SPECTRAL THEORY, HAUSDORFF DIMENSION AND THE TOPOLOGY OF HYPERBOLIC 3-MANIFOLDS

RICHARD D. CANARY, YAIR N. MINSKY, AND EDWARD C. TAYLOR

Abstract. Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Λ(M) be the supremum of λ_{0}(N) where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of N. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that Λ(M) = D(M)(2 − D(M)) if M is not handlebody or a thickened torus. We characterize exactly when Λ(M) = 1 and D(M) = 1 in terms of the characteristic submanifold of the incompressible core of M.

1. Introduction

When a closed 3-manifold admits a hyperbolic structure, this structure is unique by Mostow’s rigidity theorem [26]. It follows that any invariant of the hyperbolic structure is automatically a topological invariant. One example of this is the hyperbolic volume, which agrees with Gromov’s simplicial norm (see [15]).

In this paper we will consider geometrically derived invariants for compact 3-manifolds with boundary whose interiors admit complete hyperbolic metrics of infinite volume. In this case the hyperbolic structure is not unique, and in fact, Thurston’s geometrization theorem together with the Ahlfors-Bers quasiconformal deformation theory guarantee that there is at least a one complex dimensional space of hyperbolic structures.

2 RICHARD D. CANARY, YAIR N. MINSKY, AND EDWARD C. TAYLOR

To obtain a topological invariant in this context, one may begin with a natural geometric invariant of a hyperbolic metric, and minimize (or maximize) it over the class of all hyperbolic metrics on a given 3manifold. In particular, given a hyperbolic 3-manifold N = H^{3}/Γ we will consider the bottom λ_{0}(N) of the L^{2} spectrum of the Laplacian, and the Hausdorff dimension d(N) of the limit set L_{Γ} of Γ. (See section 2 for more precise definitions). Although these two geometric invariants seem very different, the work of Patterson, Sullivan and others has established that they are closely related.

Date: August 12, 1996.

First author partially supported by a grant from the National Science Foundation and a fellowship from the Alfred P. Sloan Foundation. Second author partially supported by an NSF Postdoctoral Fellowship. Third author partially supported by a Rackham fellowship.

Call a compact, orientable 3-manifold M hyperbolizable if its interior admits a complete hyperbolic metric. This is a topological condition: Thurston’s geometrization theorem asserts that an orientable compact 3-manifold with non-empty boundary is hyperbolizable if and only if it is irreducible and atoroidal. Given a hyperbolizable M, we will define

Λ(M) = sup λ_{0}(N)

and

D(M) = inf d(N),

where N varies over the space of all hyperbolic 3-manifolds homeomorphic to the interior of M.

Very roughly speaking, we will find that D increases with topological complexity, and Λ decreases. The purpose of this paper is to establish some quantitative aspects of this intuition. In particular we will characterize exactly which manifolds have Λ(M) = 1, the maximal possible value.

Let us introduce some topological notation. A compact irreducible manifold M has an incompressible core, which is a (possibly disconnected) submanifold with incompressible boundary, from which M is obtained by adding 1-handles. A compact irreducible 3-manifold M with incompressible boundary is called a generalized book of I-bundles if one may find a disjoint collection A of essential annuli in M such that each component R of the manifold obtained by cutting M along A is either a solid torus, a thickened torus, or homeomorphic to an I-bundle such that ∂R ∩ ∂M is the associated ∂I-bundle.

The following is a “scorecard” for the basic facts about Λ and D for hyperbolizable manifolds.

M (hyperbolizable)  Λ  D  Relation

Handlebody or thick-

ened torus

1 0

Not solid or thickened torus, but ∂M is a union of tori 0 2

Λ = D(2 − D)

Incompressible core consists of generalized books of I-bundles

1 1

Any other manifold

with boundary

0 < Λ < 1 1 < D < 2

Our main theorems fill in the first two entries in the last two rows. The rest of the entries follow from known results, as explained in section 2.

Main Theorem I: Let M be a compact, orientable, hyperbolizable 3manifold. Λ(M) = 1 if and only if every component of the incompressible core of M is a generalized book of I-bundles. Otherwise, Λ(M) < 1.

Main Theorem II: Let M be a compact, orientable, hyperbolizable 3manifold which is not a handlebody or a thickened torus. D(M) = 1 if and only if every component of its incompressible core is a generalized book of I-bundles. Otherwise, D(M) > 1.

One could think of these results as analogues of the fact that the Gromov norm of a closed, irreducible 3-manifold M is zero if and only if there exists a collection T of incompressible tori in M such that each component of M − T is a Seifert fibre space. In particular, if M has incompressible boundary, then D(M) = 1 if and only if the Gromov norm of the double of M is 0.

2. Preliminaries

In this section we will more carefully define our invariants, derive their basic properties and summarize the proof of the main theorems.

2.1. Definitions. If N is a complete orientable hyperbolic 3-manifold, then it is isometric to the quotient of hyperbolic 3-space H^{3} by a group Γ of orientation-preserving isometries. Any orientation-preserving isometry extends continuously to a conformal transformation of the sphere at infinity S_{∞}^{2} of hyperbolic 3-space. We define the domain of discontinuity Ω_{Γ} to be the maximal open subset of S_{∞}^{2} on which Γ acts discontinuously. The limit set L_{Γ} of Γ is S_{∞}^{2} − Ω_{Γ}. We define d(N) to be the Hausdorff dimension of the limit set L_{Γ} of Γ.

Define λ_{0}(N) to be the largest value of λ for which there exists a positive C^{∞} function f on N such that Δf + λf = 0. (Here Δ = div ∘ grad denotes the Laplacian. See Sullivan [30] for a discussion of why this is equivalent to other definitions of λ_{0}(N).) Note that λ_{0} ≥ 0.

It is clear that if Ñ covers N then λ_{0}(N) ≤ λ_{0}(Ñ). In particular,

λ_{0}(N) ≤ λ_{0}(H^{3}) = 1

for any hyperbolic 3-manifold N.

If M is a compact, orientable, hyperbolizable 3-manifold then we let T T (M) denote the set of complete hyperbolic 3-manifolds homeomorphic to the interior of M (T T stands for “topologically tame”; see below). Our invariants can now be written as:

Λ(M) = sup{λ_{0}(N) | N ∈ T T (M)}

and

D(M) = inf{d(N) | N ∈ T T (M)}.

We will say that an element N = H^{3}/Γ of T T (M) is geometrically finite if there exists a finite collection P of incompressible annuli and tori in ∂M such that N̂ = (H^{3} ∪ Ω_{Γ})/Γ is homeomorphic to M − P.

We say that a hyperbolic 3-manifold is topologically tame if it is homeomorphic to the interior of a compact 3-manifold. This holds by definition for each hyperbolic 3-manifold in T T (M) and we remark that it is conjectured that every hyperbolic 3-manifold with finitely generated fundamental group is topologically tame.

2.2. Basic properties of Λ and D. Remarkably, λ_{0} and the Hausdorff dimension of the limit set are intimately related. The following theorem, which records that relationship, is due to Sullivan [29] in the case that N is geometrically finite. If N is topologically tame but not geometrically finite, then Canary [11] proved that λ_{0}(N) = 0, while Bishop and Jones [4] proved that if N has finitely generated fundamental group and is not geometrically finite then d(N) = 2.

Theorem 2.1. If N is a topologically tame hyperbolic 3-manifold, then λ_{0}(N) = d(N)(2 − d(N)) unless d(N) < 1, in which case λ_{0}(N) = 1.

Thus λ_{0} completely determines d unless d < 1. The situation when d ≤ 1 was analyzed by Sullivan [28] and Braam [8] in the case where N is convex cocompact, by Canary and Taylor [13] when N is geometrically finite, and for all hyperbolic 3-manifolds with finitely generated fundamental groups by Bishop and Jones [4]:

Theorem 2.2. Let M be a compact, orientable, hyperbolizable 3-manifold. If N ∈ T T (M) and d(N) < 1, then M is a handlebody or a thickened torus. If d(N) = 1, then M is either a handlebody or an I-bundle.

