
Definition 2.4. Let N = N α,λ,τ be a model solid torus, and let x ∈ N . Then the injectivity radius of x, denoted injrad(x), is the supremal radius r such that an open metric r-ball about x is isometric to a ball B r (0) ⊂ H 3 . Since we are using open balls, the supremal radius is attained. If α = 2π and x lies on the singular core geodesic, we set injrad(x) = 0.
For > 0, the -thick part of N is We emphasize that our definition of the -thick part corresponds to injectivity radius /2 rather than . Both choices seem to be common in the literature on Kleinian groups. Our convention agrees with that of Minsky [20,21] and Brock-Canary-Minsky [7], while differing from the convention of Brock-Bromberg [5] and Namazi-Souto [22].
If M is a non-singular manifold, every essential loop through a point x ∈ M ≥ has length at least . A similar statement holds for singular tubes.
Lemma 2.5. Let N = N α,λ,τ be a model solid torus whose core is singular. That is, assume N is the quotient ofĤ 3 by Z 2 ∼ = ψ iα , ϕ λ+iτ for α = 2π. Choose a point x ∈ N and a lift x ∈Ĥ 3 . Set = 2 injrad(x). Then Similarly, if N is a nonsingular solid torus, the quotient of H 3 by Z ∼ = ϕ λ+iτ , and x ∈ H 3 is a point covering x, then Proof. We focus on the singular case, as the nonsingular case is well-known. For an arbitrary basepoint 0 ∈ H 3 , there is an isometric embedding f : Next, since injrad(x) = /2, the continuous extension of f to B /2 (0) either hits the core Σ or fails to be one to one. If it fails to be one to one, then a lifted ball is tangent to a translate of itself under a nontrivial element η ∈ Z 2 . On the other hand, if it meets the core Σ, then there is a point z ∈ f (B /2 (0)) ∩ Σ. The preimage z of z is fixed by an elliptic subgroup ψ iα ⊂ Z 2 . Thus a translate of the lifted ball by the generator η = ψ iα will be tangent to the ball.
In either case, there are two distinct lifts of the ball, namely B /2 ( x) and B /2 (η x), that are tangent inĤ 3 . Therefore, d( x, η x) = , and the minimum over all group elements must be at most . The minimum is attained because Z 2 acts discretely.
Definition 2.8. If N is a nonsingular tube and ≥ λ, we define the power for to be any n ∈ N so that the deck transformation η = ϕ n realizes the minimum in Equation (2.7). If N is a singular tube and > 0, we define the power for to be any n ∈ N ∪ {0} so that the deck transformation η = ϕ n ψ m realizes the minimum in Equation (2.6), for some m ∈ Z. The power is uniquely defined for almost every .
We remark that the power for is genuinely a function of and N , and does not depend on the choice of x ∈ N such that injrad(x) = /2. This is because for every > 0 and every loxodromic η ∈ Isom(Ĥ 3 ), the set {y ∈Ĥ 3 : d(y, ηy) = } is a Euclidean plane at fixed radius from the core geodesicσ (or equal toσ, or empty). The quotient of this set in N is equidistant from the core of N . This is a distinctly 3-dimensional phenomenon: already for η ∈ Isom(H 4 ), the set of points moved by distance can be far more complicated [26].

Tube radii
We now provide a bit of background for tube radii in hyperbolic 3-manifolds, well-known to the experts. Let N = N α,λ,τ be a model solid torus, as in Definition 2.3. Our eventual goal is to bound distances between the boundaries of the -thin and δ-thin tubes N ≤ and N ≤δ , independently of α, λ, and τ . In this section, we derive a formula for the radius of one such tube.
Definition 3.1. Let > 0, and assume the -thin part N < is non-empty. Then T = ∂N ≤ is a torus consisting of points whose injectivity radius is exactly /2. All of the points of T lie at the same radius from the core geodesic Σ. We denote this radius by r( ) = r α,λ,τ ( ).
We let T r denote the equidistant torus at radius r from the core of N . Subscripts denote radius, while superscripts denote thickness. Thus Given 0 < δ < , where N <δ = ∅, we define the distance d(δ, ) between T δ and T as follows: Our next goal is to find a formula for r( ) = r α,λ,τ ( ) in terms of complex lengths of isometries stabilizing the singular geodesic ofĤ 3 . Definition 3.3. Let ϕ λ+iτ be an isometry ofĤ 3 , fixing the singular geodesicσ, with complex length λ + iτ . For ≥ λ, define the translation radius, denoted trad λ,τ ( ), to be the value of r such that ϕ λ+iτ translates all points of the form (r, ζ, θ) ∈Ĥ 3 by distance .