The above two theorems assure us that D(M) and Λ(M) are essentially the same invariant:

Corollary 2.3. If M is a compact, orientable, hyperbolizable 3-manifold, then Λ(M) = D(M)(2 − D(M)) unless M is a handlebody or a thickened torus.

Theorem 2.2 also has the following consequence:

Corollary 2.4. If M is a compact, orientable, hyperbolizable 3-manifold which is not a handlebody or a thickened torus, then D(M) ≥ 1.

We note that it follows from work of Beardon [2] that if M is a handlebody, then D(M) = 0 and therefore Λ(M) = 1. If M is a thickened torus and N = H^{3}/Γ ∈ T T (M), then L_{Γ} is a single point, so again D(M) = 0 and Λ(M) = 1. This completes the first row of the scorecard.

The next proposition completes the second row:

Proposition 2.5. Let M be a compact, orientable, hyperbolizable 3manifold which is not a solid torus or a thickened torus. Then D(M) = 2 if and only if any boundary component of M is a torus.

Proof. We first suppose that any boundary component of M is toroidal. This implies that if N ∈ T T (M) then N = H^{3}/Γ has finite volume (see Proposition D.3.18 in [3]). In this case L_{Γ} = S_{∞}^{2}, which implies that d(N) = 2 and hence D(M) = 2.

We now suppose that M has a non-toroidal boundary component. In this case, N has infinite volume for any N ∈ T T (M). Thurston’s geometrization theorem (see [24]) guarantees there is a geometrically finite manifold N in T T (M). Sullivan [29] and Tukia [37] proved that if N is geometrically finite and has infinite volume, then d(N) < 2. Hence D(M) < 2.

It is also useful to note that D(M) and Λ(M) behave monotonically under passage to covers.

Proposition 2.6. Let M and M′ be compact, orientable, hyperbolizable 3-manifolds, such that the interior of M′ covers the interior of M. Then D(M) ≥ D(M′) and Λ(M) ≤ Λ(M′).

Proof. If N = H^{3}/Γ is any hyperbolic 3-manifold homeomorphic to the interior of M, then it has a cover N′ = H^{3}/Γ′ which is homeomorphic to the interior of N′. Since Γ′ ⊂ Γ, L_{Γ′} ⊂ L_{Γ}, so d(N) ≥ d(N′). The assertion that D(M) ≥ D(M′) then follows immediately from the definition of our invariant D.

The proof of the assertion that Λ(M) ≤ Λ(M′) is similar.

2.3. Outline of proof of the main theorems. We will break the argument up into several steps. First we reduce to considering manifolds with incompressible boundary.

We say that M is obtained from two manifolds M_{0} and M_{1} by adding a 1-handle if M is obtained from M_{0}, M_{1} and D^{2} × [0, 1] by identifying D^{2} × {i} with an embedded disk in ∂M_{i} (for i = 0, 1.) M is said to be obtained from M_{0} by adding a 1-handle if M is obtained from M_{0} and D^{2} × [0, 1] by identifying D^{2} × {0} and D^{2} × {1} with disjoint embedded disks in ∂M_{0}. M is said to be obtained from {M_{1}, …, M_{n}} by adding 1-handles if it is obtained by applying the above two topological operations finitely many times using the manifolds {M_{1}, …, M_{n}} as building blocks.

Bonahon [5] and McCullough-Miller [22] showed that if M is a compact irreducible 3-manifold, then there exists a collection {M_{1}, …, M_{n}} of submanifolds of M such that M is obtained from {M_{1}, …, M_{n}} by adding 1-handles and each M_{i} has incompressible boundary. (The boundary of a 3-manifold M is incompressible if the fundamental group of any component injects in π_{1}(M)). The union ∪M_{i} is called the incompressible core of M. If M is a handlebody its incompressible core is a ball, and otherwise we will assume that no component of the incompressible core is a ball. With this convention, the incompressible core is unique up to isotopy.

In section 3 we will use work of Patterson [27] to show that one may analyze our invariants on M simply by studying the invariants of the components of the incompressible core of M:

Theorem 2.7. Let M be a compact, orientable, hyperbolizable 3-manifold. If {M_{1}, …, M_{n}} are the components of the incompressible core of M, then

Λ(M) = min{Λ(M_{1}), …, Λ(M_{n})}.

The idea is to “pull apart” the groups uniformizing the components of the incompressible core. For example, if M is obtained from M_{0} and M_{1} by adding a 1-handle, we first find N_{0} ∈ T T (M_{0}) and N_{1} ∈ T T (M_{1}) with λ_{0}(N_{i}) near Λ(M_{i}). We then construct a sequence {N^{i}} of hyperbolic 3-manifolds homeomorphic to the interior of M by removing half-spaces from N_{0} and N_{1} which lie farther and farther from a fixed point in each, and gluing the boundary planes together. Patterson’s result is used to show that {λ_{0}(N^{i})} converges to min{λ_{0}(N_{0}), λ_{0}(N_{1})} and hence that Λ(M) = min{Λ(M_{0}), Λ(M_{1})}.

We now turn to 3-manifolds with incompressible boundary, for which we will need a bit more notation. If X is an annulus or torus, we say that a map f : (X, ∂X) → (M, ∂M) is essential if f_{∗} : π_{1}(X) → π_{1}(M) is injective and f is not properly homotopic to a map of X into M with image in ∂M. We will say that an embedding f : R → M of an Ibundle R into M is admissible if f^{−1}(∂M) is the associated ∂I-bundle of R. We will say that an embedding f : R → M of a Seifert-fibred space R into M is admissible if f^{−1}(∂M) is a collection of fibres in ∂S. In either case, we will say that f is essential if f_{∗} : π_{1}(R) → π_{1}(M) is injective and whenever X is a component of ∂R − f^{−1}(∂M) then f|_{X} is an essential map of a torus or annulus into M.

A compact submanifold Σ of M is said to be a characteristic submanifold if Σ consists of a minimal collection of admissibly embedded essential I-bundles and Seifert fibre spaces with the property that every essential, admissible embedding f : R → M of a Seifert fibre space or Ibundle into M is properly homotopic to an admissible map with image in Σ. Jaco-Shalen [17] and Johannson [18] showed that every compact, orientable, irreducible 3-manifold with incompressible boundary contains a characteristic submanifold and that any two characteristic submanifolds are isotopic. Hence, we often speak of the characteristic submanifold Σ(M) of M. If M is hyperbolizable then every Seifert fibred component of Σ(M) is homeomorphic to either a solid torus or a thickened torus (see Morgan [24]).

We recall that a compact irreducible 3-manifold M with incompressible boundary is called a generalized book of I-bundles if one may find a disjoint collection A of essential annuli in M such that each component R of the manifold obtained by cutting M along A is either a solid torus, a thickened torus, or homeomorphic to an I-bundle such that ∂R ∩∂M is the associated ∂I-bundle. One may check that a compact, orientable, irreducible 3-manifold with incompressible boundary is a generalized book of I-bundles if the closure of any component of M − Σ(M) is homeomorphic to a solid torus or a thickened torus, and every component of Σ(M) is a solid torus, a thickened torus, or an I-bundle (see Proposition 4.3 in [14] for the corresponding characterization of books of I-bundles.)

In section 4 we prove:

Theorem 2.8. If M is a hyperbolizable generalized book of I-bundles, then Λ(M) = 1.

The proof is by explicit construction. We build a hyperbolic structure for M by piecing together structures on each of the I-bundle pieces, and show that if the parameters are chosen appropriately (essentially “pulling apart” the I-bundles) the Hausdorff dimension d can be made arbitrarily close to 1. The result then follows via the connection between D and Λ.

In section 6 we prove:

Theorem 2.9. If M is a compact, orientable, hyperbolizable 3-manifold with incompressible boundary which is not a generalized book of Ibundles, then Λ(M) < 1.