We also need to compute the translation radius of a loxodromic isometry ϕ λ+iτ acting on H 3 instead ofĤ 3 . Since angles in H 3 are only defined modulo 2π, this radius coincides with the translation radius of ϕ λ+i(τ mod 2π) acting onĤ 3 , namely trad λ, τ mod 2π ( ).
The usage (τ mod 2π) can be generalized to angles modulo other numbers.
Definition 3.4. Given a ∈ R and b > 0, we define (a mod b) to be In many situations, the translation radius can be computed in closed form.
The translation radius trad λ,τ ( ) is related to the radius of the -thin part of N = N α,λ, , but they are not necessarily identical. Example 3.9. Set α = 2π, λ = 0.1, and τ = π. Then N = N α,λ,τ is a quotient of H 3 . The generator ϕ = ϕ λ,τ of π 1 N ∼ = Z will translate any point x ⊂ H 3 along the invariant geodesic σ, and also rotate it by π about σ. When < 0.1, we have N ≤ = ∅, or equivalently trad λ,τ ( ) = −∞. When 0.1 ≤ ≤ 0.2, the map ϕ n for n ≥ 2 will translate any point of H 3 , including points of σ, by a distance larger than . Thus, the radius of N ≤ is governed by ϕ alone: that is, r( ) = trad λ,τ ( ) and the power for is 1.
Finally, ϕ n for n > 2 has both translational and rotational part larger than that of ϕ 2 , so ϕ n will definitely move any point of H 3 further than ϕ 2 . In fact, for every ≥ 0.201, the radius of N ≤ is governed by ϕ 2 : that is, r( ) = trad 2λ,2τ ( ) and the power for is 2. See Figure 3.
The above example is instructive in two ways. First, it illustrates that the power for is locally constant but jumps when crosses certain isolated values. Additional examples of this phenomenon are described in Section 4. Second, the displayed equations above show that while = 2injrad(x) is determined by taking a minimum distance over all nonzero powers of ϕ (see Lemma 2.5), the tube radius r( ) is determined by taking a maximum value of trad nλ,nτ ( ) over all nonzero powers. Proposition 3.10 makes this idea precise, for singular as well as nonsingular tubes.
Before moving on, we note that the power for can only jump upward as increases; see Remark 5.4. When is fixed but λ + iτ varies, the jumping behavior of powers follows the combinatorics of the Farey graph, as illustrated in Figure 1. The jumping behavior also explains why the function r α,λ,τ ( ) and the related function d α,λ,τ (δ, ) have points of non-differentiability, as visible in Figure 2.
We are now ready to give a formula for the radius of the -tube N ≤ . The following is the main result of this section. Proposition 3.10. Let N = N α,λ,τ be a model solid torus of cone angle α ≤ 2π. For any such that N ≤ = ∅, the radius r( ) = r α,λ,τ ( ) of the tube N ≤ can be computed as follows.
Remark 3.11. In each case of Proposition 3.10, it suffices to take a maximum over a finite set. Indeed, when nλ > , the translation length of ϕ n is sure to be larger than . Thus, by Remark 3.8, all values of n larger than /λ will contribute −∞ to the set over which we are taking a maximum, and can therefore be ignored. The integer n ∈ N ∪ {0} that realizes the maximum is the same as the power for , defined in Definition 2.8. This shows that r( ) is actually a maximum rather than a supremum.
Proof of Proposition 3.10. First, assume that N is nonsingular. In this case, the solid torus N can be described as N 2π,λ,τ = H 3 / ϕ λ+iτ ∼ = H 3 /Z. Let x ∈ N be a point such that 2injrad(x) = , and let x be a preimage in H 3 . By Lemma 2.5, we have Let m ∈ Z − {0} be a power realizing the minimum. Without loss of generality, we may assume m > 0. Then, for every n ∈ N, we have d( x, ϕ n x) ≥ d( x, ϕ m x). In other words, a point y ∈ H 3 that ϕ n moves by distance would have to be closer to the core geodesic than x is, hence trad mλ, mτ mod 2π ( ) ≥ trad nλ, nτ mod 2π ( ). (This includes the possibility that no such point y exists: that is, trad nλ,nτ mod 2π ( ) = −∞; see Remark 3.8.) We conclude that r 2π,λ,τ ( ) = trad mλ, mτ mod 2π ( ) = max n∈N trad nλ, nτ mod 2π ( ) . where both sides might be −∞ as in Remark 3.8. Now, assume that N is a singular tube of cone angle α < 2π. In this case, N can be described as N α,λ,τ =Ĥ 3 /Z 2 , where Z 2 = ϕ, ψ = ϕ λ+iτ , ψ iα . Let x ∈ N be a point such that 2injrad(x) = , and let x be a preimage inĤ 3 . As above, Lemma 2.5 describes as a minimum of d( x, η x) over all choices of η ∈ Z 2 − {0}. We begin by restricting the values of η that need to be considered.