Here is an outline of the proof in the case that M is acylindrical. Arguing by contradiction, we assume the existence of a sequence N_{i} ∈ T T (M) with λ_{0}(N_{i}) → 1. By a theorem of Thurston the deformation space of M is compact and we can extract a limit manifold N which is homeomorphic to the interior of M with λ_{0}(N) = 1. This contradicts the assumption that M is not a handlebody, thickened torus or Ibundle, by Theorem 2.2.

If M is not acylindrical, we must apply the Jaco-Shalen-Johannson characteristic submanifold theory, and Thurston’s relative compactness theorem. In general, the limit manifold we obtain will be homeomorphic to a submanifold of M, rather than to M itself.

Theorems 2.7, 2.8 and 2.9 combine to give a complete proof of Main Theorem I. Main Theorem II follows immediately from Main Theorem I and corollary 2.3.

In section 7 we will make further comments and conjectures about the invariants and related quantities.

3. Reduction to the incompressible case

In this section we prove Theorem 2.7, which assures us that our invariants are determined by their value on the components of the incompressible core. We do this by showing that if M is obtained by adding a 1-handle to M_{0} and M_{1} (or by adding a 1-handle to M_{0}) then Λ(M) = min{Λ(M_{0}), Λ(M_{1})} (or Λ(M) = Λ(M_{0})).

The topological operation of adding a 1-handle is realized geometrically by Klein combination. Throughout this section we work in the ball model of H^{3}; the Euclidean boundary in this model is S^{2}, and \overline{H^{3}} = H^{3} ∪ S^{2}. If F ⊂ H^{3} is a (convex) fundamental polyhedron of Γ, then int(F) denotes the interior of F, \overline{F} denotes the Euclidean closure of F, and F^{c} = H^{3} − F. We refer the reader to sections IV.F and VI.A of [21] for a full discussion of fundamental polyhedra.

Theorem 3.1. (Klein Combination, Theorem VII.A.13 in [21]) Let Γ_{0} and Γ_{1} be discrete subgroups of PSL_{2}(C). Suppose there are (convex) fundamental polyhedra F_{i} for Γ_{i} (i = 0, 1), so that int(F_{0}) ∪ int(F_{1}) = H^{3}. Then the group Γ generated by Γ_{0} and Γ_{1} is discrete and is isomorphic to Γ_{0} ∗Γ_{1}. Moreover, F = F_{0} ∩F_{1} is a fundamental polyhedron for Γ.

If Γ_{0} and Γ_{1} satisfy the hypotheses of Theorem 3.1 we will say that they are Klein-combinable.

The main tool in the proof of Theorem 2.7 is a result of Patterson. Let Γ_{0} and Γ_{1} be two Klein-combinable groups and suppose N_{i} = H^{3}/Γ_{i}. The intuitive content of Patterson’s result is that, if we “pull Γ_{1} away from Γ_{0}” by a suitable sequence of conjugations h_{k}Γ_{1}h_{k}^{−1}, then λ_{0} of the quotient of the combination of Γ_{0} and h_{k}Γ_{1}h_{k}^{−1} approaches min{λ_{0}(N_{0}), λ_{0}(N_{1})}.

The statement we give is a version of Theorem 1 in [27]. (Patterson actually proves his result for the critical exponents of the groups involved. However, see [30], the critical exponent of Γ determines λ_{0}(H^{3}/Γ) and we have translated Patterson’s result into a result about λ_{0}.) Let |g′(x)| denote the Euclidean norm of the derivative of g at x, where g ∈ PSL_{2}(C) and x ∈ H^{3}. Let d(A_{1}, A_{2}) denote the Euclidean distance between sets A_{1} and A_{2} in \overline{H^{3}}.

Theorem 3.2. (Patterson) Let Γ_{0} and Γ_{1} be discrete, torsion-free subgroups of PSL_{2}(C) with convex fundamental polyhedra F_{0} and F_{1}. Let N_{i} = H^{3}/Γ_{i}. Suppose there is a sequence {h_{k}}_{k∈Z_{+}} in PSL_{2}(C) so that int(F_{0}) ∪ h_{k}(int(F_{1})) = H^{3} for all k and

\frac{sup_{w∈F_{1}^{c}} |hk^{′}(w)|}{d(F_{0}^{c}, h_{k}F_{1}^{c})} → 0

as k → ∞. Let Γ^{k} be the discrete group generated by Γ_{0} and h_{k}Γ_{1}h_{k}^{−1} and let N^{k} = H^{3}/Γ^{k}. Then

lim_{k→∞} λ_{0}(N^{k}) = min{λ_{0}(N_{0}), λ_{0}(N_{1})}.

We begin by studying the case where M is obtained from M_{0} and M_{1} by adding a 1-handle. Combining Theorems 3.1 and 3.2, we obtain:

Proposition 3.3. Let M_{0} and M_{1} be two compact, orientable, hyperbolizable 3-manifolds. If M is hyperbolizable and is obtained from M_{0} and M_{1} by adding a 1-handle then Λ(M) = min{Λ(M_{0}), Λ(M_{1})}.

Proof. We first note that Proposition 2.6 implies that

Λ(M) ≤ min{Λ(M_{0}), Λ(M_{1})},

since the interior of M is covered by both the interior of M_{0} and the interior of M_{1}.

Without loss of generality assume Λ(M_{0}) = min{Λ(M_{0}), Λ(M_{1})}. Then Proposition 2.5 ensures that Λ(M_{0}) > 0. Fix a positive ϵ < Λ(M_{0}), and choose N_{i} = H^{3}/Γ_{i} ∈ T T (M_{i}) so that λ_{0}(N_{1}) ≥ λ_{0}(N_{0}) > Λ(M_{0}) − ϵ > 0. Canary [11] showed that if a complete hyperbolic 3manifold is topologically tame but not geometrically finite, then λ_{0} = 0. Therefore, N_{0} and N_{1} are both geometrically finite.

It follows that there exist homeomorphisms ψ_{i} : N̂_{i} → M_{i}−P_{i}, where N̂_{i} = (H^{3} ∪ Ω_{Γ_{i}})/Γ_{i} and P_{i} is a collection of disjoint incompressible annuli and tori in ∂M_{i}. Because M is hyperbolizable, the attaching disks of the 1-handle are not contained in toroidal boundary components of M_{i}. Hence, we may choose disks D_{i} in M_{i}−P_{i}, such that M is obtained from M_{0} and M_{1} by attaching a 1-handle to the disks D_{0} and D_{1}. We can also assume that the pre-images ψ_{i}^{−1}(D_{i}) are “round” disks (i.e. they are the quotients of round disks in Ω_{Γ_{i}}.)

We choose lifts D̃_{i} of ψ^{−1}(D_{i}) to Ω_{Γ_{i}}. We may assume, by conjugating Γ_{1}, that the round disks D̃_{0} and D̃_{1} intersect only along their common boundary circle J. One can find (convex) fundamental polyhedra F_{i} for Γ_{i} such that D̃_{i} are contained in the interiors of the intersection of the Euclidean closures of F_{i} with S^{2}. Therefore H_{i} ⊂ int(F_{i}), where H_{i} denotes the closed half-space for which \overline{H_{i}} ∩ S^{2} = D̃_{i}. Thus, Γ_{0} and Γ_{1} are Klein-combinable. Since F_{0}∩F_{1} is a fundamental polyhedron for the group Γ generated by Γ_{0} and Γ_{1}, we see that N = H^{3}/Γ ∈ T T (M).

Fix points z_{i} in the interior of D̃_{i}. Let γ be a hyperbolic Möbius transformation with z_{0} as its attracting fixed point and z_{1} as its repelling fixed point. We may further choose γ so that γ(D̃_{0}) is contained in the interior of D̃_{0}. We will apply Theorem 3.2 to the sequence {h_{k} = γ^{k}}.

Note that γ^{k}(D̃_{0}) ⊂ γ(D̃_{0}) ⊂ intD̃_{0} for all k ≥ 1 which implies that γ^{k}(H_{0}) ⊂ γ(H_{0}) ⊂ H_{0}. Since γ^{k}(F_{1}) is a fundamental polyhedron for γ^{k}Γ_{1}γ^{−k} and γ^{k}(F_{1})^{c} is contained in γ^{k}(H_{0}) ⊂ int(H_{0}) ⊂ F_{0}, we see that Γ_{0} and γ^{k}Γ_{1}γ^{−k} are Klein-combinable and that F_{0} ∩ γ^{k}(F_{1}) is a fundamental polyhedron for the group Γ^{k} generated by Γ_{0} and γ^{k}Γ_{1}γ^{−k}. In particular, one may readily observe that N^{k} = H^{3}/Γ_{k} ∈ T T (M) for all k.

Since F_{0}^{c} ⊂ H_{1} and γ^{k}(F_{1})^{c} ⊂ γ^{k}(H_{0}) ⊂ γ(H_{0}) ⊂ int(H_{0}), we see that d(F_{0}^{c}, γ^{k}(F_{1}^{c})) ≥ δ for all k ≥ 1, where δ = d(H_{1}, γ(H_{0})) > 0.

One may easily check that {(γ^{k})′} converges to 0 uniformly on all compact subsets of \overline{H^{3}} − \{z_{1}\}. Since F_{1}^{c} ⊂ H_{0} and \overline{H_{0}} is a compact subset of \overline{H^{3}} − \{z_{1}\}, then sup_{w∈F_{1}^{c}} |(γ^{k})′(w)| converges to 0.

We may combine the observations above to establish that

\frac{sup_{w∈F_{1}^{c}} |(γ^{k})′(w)|}{d(F_{0}^{c}, h_{k}F_{1}^{c})} → 0.

Theorem 3.2 then allows us to conclude that

lim_{k→∞} λ_{0}(N^{k}) = min{λ_{0}(N_{0}), λ_{0}(N_{1})} ≥ Λ(M_{0}) − ϵ.

Therefore, Λ(M) ≥ min{Λ(M_{0}), Λ(M_{1})} − ϵ. Since ϵ can be arbitrarily small, this completes the proof of Proposition 3.3.

We will also need the following direct analogue of Theorem 1 in [27], which can be deduced from Patterson’s arguments.

Theorem 3.4. (Patterson) Let Γ_{0} be a torsion-free discrete subgroup of PSL_{2}(C) with convex fundamental polyhedron F_{0}, and let {h_{k}} be an infinite sequence in PSL_{2}(C). Suppose that F_{k} is a convex fundamental polyhedron for ⟨h_{k}⟩ such that int(F_{0})∪int(F_{k}) = H^{3} for all k, and there exists a δ > 0 so that d(F_{0}^{c}, F_{k}^{c}) ≥ δ. Also, assume there exists a w lying in F_{0} ∩ F_{k} for all index k, such that for any fixed s > 0,

∑_{j≠0} |(h_{k}^{j})′(w)|^{s} → 0

as k → ∞. Denote by Γ^{k} the discrete group generated by Γ_{0} and ⟨h_{k}⟩, and let N_{0} = H^{3}/Γ_{0} and N^{k} = H^{3}/Γ^{k}. Then

lim_{k→∞} λ_{0}(N^{k}) = λ_{0}(N_{0}).

Theorem 3.4 allows us to handle the case where M is obtained from M_{0} by adding a 1-handle.

Proposition 3.5. Let M_{0} be a compact, irreducible, hyperbolizable 3manifold with non-empty boundary. If M is a compact hyperbolizable 3-manifold obtained by adding a 1-handle to M_{0}, then Λ(M) = Λ(M_{0}).

Proof. As in Proposition 3.3, we observe that Proposition 2.6 implies that Λ(M) ≤ Λ(M_{0}) and Proposition 2.5 guarantees that Λ(M_{0}) > 0.

Fix a positive ϵ < Λ(M_{0}), and choose N_{0} ∈ T T (M_{0}) such that λ_{0}(N_{0}) ≥ Λ(M_{0})−ϵ. As before, by [11], N_{0} is necessarily geometrically finite. Thus, there exists a collection P of disjoint incompressible annuli and tori in ∂M_{0} and a homeomorphism ψ : N̂_{0} → M_{0} − P, where N̂_{0} = (H^{3} ∪ Ω_{Γ_{0}})/Γ_{0}. Since M is hyperbolizable, the attaching disks of the 1-handle are not contained in toroidal boundary components of M. Hence, we may choose disks D_{0} and D_{1} in ∂M_{0} − P such that M is formed by attaching a 1-handle to the disks D_{0} and D_{1}. Since the interior of F̄ ∩ Ω_{Γ_{0}} is a fundamental domain for the action of Γ_{0} on Ω_{Γ_{0}} (Proposition VI.A.3 in [21]), we may further choose D_{0} and D_{1} so that there are lifts D̃_{0} and D̃_{1} of ψ^{−1}(D_{0}) and ψ^{−1}(D_{1}) which are round disks in the interior of F̄ ∩ Ω_{Γ_{0}}.

Find a loxodromic element γ that takes the exterior of D̃_{0} to the interior of D̃_{1}. Let H_{i} be the closed half-spaces whose Euclidean closures intersect S^{2} in D̃_{i}. Then the region F_{k} = H^{3} − (H_{0} ∪ γ^{k−1}(H_{1})) is a fundamental polyhedron for ⟨γ^{k}⟩. Since int(F_{k}) ∪ int(F_{0}) = H^{3}, Γ_{0} and ⟨γ^{k}⟩ are Klein-combinable and F_{k} ∩ F_{0} is a convex fundamental polyhedra for the group Γ^{k} generated by Γ and ⟨γ^{k}⟩. It is now easy to check that N_{k} = H^{3}/Γ^{k} ∈ T T (M).

We will apply Theorem 3.4 with {h_{k} = γ^{k}}. Let δ = d(F_{0}^{c}, H_{0}∪H_{1}) > 0 (recall H_{i} ⊂ F_{0}). Because the H_{i} are disjoint and F_{k}^{c} ⊂ H_{0} ∪H_{1}, then

d(F_{0}^{c}, F_{k}^{c}) ≥ δ

for all k.

Fix w ∈ F_{0} ∩ F_{k} for all k > 0 and fix s > 0. It is well-known that ∑_{j≠0} |(γ^{j})′(w)|^{s} is finite. It follows immediately that ∑_{j≠0} |(h_{k}^{j})′(w)|^{s} = ∑_{j≠0} |(γ^{jk})′(w)|^{s} converges to 0 as k converges to ∞.

Thus, Theorem 3.4 implies that

lim_{k→∞} λ_{0}(N^{k}) = λ_{0}(N_{0}).

Therefore, Λ(M) > Λ(M_{0}) − ϵ. Since ϵ > 0 was chosen arbitrarily, we have completed the proof of proposition 3.5.

Notice that one need only apply Propositions 3.3 and 3.5 finitely many times in order to prove Theorem 2.7.

4. Generalized books of I-bundles

In this section we will prove Theorem 2.8, which says that Λ(M) = 1 for any hyperbolizable generalized book of I-bundles M. The key step in the proof is:

Theorem 4.1. If M is a hyperbolizable generalized book of I-bundles then for any α > 1, there exists a hyperbolic manifold N homeomorphic to int(M) with d(N) < α.

Proof of Theorem 2.8. Let M be a hyperbolizable generalized book of I-bundles which is not a thickened torus. Theorem 4.1 implies that D(M) ≤ 1. On the other hand, since M is not a handlebody or a thickened torus, Corollary 2.4 guarantees that D(M) ≥ 1. Hence, D(M) = 1 and we conclude that Λ(M) = 1 by applying Corollary 2.3. If M is a thickened torus we have already observed that Λ(M) = 1.

The remainder of the section is taken up with the proof of Theorem 4.1.