Fix n > 0, and consider all isometries of the form η = ϕ n ψ m as m varies over Z. All of these isometries translate the singular geodesic σ ⊂Ĥ 3 by the same distance, so d( x, η x) will be smallest when the rotational angle is smallest (in absolute value). Equivalently, the translation radius will be largest when the rotational angle is smallest. Thus max m∈Z trad nλ,nτ +mα ( ) = trad nλ,nτ mod α ( ).

Examples demonstrating sharpness
In Section 7 below, we will prove lower bounds on the radius of an -tube in terms of and the core length λ. Quantitatively, the lower bound on cosh r( ) will be on the order of / √ λ. The following family of examples, suggested by Ian Biringer, shows that an O( / √ λ) bound is in fact optimal. As a consequence, we show in Theorem 4.6 that the lower bound of Theorem 1.1 is sharp up to additive error.
Proposition 4.1. For any n ≥ 4, let N n = N 2π,λ,τ be a nonsingular model solid torus whose core geodesic has complex length Then, for every in the range 1.016/n ≤ ≤ 0.3, the tube radius in N n satisfies Plugging the minimal and maximal values of into the above estimate shows that the radii appearing in Proposition 4.1 range from arccosh(1.016) = 0.1767 . . . to arccosh(0.3n), which goes to ∞ with n. We suspect that similar behavior cannot hold for very small radii.
The proof of Proposition 4.1 needs the following easy monotonicity result, which will be used repeatedly below.
Lemma 4.2. Let a, b ∈ R be constants, and consider the function Then g(x) is strictly increasing in x if b < a and strictly decreasing in x if a < b.
Proof. Compute the derivative. Alternately, consider g as a linear fractional transformation of RP 1 , which preserves orientation if and only if det 1 −a 1 −b > 0.
Proof of Proposition 4.1. Fix ≥ √ λ = 1/n. Let r( ) = r 2π,λ,τ ( ) be the radius of the -tube, as in Definition 3.1. We begin by proving a lower bound on r( ), using Proposition 3.10: We will eventually show that, for ≥ 1.016/n, the above inequality is actually equality. This amounts to showing that n is indeed the power for (see Definition 2.8). This requires a few estimates.
First, we get a tight two-sided estimate on the quantity in (4.3). Assume that 1/n = √ λ < ≤ 0.3. The monotonicity of the function h(x, y) in Lemma A.3 implies that Multiplying all of this by 2 /λ and taking square roots gives as claimed in the statement of the Proposition. Now, assume that ≥ 1.016/n. Then (4.3) and (4.4) combine to give which implies (4.5) tanh 2 r( ) > 0.0312.
Next, we claim that (for any ≥ λ) the only possible powers for are either 1 or n. This can be seen from Equation (3.7), substituting τ = 2π/n: When k ∈ nZ, the subtracted term is cos(2πk/n) = 1, hence constant, and the denominator is smallest for k = n. When k / ∈ nZ, we have cos(2πk/n) ≤ cos(2π/n), hence Lemma 4.2 gives Thus only k = 1 or k = n can give a maximal value of trad. Finally, we claim that n is indeed the power for . Let ϕ = ϕ λ,τ be the loxodromic isometry of H 3 that generates the deck group of N n . Fix r = r( ), and let T r be the lift to H 3 of the equidistant torus T r ⊂ N n . For x ∈ T r , let d 1 = d( x, ϕ x) and d n = d( x, ϕ n x). By Lemma 3.5, cosh d 1 = cosh λ cosh 2 r − cos τ sinh 2 r = cosh 1 n 2 cosh 2 r − cos 2π n sinh 2 r cosh d n = cosh(nλ) cosh 2 r − cos(nτ ) sinh 2 r = cosh 1 n cosh 2 r − cos(0) sinh 2 r hence cosh d n − cosh d 1 = cosh 1 n − cosh 1 n 2 cosh 2 r − 1 − cos 2π n sinh 2 r.