Proof of Theorem 4.1. The characteristic submanifold of M is a union of solid tori, thickened tori, and I-bundles whose bases have negative Euler characteristic. For each I-bundle the subbundle over the boundary of the base surface is a union of annuli, which are glued to the boundary of a solid torus or thickened torus (for a thickened torus, note that only one of its boundaries participates in the gluing). The union of the bases of the I-bundles, with boundaries glued together inside each solid torus, and torus boundaries of the thickened tori, comprise a “spine” for M, that is a 2-complex of which M is a regular neighborhood.

We shall put a hyperbolic structure on the interior of M, for which each I-bundle base determines a Fuchsian or extended Fuchsian group, and each thickened torus corresponds to a rank-2 parabolic group. (We recall that a discrete subgroup of PSL_{2}(C) is Fuchsian if it preserves a half-space in H^{3} and extended Fuchsian if it has a Fuchsian subgroup of index 2. In either case there is a totally geodesic hyperplane preserved by the group.) By changing the parameters of this construction we will obtain Hausdorff dimension arbitrarily close to 1.

For each solid torus, the cores of annuli glued to it describe some number m of parallel (p, q) curves, for some relatively prime p, q (where (1, 0) denotes a meridian and p is well-defined mod q). Consider the following hyperbolic structure on this solid torus: Begin with a geodesic L in H^{3} which is the boundary of mq half-planes equally spaced around it (more generally the angles between them can vary, but we will avoid this for ease of exposition). Let γ be a loxodromic with axis L, translation distance ℓ/q, for some (small) ℓ > 0, and rotation angle 2πp/q. The quotient of a neighborhood of L by γ is a solid torus, which the quotients of the half-planes meet in a collection of annuli with boundaries glued together at the core. The intersection of these annuli with the torus boundary give the m desired (p, q) curves.

For each thickened torus we choose (large) d > 0 and consider a horoball in H^{3} with a rank 2 parabolic group acting so that a fundamental domain on the boundary is a rectangle with one sidelength μ_{0} and one sidelength md > 0. In the horoball we consider m planes orthogonal to the boundary, parallel to the μ_{0} side, and equally spaced (by distance d along the boundary). In the quotient these give m parallel cusps with boundary length μ_{0}. Here μ_{0} denotes a fixed number less than the Margulis constant for H^{2}.

Choose a list of parameters {ℓ_{i}} for the solid tori and {d_{i}} for the thickened tori. For each base surface S, let S′ denote S minus the boundary components that attach to thickened tori, and choose a finitearea hyperbolic structure on S′ so that a neighborhood of each missing boundary component is a cusp, and each remaining boundary component that glues to a solid torus with parameter ℓ_{i} is a geodesic of length ℓ_{i}. For each base surface we can find a Fuchsian or extended Fuchsian group such that the convex core of its quotient realizes the given hyperbolic structure. (The convex core of a hyperbolic manifold is the quotient of the convex hull of the limit set by the associated group action.) Note that the boundary components correspond to pure translations. We then truncate each cuspidal end so that the boundaries corresponding to thickened tori are horocycles of length μ_{0}. We may obtain an incomplete structure on each I-bundle by considering the embedding of our truncated region in the full quotient of the associated Fuchsian or extended Fuchsian group.

For each solid torus we can then identify neighborhoods of the corresponding boundaries of I-bundle bases to the annuli arranged around its core, and for each thickened torus we can glue the horocycles to the boundaries of the cusps embedded in the horoball. This extends consistently to the thickenings of the I-bundle bases so that we obtain an (incomplete) hyperbolic structure on the interior of M, in which each Ibundle base is totally geodesic. With proper choice of the parameters, we will show that this gives rise to a complete structure.

The developing map for this structure maps the universal cover M̃ to H^{3} by a locally isometric immersion (see e.g. Benedetti-Petronio [3] §B.1). Let Γ denote the holonomy group. Each component of the lift S̃ of a base surface S maps to a totally geodesic subset of H^{3}. These subsets, which we will call flats, are arranged in a “tree”, in this sense: For a given flat F, at each lift of a geodesic boundary of its base surface there is a collection of mq − 1 other flats, equally spaced (where m is the number of annuli glued to the corresponding torus, and (p, q) describes the slopes of these annuli, as above). At each parabolic fixed point corresponding to one of the boundaries glued to a thickened torus, there is an bi-infinite sequence of other flats, arranged with equal spacing around a horoball based at this point. The corresponding graph of adjacencies is a tree (of infinite valence).

4.1. Discrete holonomy. Let ℓ_{0} = max ℓ_{i} and d_{0} = min d_{i}. Let θ_{0} = min 2π/q_{i}m_{i} where {m_{i}} and {(p_{i}, q_{i})} describe the gluings for the solid tori. We will show that, if ℓ_{0} is sufficiently small and d_{0} sufficiently large, the holonomy group Γ is discrete, and the quotient manifold is homeomorphic to M.

We first make the following geometric observation, which is a standard type of fact for broken geodesics in H^{n}.

Lemma 4.2. Given θ ∈ (0, π] there exists K ≥ 0 such that the following holds. Let γ be a broken geodesic in H^{3} composed of a chain of n segments γ_{1}, …, γ_{n} of lengths k_{i} > K that meet at angles θ_{i} ≥ θ. Let P_{i} denote the orthogonal bisecting plane to γ_{i}. Then the P_{i} are all disjoint, and each P_{j} separates P_{i} and γ_{i} from P_{k} and γ_{k} whenever i < j < k. Furthermore dist(P_{i}, P_{i+1}) ≥ \frac{1}{2}(k_{i} + k_{i+1}) − K.

Proof. Choose K by the formula

cosh^{2} K/2 = \frac{2}{1 − cos θ}.

A little hyperbolic trigonometry shows that if two segments meet at their endpoints at angle θ then the planes orthogonal to the segments at a distance K/2 from the intersection point meet at a single point at infinity, and if the angle is greater than θ the planes are disjoint.

Now consider for each segment of γ the family of planes orthogonal to it, excluding the ones closer than K/2 to either endpoint (a nonempty family since k_{i} > K). The planes meeting any segment thus separate the planes meeting the previous segment from those meeting the next segment, and the distance between the first and last plane for segment i is k_{i} − K. The statement for the bisecting planes follows from this.

Recall that the μ-thin part of a flat F denotes the points where some element of the stabilizer of F acts with translation μ or less. If μ is smaller than the Margulis constant, this set consists of a union of disjoint pieces, each of which is either a horodisk around a parabolic fixed point or a neighborhood of an axis of a translation. The μ-thick part is the complement of the μ-thin part.

Given any two points x, y in two flats F, F′, let F = F_{1}, …, F_{n} = F′ denote the sequence of flats in the tree connecting them. Each F_{i} and F_{i+1} share either a geodesic boundary or a parabolic fixed point at infinity, called F_{i} ∩ F_{i+1} in either case. There is a chain of geodesics {α_{i}} connecting x to y such that α_{i} ⊂ F_{i}, and α_{i} meets α_{i+1} at F_{i}∩F_{i+1} (possibly at infinity). The chain is uniquely determined by the condition that, whenever F_{i} ∩ F_{i+1} is a geodesic boundary, α_{i} meets it orthogonally. Whenever α_{i}, α_{i+1} meet in a parabolic point, adjust them as follows: Truncate each at the point where it enters the μ_{1}-horoball of the corresponding parabolic group (μ_{1} < μ_{0} will be determined shortly), and join the new endpoints with a geodesic, which we note makes an angle greater than π/2 with α_{i} and α_{i+1}. When i = n − 1, α_{n} may be entirely contained in the μ_{1}-horoball, and in that case we remove α_{n} entirely and join y directly to the truncated α_{n−1}. Call the resulting chain of geodesics γ_{x,y}.