Using calculus, we check that when n ≥ 4, where the last inequality is by (4.5). Multiplying both sides by cosh 2 r gives hence cosh d n < cosh d 1 . This means that ϕ n translates points of T r by less than ϕ for all radii satisfying (4.5), hence n is the power for . We conclude that the inequality (4.3) is equality, as desired.
We use Proposition 4.1 to prove Theorem 4.6, below. Recall from Section 1.3 that the upper bound of Theorem 4.6 differs from the lower bound of Theorem 1.1 by an additive error of less than 2.2.

Distance between tubes: upper bound
In this section we prove the upper bound of Theorem 1.1. This result requires some lemmas, the first of which is also used in the lower bound. Proof. Let h = d(x, y). If h ≥ /2, there is nothing to prove. Thus we may assume that h < /2. By Definition 2.4, there is an embedded ball B = B /2 (y) that is isometric to a ball in H 3 . Since h < /2, we have x ∈ B. By the triangle inequality, there is an embedded ball The following lemma controls the type of isometry that realizes injectivity radius in singular tubes. Recall that T = ∂N ≤ ⊂ N denotes the equidistant torus consisting of points whose injectivity radius is exactly /2. Lemma 5.2. Consider a singular solid torus N = N α,λ,τ of cone angle α < 2π, and let 0 < δ < . Suppose that, for q ∈ T ⊂ N , the injectivity radius injrad(q) is realized by an elliptic isometry ψ iα (compare Lemma 2.5). Then, for every p ∈ T δ , the injectivity radius injrad(p) is also realized by the same elliptic isometry ψ iα .
Proof. There are two cases: α < π and α ≥ π. We consider the latter case first.
Suppose that π ≤ α < 2π, and let q ∈ T . By Proposition 3.10, we must have r( ) = /2, hence there is a point of intersection z ∈ Σ ∩ B /2 (q), where Σ is the singular core of N . Let α be the geodesic segment of length /2 from q to z.
We consider a, b, a , b to be constants in the following argument, because the integer n will stay fixed. Note that b < b and a < a . The inequality (5.3) can be rewritten as This, in turn, can be rewritten as Since a < a , Lemma 4.2 implies that g(x) = x−a x−a is a strictly decreasing function of x. Now, for any 0 < δ < , set y = cosh δ. We have We conclude that This means that for every n ∈ N, we have that trad nλ,nτ mod α (δ) is strictly smaller than trad 0,α (δ). Thus r(δ) = trad 0,α (δ), as desired.
Now, (5.5) combined with the monotonicity of the function g(x) of Lemma 4.2 (in the opposite direction compared to the last proof) will imply that (5.5) also holds with instead of δ. Since we do not need this statement, we omit the details.
We can now prove the following statement, which will quickly imply the upper bound of Theorem 1.1.
Proposition 5.6. Suppose 0 < δ < and 0 < α ≤ 2π. Then, in any model solid torus N = N α,λ,τ such that N ≤δ = ∅, we have Equality holds if and only if the injectivity radii of T δ and T are realized by the same loxodromic isometry, whose rotational part is trivial. In particular, if α = 2π and τ = 0, then equality holds.
Proof. We consider three cases, depending on the value of α and the power for . First, suppose that the power for is m ∈ N, hence the injectivity radius of T is realized by a loxodromic isometry ϕ m λ+iτ . In the following computation, using Proposition 3.10, the ellipsis (· · · ) denotes any elliptic terms that arise when α < 2π. Observe that the first inequality is equality precisely when m is also the power for δ. The second inequality is equality precisely when (mτ mod α) = 0: that is, when the realizing isometry has trivial rotational part. Next, suppose that the injectivity radius of T is realized by an elliptic isometry ψ iα , and furthermore 0 < α < π. Then Lemma 5.2 says that the injectivity radius of T δ is realized by the same elliptic. Thus we may compute as above: cosh r α,λ,τ ( ) cosh r α,λ,τ (δ) = cosh trad 0,α ( ) cosh trad 0,α (δ) The inequality is strict because 0 < α < π, hence cos α < 1. Finally, suppose that the injectivity radius of T is realized by an elliptic isometry ψ iα , and furthermore π ≤ α < 2π. Then Lemma 5.2 says that the injectivity radius of T δ is realized by the same elliptic. Furthermore, r( ) = /2 and r(δ) = δ/2. Thus cosh 2 (r α,λ,τ ( )) cosh 2 (r α,λ,τ (δ)) = cosh 2 ( /2) cosh 2 (δ/2) Again, the inequality is strict in this case.