Suppose that x is in the μ_{0}-thick part of F. We claim that, given any k, if ℓ_{0} and μ_{1} are sufficiently small and d_{0} is sufficiently large, each segment of γ_{x,y}, except possibly the last, has length at least k. By the collar lemma for hyperbolic surfaces, if ℓ_{0} and μ_{1} are sufficiently small then the μ_{0}-thick part of each quotient surface is separated from its boundary by at least c log μ_{0}/ℓ_{0}, and from the μ_{1}-thin parts of the cusps by c log μ_{0}/μ_{1}, for a fixed constant c. This bounds from below the length of each segment in γ_{x,y}, except possibly the last segment containing y, and the additional segments added in horoballs. Each segment of the latter type has length at least c log d_{0}μ_{1}/μ_{0} (for a constant c) since the horospherical distance between flats on the boundary of the μ_{0}-horoball is a multiple of d_{0} by construction. Thus, choosing d_{0} large enough (after the choice of μ_{1} is made) this gives a high lower bound for the horoball segments, and establishes our claim.

Any two segments meet at angle at least θ = min{θ_{0}, π/2}, so let K = K(θ) be the constant given in Lemma 4.2 and suppose k ≥ 2K and ℓ_{0}, d_{0} and μ_{1} are determined as above. Lemma 4.2 then provides a sequence of planes with definite spacing that separate x from y.

In particular we can deduce that any two flats which are non-adjacent in the tree are disjoint, and more generally, fixing x in the μ_{0}-thick part of F, for any path F = F_{1}, …, F_{n} of successively adjacent flats in the tree, that

dist(x, F_{n}) ≥ (n − 2)(k − K) + k − K/2.

In particular the entire tree of flats is (properly) embedded in H^{3}, and therefore Γ is discrete.

It remains to show that N = H^{3}/Γ is homeomorphic to M. Bonahon’s theorem [6] guarantees that N is topologically tame (we could also deduce this directly by showing that Γ is geometrically finite). Since a neighborhood of the tree of flats embeds, it must be the homeomorphic developing image of a neighborhood of the lift to the universal cover of the spine of M. It follows that M embeds in N, by a map which is a homotopy equivalence. By a theorem of McCullough-Miller-Swarup [23], this implies that N is homeomorphic to the interior of M.

Remark: Another approach to this construction is by means of Klein-Maskit combinations (see Maskit [20].)

4.2. Hausdorff dimension. We next show that, with further restrictions on ℓ_{0} and d_{0}, we can obtain upper bounds on Hausdorff dimension. This will be done directly, by exhibiting an appropriate family of coverings.

Choose one flat F_{0} as the root of the tree. Choose a point x_{0} in the μ_{0}-thick part of F_{0}. Normalize the picture in the upper half-space model so that the plane H_{0} containing F_{0} is a hemisphere meeting the complex plane in the unit circle C_{0}, and so that x_{0} is the point (0, 0, 1). Each child F′ of F_{0} in the tree structure is of one of two types: Type (1): if F′ meets F_{0} along a geodesic L, F′ is contained in a half-plane H′ which meets the complex plane in an arc C′ of a circle. There are a finite number (at most 2π/θ_{0}) of other flats adjoined at L. Type (2): If F′ meets F_{0} along a parabolic fixed point, it is part of a sequence of flats {F_{n}}_{n∈Z} meeting at that point, where F_{n} are all children of F_{0} for n ≠ 0. These meet the plane in a family of concentric circles {C_{n}} tangent at the same point. These divide into C_{0} itself, and the circles outside and inside. Call the set of outer ones (and of inner ones) an “earring”.

The same description holds for the children of any flat. We thus get a family of circular arcs {C} arranged in a tree structure with root C_{0}. (Note that for type (1) flats the arc is just the portion of a circle where a half-plane meets C, whereas for type (2) it is a whole circle.) For any C let s(C) denote the set of its children, and similarly s^{2}(C) = s(s(C)), etc. Let r(C) denote the diameter of C, which of course is uniformly comparable to the length of C.

The limit set L_{Γ} of Γ is contained in the closure L̂_{Γ} of the union of these arcs C.

Fix any positive ρ < 1/2. We claim that we can choose the parameter d_{0} sufficiently large and ℓ_{0} sufficiently small, so that the following holds (where c_{0} is a fixed constant):

• For any C and D ∈ s(C),

r(D) ≤ ρr(C).  (4.1)

• If D_{1}, D_{2}, … are the nested circles in an earring, with D_{1} the outermost, then

r(D_{n}) ≤ c_{0} \frac{r(D_{1})}{n}.  (4.2)

Proof of (4.1). Let μ_{2} < μ_{0} be a constant to be chosen later. Let H and H′ be the hemispheres containing C and D, respectively. They meet either in a geodesic g or at a parabolic fixed point. Let J(H, H′) denote the component of the μ_{2}-thin part associated to the intersection in either case. Let x be the top point of H (in the upper half-space). Suppose that x is outside J(H, H′). Then the geodesic chain γ_{x,y} for any y ∈ H′, constructed as in §4.1 but with μ_{2} taking the place of μ_{0}, has initial segment γ_{1} of length at least k, where k can be made arbitrarily large by making ℓ_{0}/μ_{2} and μ_{1}/μ_{2} small, and d_{0} large. Given these choices, we conclude via Lemma 4.2 that all of H′ is separated from x by the bisecting hemisphere of γ_{1}, which is distance at least k/2 from x, and is thus of Euclidean diameter at most ce^{−k/2} diam(H) for a fixed c. This gives the desired bound on r(D), if k is chosen so that ce^{−k/2} ≤ ρ.

It remains to show that, with appropriate choice of μ_{2}, the top x of each H is outside J(H, H′) for any child H′ of H. For the root of the tree this holds by our normalization. We argue by induction. Suppose that H_{1}, H_{2}, H_{3} are hemispheres such that H_{i} is the parent of H_{i+1} and x_{i} are the tops of H_{i}. If the inductive hypothesis holds for H_{1} then, in particular, diam(H_{2}) ≤ diam(H_{1}) by the above paragraph.

It follows that dist(x_{2}, H_{1}) is bounded from above by a fixed number a. However, if x_{2} were contained in J(H_{2}, H_{3}) then there would be a separation between x_{2} and J(H_{1}, H_{2}) of at least c log(μ_{0}/μ_{2}), since J(H_{1}, H_{2}) and J(H_{2}, H_{3}) meet H_{2} in two distinct, and hence disjoint, components of the μ_{0} thin part. Assuming that μ_{2} is sufficiently short this distance is long enough that we can construct a geodesic chain γ_{x_{2},y} from x_{2} to any y ∈ H_{1} with long initial segment, and apply Lemma 4.2 to conclude dist(x_{2}, H_{1}) > a, a contradiction. Thus there is an a-priori choice of μ_{2} which guarantees that x_{2} will be outside J(H_{2}, H_{3}), and we are done by induction.

Proof of (4.2). We observe that since, by (4.1), D_{1} is at most half the size of the parent C, we may re-normalize, by a Möbius transformation whose derivative is within a universally bounded ratio of a constant on all of the D_{i}, so that C becomes a straight line meeting the D_{i} at the origin. The D_{i}, and C, are then taken by the map z ↦ 1/z to a sequence of equally spaced parallel lines. An easy computation gives (4.2), where the constant c_{0} comes from the initial re-normalization.

We now claim the following, for any 2 ≥ α > 1:

∑_{D∈s(C)} r(D)^{α} ≤ a_{0}ρ^{α−1} \frac{ζ(α)}{2^{α−1} − 1}r(C)^{α}  (4.3)

where a_{0} is a fixed constant and ζ(α) = ∑ 1/n^{α} is the usual Zeta function.

To prove this consider first the children of type (1): these are arranged in groups of bounded number which subtend a common interval on C, and any two such intervals are disjoint. The lower bound θ_{0} on the angle at which any such child meets C implies that its diameter is comparable to the diameter of the interval. The sum of lengths of intervals is at most the length of C, so there is some constant a_{1} such that ∑ r(D) ≤ a_{1}r(C), for D of type (1). Since also r(D) ≤ ρr(C), we make the following observation: If ∑ x_{i} ≤ ax and each x_{i} ≤ ρx, then ∑ x_{i}^{α} ≤ ∑ x_{i}(ρx)^{α−1} ≤ aρ^{α−1}x^{α}. Thus we can bound the contribution of type (1) children by:

∑_{type 1} r(D)^{α} ≤ a_{1}ρ^{α−1}r(C)^{α}.  (4.4)

For the children of type (2) the sum of lengths is infinite and we must take more care. Consider first an earring D_{1}, D_{2}, … with D_{1} outermost. By (4.2), we have

∑_{n=1}^{∞} r(D_{n})^{α} ≤ c_{0}^{α}ζ(α)r(D_{1})^{α}.  (4.5)

At each parabolic point p on C there are two earrings (inside and outside the circle). Let D_{p} denote the outermost circle of the outside earring. Clearly it just remains to bound ∑_{p} r(D_{p})^{α}.

Note first that all the D_{p} are disjoint, by the argument of §4.1 showing that the tree of flats embeds. It follows, we claim, for any δ > 0, that

∑_{\frac{r(Dp)}{r(C)} ∈[δ/2,δ]} r(D_{p})^{α} ≤ a_{2}δ^{α−1}r(C)^{α}.  (4.6)

Each D_{p} projects radially to an interval on C, and the condition that diameters lie in [r(C)δ/2, r(C)δ], together with disjointness of the D_{p}, means that these intervals cover C with multiplicity at most 2. This implies ∑ r(D_{p}) ≤ a_{2}r(C) for this subset with some constant a_{2}, and (4.6) follows, using the same observation as for the type (1) children.

Summing over δ = ρ/2^{k} for k = 0, 1, …, we obtain a bound for the sum over all outer circles of (outside) earrings:

∑_{p} r(D_{p})^{α} ≤ a_{2}r(C)^{α} ∑_{k=0}^{∞} (\frac{ρ}{2^{k}} )^{α−1}

≤ \frac{2a_{2}ρ^{α−1}}{2^{α−1} − 1}r(C)^{α}.  (4.7)

The same argument works for the outer circles of earrings contained inside C, doubling our bound. Combining with the bound (4.5) for the sum over each earring, we obtain the inequality (4.3) for the sum over type (2) children. Now combining with (4.4) we get (4.3) over all the children of C (note that in (4.4) the factor ζ(α)/(2^{α−1} − 1) does not appear, but this does not matter since it has a positive lower bound when α ∈ (1, 2] ).

We shall now define, for any ϵ_{0} > 0, a covering of the closure L̂_{Γ} of the union of arcs C, by balls of radius less than or equal to ϵ_{0}.

For each arc C we shall inductively assign a number ϵ(C) with which to cover C. Let ϵ(C_{0}) = ϵ_{0}. For a child D of C let

ϵ(D) = \frac{r(D)}{ρr(C)}ϵ(C).

Note that ϵ(D) ≤ ϵ(C) by (4.1). Furthermore we observe by induction that if C ∈ s^{j}(C_{0}), the j-th level of the tree, then

ϵ(C) = \frac{r(C)}{ρ^{j}r(C_{0})}ϵ_{0} ≤ ϵ_{0}.

Recall that ρ < 1/2. If D is a child of C and T (D) is the union of all arcs in the subtree whose root is D, we immediately have by (4.1) that T (D) is contained in a ball of radius

∑_{k=0}^{∞} ρ^{k}r(D) < 2r(D)

around any point of D. Thus there is some fixed a_{3} for which there exists a covering of C by a_{3}r(C)/ϵ(C) balls of radius ϵ(C), which also covers the closure of any subtree descended from C with root D, provided r(D) ≤ ϵ(C)/2.

Now given k > 0 let ϵ_{0} = ρ^{k}r(C_{0}) and let U_{k} denote the covering which is the union of these coverings for all C in levels 0 through k. We claim that U_{k} is in fact a covering of all of L̂_{Γ}. For, if D ∈ s(C) and C is at level k, r(D) ≤ ρr(C), and ϵ(C) = (r(C)/ρ^{k}r(C_{0}))ϵ_{0} = r(C). Thus the covering for C covers the closure of T (D), as above.

We shall now compute the α-dimensional mass of this covering. Let M_{k} denote the sum ∑ r_{i}^{α} over the balls of U_{k}, where r_{i} denotes the radius of the i-th ball and let M_{k}(C) for any arc C denote the sum over just the subset of balls covering C.

For each C we have

M_{k}(C) = a_{3}(r(C)/ϵ(C))ϵ(C)^{α} = a_{3}r(C)ϵ(C)^{α−1}.

If D ∈ s(C), we get

M_{k}(D) = a_{3}r(D) (\frac{r(D)ϵ(C)}{ρr(C)})^{α−1}

= a_{3} (\frac{ϵ(C)}{ρr(C)})^{α−1} r(D)^{α}.

Summing over all D ∈ s(C) and using (4.3),

∑_{D∈s(C)} M_{k}(D) ≤ a_{3} (\frac{ϵ(C)}{ρr(C)})^{α−1} a_{0}ρ^{α−1} \frac{ζ(α)}{2^{α−1} − 1}r(C)^{α}

= a_{0} \frac{ζ(α)}{2^{α−1} − 1}a_{3}r(C)ϵ(C)^{α−1}

= A(α)M_{k}(C).  (4.8)

Where we abbreviate A(α) = a_{0}a_{3}ζ(α)/(2^{α−1} − 1). Applying this inductively, we get

∑_{D∈s^{j}(C_{0})} M_{k}(D) ≤ A(α)^{j}M_{k}(C_{0})  (4.9)

For any j ≤ k. Let us assume A(α) ≥ 2 (since we may always enlarge a_{0}). Then ∑_{j=0}^{k} A(α)^{j} ≤ A(α)^{k+1}, and summing up (4.9) over levels 0 through k for the covering U_{k}, we get

M_{k} ≤ A(α)^{k+1}M_{k}(C_{0}).  (4.10)

By the choice of ϵ_{0} = r(C_{0})ρ^{k}, we have

M_{k}(C_{0}) = a_{3}r(C_{0})^{α}(ρ^{α−1})^{k}.

Thus, given any α > 1, we may choose ρ small enough that ρ^{α−1} < 1/A(α), and then lim_{k→∞} M_{k} = 0, and L̂_{Γ} (hence L_{Γ}) has zero Hausdorff measure in dimension α. Thus d(H^{3}/Γ) ≤ α, which concludes the proof of Theorem 4.1.

5. More preliminaries

In this section we recall more of the background which will be needed to handle manifolds with incompressible boundary which are not books of I-bundles. We first recall Bonahon’s theorem about topological tameness and some basic facts about geometric convergence. In section 5.3 we recall Thurston’s relative compactness theorem and prove an unmarked version of it which will be the key technical tool in the proof of theorem 2.9.

5.1. Bonahon’s theorem. Bonahon [6] proved that a hyperbolic 3manifold H^{3}/Γ is topologically tame if Γ satisfies Bonahon’s condition (B), which is the following: whenever Γ = A ∗ B is a non-trivial free decomposition of Γ, there exists a parabolic element of Γ which is not conjugate to an element of either A or B.

Bonahon provided the following topological interpretation of his condition (B) (see Proposition 1.2 in [6].)

Lemma 5.1. Let Γ be a torsion-free Kleinian group. Suppose that M is a compact 3-manifold, P is a collection of homotopically nontrivial annuli in ∂M, no two of which are homotopic in M and every component of ∂M − P is incompressible in M. Then, if there exists an isomorphism ϕ : π_{1}(M) → Γ such that ϕ(g) is parabolic if g is conjugate to an element of π_{1}(P), then Γ satisfies Bonahon’s condition (B).

5.2. Geometric convergence. We say that a sequence of Kleinian groups {Γ_{j}} converges geometrically to a Kleinian group Γ if for every γ ∈ Γ, there exists a sequence {γ_{j} ∈ Γ_{j}} converging to γ and if any accumulation point of any sequence {γ_{j} ∈ Γ_{j}} is contained in Γ. We will need the following observation, whose proof we sketch. For a complete argument, from a slightly different point of view, see Taylor [32].