We can now prove the upper bound of Theorem 1.1, including its sharpness.
Next, let us analyze when equality can hold. Assuming δ < , hence r(δ) < r( ), Lemma A.1 implies that the first inequality of (5.8) will be equality if and only if r(δ) = 0. But r(δ) ≥ δ/2 > 0 when N is singular, hence equality cannot hold for singular tori.
From now on, assume that α = 2π, hence N is nonsingular. By Lemma A.1, the first inequality of (5.8) will be equality precisely when r(δ) = 0, hence when λ = δ and the realizing isometry is a generator of π 1 (N ) = Z. By Proposition 5.6, the second inequality of (5.8) is equality precisely when this realizing isometry has rotational part τ = 0.

Euclidean bounds
We now begin working toward the lower bound of Theorem 1.1. In that argument, we will need to estimate distances on the torus T that forms the boundary of an -tube. The torus T ⊂ N inherits a Euclidean metric, as does its preimage T ⊂Ĥ 3 , and we may use that metric to measure the distance between points. In this section, we consider how the Euclidean distance between points on T relates to the actual hyperbolic distance between these points inĤ 3 .
For points p and q in an equidistant torus T = T r ⊂Ĥ 3 , define d E (p, q) to be the distance between them in the Euclidean metric on T r . Note that if p = (r, 0, 0) and q = (r, ζ, θ) in cylindrical coordinates, then (2.2) gives (6.1) d E (p, q) 2 = ζ 2 cosh 2 r + θ 2 sinh 2 r.
Lemma 6.2. Let T r ⊂Ĥ 3 be a plane at fixed distance r > 0 from the singular geodesicσ. Let p, q ∈ T r be points whose θ-coordinates differ by at most A ≤ π and whose ζ-coordinates differ by at most B. Then Proof. Suppose without loss of generality that the cylindrical coordinates of p are (r, 0, 0) and the coordinates of of q are (r, ζ, θ), where |θ| ≤ A. We can now compute using Lemma 3.5: If either ζ = 0 or θ = 0, equation (6.3) should be interpreted by substituting the limiting value of 1/2: The monotonicity of the functions in (6.4) leads to the following chain of inequalities: To get the lower bound of the lemma, we return to (6.3), and substitute the lower bound (1 − cos A)/(A 2 ) for both (1 − cos θ)/(θ 2 ) and (cosh ζ − 1)/(ζ 2 ). The upper bound follows similarly.
The last result should be interpreted in light of Lemma A.3: so long as d(p, q) is bounded above, (cosh d(p, q) − 1) is not too different from 1 2 d(p, q) 2 . Thus Lemma 6.2 can be interpreted as saying that the Euclidean and hyperbolic distances are quite similar. The next lemma makes this idea precise, in a special case.
Lemma 6.5. Let T r ⊂Ĥ 3 be a plane at fixed distance r > 0 from the singular geodesicσ. Let p, q ∈ T r be points whose θ-coordinates differ by at most A ≤ π. Suppose that d(p, q) ≤ 0.3. Plugging (6.6) and (6.7) into Lemma 6.2 gives , which simplifies to the desired result.

The depth of a tube
The next step in the proof of Theorem 1.1 is Proposition 7.1, which provides a lower bound on the radius of an -tube. Note that by Proposition 4.1, the estimate of Proposition 7.1 is sharp up to multiplicative error in cosh r( ), hence up to additive error in r( ).
Proposition 7.1. Let N = N α,λ,τ be a model solid torus whose core curve has cone angle 0 < α ≤ 2π and length 0 < λ ≤ 2.97. Suppose that N ≤ = ∅. Then the tube radius r( ) satisfies The proof breaks into separate cases, depending on whether N is singular or nonsingular. We will handle singular tubes first.
Lemma 7.2. Let N = N α,λ,τ be a model solid torus whose core has cone angle α < 2π. Then Proof. Let T r ⊂ N be the torus at radius r from the core, and let T r be is preimage inĤ 3 . By Equation (2.2), the area of T r is (7.3) area(T r ) = αλ sinh r cosh r = αλ 2 sinh 2r. Now, suppose that an (arbitrary) point x ∈ T r has injrad(x) = /2. Let x ∈ T r be a lift of x. By Lemma 2.5, every non-trivial deck transformation η ∈ Z 2 − {0} satisfies This includes elliptic isometries ofĤ 3 , because N is singular. Observe that Euclidean distance along T r is greater than hyperbolic distance, because tubes are strictly convex (compare Lemma 6.5). Thus This means x is the center of a Euclidean disk of radius /2, disjoint from all of its translates by the deck group. Projecting down to T r gives an embedded Euclidean disk of radius /2. Since a packing of R 2 by isometric disks has density at most π/(2 √ 3), we have Combining this result with Equation ( Lemma 7.5. Consider a nonsingular tube whose core geodesic has complex length λ + iτ , where 0 < λ ≤ 2.97. Then there is an integer m ≥ 1 such that Proof. This is a restatement of [9,Lemma 3.4]. To convert their result to the form stated above, one needs the identity which is readily verified. See [9,Equation (3.10)].
Using this, we can prove the nonsingular case of Proposition 7.1.
Proof of Proposition 7.1. If N is a nonsingular tube whose core has length λ ≤ 2.97, the proposition holds by Lemma 7.6. Meanwhile, if N is a singular tube whose core curve has cone angle α < 2π, Lemma 7.2 implies

Distance between tubes: lower bound
In this section, we prove the lower bound of Theorem 1.1. The proof breaks into two cases: shallow and deep. A tube is said to be shallow if its radius is bounded above by some constant denoted r max . Similarly, a tube is said to be deep if its radius is bounded below by some constant denoted r min . The optimal values of r min and r max will be determined later.
The following lemma gives a bound for shallow tubes.
To prove Lemma 8.2, we need to compute how fast the Euclidean injectivity radius changes.
Lemma 8.3. Let 0 < r < R, and consider equidistant tori T r , T R ⊂ N α,λ,τ . Let c r ⊂ T r be a rectifiable curve, and let c R ⊂ T R be the cylindrical projection of c r to T R . Then Proof. Let (r, ζ(t), θ(t)) be a parametrization of c r , where t ∈ [0, 1]. By (2.2), the distance element on T r satisfies ds 2 = cosh 2 r dζ 2 + sinh 2 r dθ 2 .
Thus we may compute that (c r ) = where the inequality in the next-to-last line is Lemma A.2.
A similar computation proves the other inequality in the lemma.
Proof of Lemma 8.2. Let x ∈ T δ ⊂ N . Then, by Lemma 2.5, there is a lift x ∈Ĥ 3 and a deck transformation η such that d( x, η x) = δ. Furthermore, η is a loxodromic if N is nonsingular.
The points x, η( x) are connected by a Euclidean geodesic arc c δ ⊂ T δ . Projecting back down to N , we have a Euclidean geodesic c δ whose length satisfies where the inequality is by Lemma 6.5. Let c ⊂ T be the cylindrical projection of c δ to T . Since every point of T has injectivity radius /2, we know that (c ) ≥ . Therefore, δ < (c ) 0.634 (c δ ) ≤ sinh r( ) 0.634 sinh r(δ) < e r( ) 2 · 0.634 · e −r(δ) · e r(δ) sinh r(δ) ≤ e r( )−r(δ) 1.268 · e r min sinh r min .
Here, the first inequality uses (8.4), the second inequality is Lemma 8.3, the third inequality uses the definition of sinh r, and the fourth inequality uses the monotonicity of e −r sinh r.
Taking logarithms completes the proof.
To complete the proof of Theorem 1.1, we need one more elementary lemma. Proof. It is clear from the definition that j(δ, ) is increasing in . Thus the maximum over Q occurs on the arc of ∂Q where = 0.3. On this arc, we compute ∂ ∂δ j(δ, ) and find that j(δ, ) has a single interior critical point at δ = 0.0093026 . . ., with maximal value j(δ, ) = 0.042357 . . ..
We can now complete the proof of the main theorem.
Proof of Theorem 1.1. The upper bound of the theorem is proved in Proposition 5.7.
We will use Lemma 8.5 to show that the lower bound of Lemma 8.2 is stronger than (8.6). We compute as follows, starting from Lemma 8.5: with equality if and only if r = 0 or r = s.
Note that we will have equality if and only if r = 0 or h = 0, as desired. Then f (x) strictly increasing in x, while h(x, y) is increasing in y and decreasing in x.