Lemma 5.2. Suppose that Γ_{i} is a sequence of torsion-free Kleinian groups converging geometrically to a torsion-free Kleinian group Γ. Let N_{i} = H^{3}/Γ_{i} and N = H^{3}/Γ. Then

lim sup λ_{0}(N_{i}) ≤ λ_{0}(N).

Proof. Let {N_{j}} be a subsequence of {N_{i}} such that {λ_{0}(N_{j})} converges to L = lim sup λ_{0}(N_{i}). Let f_{j} be a positive C^{∞}-function such that Δf_{j} + λ_{0}(N_{j})f_{j} = 0 and let f̃_{j} be the lift of f_{j} to a map f̃_{j} : H^{3} → R. We may scale f̃_{j} so that f̃_{j}(0) = 1 where 0 denotes the origin of H^{3}. Yau’s Harnack inequality [38] and basic elliptic theory guarantee that there is a subsequence of {f̃_{j}} which converges to a positive C^{∞}- function f̃ such that Δf̃ + Lf̃ = 0. Since f̃_{j} was Γ_{j}-invariant and Γ_{j} converges geometrically to Γ, f̃ is Γ-invariant and hence descends to a C^{∞}-function on N such that Δf + Lf = 0. It follows that λ_{0}(N) ≥ L.

5.3. Deformation theory of Kleinian groups. In the proof of Theorem 2.9 we will consider sequences of Kleinian groups in an algebraic deformation space. Let us therefore introduce the following terminology. If G is a group, let D(G) denote the space of discrete, faithful representations of G into PSL_{2}(C). Let AH(G) = D(G)/PSL_{2}(C) where PSL_{2}(C) acts by conjugation of the image. If c is a loop in M represented by the element g ∈ π_{1}(M) and ρ ∈ AH(π_{1}(M)), then l_{ρ}(c) is the translation length of ρ(g) if ρ(g) is hyperbolic and 0 if ρ(g) is parabolic.

We will use the following basic lemma which relates convergence in D(G) and geometric convergence (see [19]).

Lemma 5.3. Suppose that G is a torsion-free, non-abelian group and {ρ_{j}} is a sequence in D(G) which converges to ρ ∈ D(G). Then there is a subsequence of {ρ_{j}(G)} which converges geometrically to a torsionfree Kleinian group Γ which contains ρ(G).

In order to state Thurston’s relative compactness theorem we need to introduce the window W of a compact hyperbolizable 3-manifold M with incompressible boundary. The window W consists of the I-bundle components of the characteristic submanifold Σ(M) together with a thickened neighborhood of every essential annulus in ∂Σ(M) − ∂M which is not the boundary of an I-bundle component of Σ(M). The window is itself an I-bundle over a surface w; w is called the window base.

The following is Thurston’s relative compactness theorem for hyperbolic structures on M (see [36] or Morgan-Shalen [25].)

Theorem 5.4. Let M be a compact, orientable, hyperbolizable 3-manifold with incompressible boundary and window W. If G is any subgroup of π_{1}(M) which is conjugate to the fundamental group of a component of M − W whose closure is not a thickened torus, then the image of the induced map from AH(π_{1}(M)) into AH(G) has compact closure.

We wish to extend this theorem to the following “unmarked” version of Thurston’s theorem.

Theorem 5.5. Let {ρ_{i}} be a sequence in AH(π_{1}(M)). We may then find a subsequence {ρ_{j}}, a sequence of elements {ϕ_{j}} of Out(π_{1}(M)), and a collection x of disjoint, non-parallel, homotopically non-trivial simple closed curves in the window base w such that if G is any subgroup of π_{1}(M) which is conjugate to the fundamental group of a component of M − X whose closure is not a thickened torus (where X is the total space of the I-bundle over x), then {ρ_{j} ∘ ϕ_{j}|_{G}} converges in AH(G). Moreover, if c is a curve in x, then {l_{ρ_{j}∘ϕ_{j}}(c)} converges to 0.

The idea of the proof is the following. By Thurston’s theorem, the restrictions of the representations to the complementary components of the window have convergent subsequences, and in particular the lengths of the window boundaries are bounded. Now for each element of the sequence we represent each component of the window base as a pleated surface with geodesic boundary, and observe that the hyperbolic structures induced on these surfaces, after appropriate remarkings and restriction to a subsequence, either converge or develop cusps. In the latter case we cut along the curves which become cusps (these are the family x), and argue that the representations restricted to the remaining components converge up to subsequence. Finally, we reglue along the window boundaries which did not converge to cusps, and again after restriction to a subsequence obtain the desired convergence.

Proof. The following proposition states the corresponding fact for hyperbolic surfaces. This fact is reasonably standard, and one may construct a proof using techniques described in Abikoff [1] and Harvey [16].

Proposition 5.6. Let S be a compact surface with boundary and let {ψ_{i}} be a sequence of discrete faithful representations of π_{1}(S) into Isom(H^{2}). If there exists K such that l_{ψ_{i}}(∂S) ≤ K for all i, then there exists a subsequence {ψ_{k}} of {ψ_{i}}, a collection y of disjoint, homotopically distinct curves in S and a collection of homeomorphisms h_{k} : S → S which are the identity on ∂S such that if R is a component of S − y, then {ψ_{k} ∘ (h_{k})_{∗}|_{π_{1}(R)}} converges in AH(π_{1}(R)). Moreover, {l_{ψ_{k}∘(h_{k})_{∗}}(y_{0})} converges to 0, for any component y_{0} of y.

We begin by using Proposition 5.6 to choose x, a subsequence {ρ_{k}} of {ρ_{i}}, and a sequence of homeomorphisms {f_{k} : w → w} such that if R is a component of w−x, then {ρ_{k} ∘(f_{k}|_{R})_{∗}} converges in AH(π_{1}(R)).

Let w_{1}, …, w_{m} be the components of w. We choose the restriction of x to each component inductively, at the same time passing to subsequences. We inductively assume that we have chosen a subsequence {ρ_{k}} and the restriction of x and f_{k} to w_{1} ∪ ⋯ w_{l−1} such that if R is a component of (w_{1} ∪ ⋯ ∪ w_{l−1}) − x, then {ρ_{k} ∘ (f_{k}|_{R})_{∗}} converges in AH(π_{1}(R)).

If w_{l} is an annulus or a Möbius strip, let c_{l} be a core curve of w_{l}. Thurston’s relative compactness theorem implies that l_{ρ_{i}}(c_{l}) is bounded above. We may then pass to a subsequence of {ρ_{k}}, again called {ρ_{k}}, such that {ρ_{k}|_{π_{1}(w_{l})}} converges in AH(Z). We include ∂w_{l} in x if and only if {l_{ρ_{k}}(c_{l})} converges to 0. Let the restriction of f_{k} to w_{l} be the identity map.

Now suppose that w_{l} is not an annulus or a Möbius strip, in which case w_{l} has negative Euler characteristic. We may pass to a subsequence of {ρ_{k}}, again called {ρ_{k}}, such that if z is any boundary component of w_{l} then either l_{ρ_{k}}(z) = 0 for all k or l_{ρ_{k}}(z) ≠ 0 for any k. Let w′_{l} be obtained from w_{l} by removing any component z of ∂w_{l} such that l_{ρ_{k}}(z) = 0 for all k. One can then find, for each k, a complete finite-area hyperbolic metric τ_{k} on w′_{l} with geodesic boundary, and a path-wise isometry r_{k} : (w′_{l}, τ_{k}) → N_{k} in the homotopy class determined by the restriction of ρ_{k} to π_{1}(w_{l}), such that that r_{k}(∂w′_{l}) is a collection of closed geodesics. (A map between Riemannian manifolds is a pathwise isometry if any rectifiable path in the domain is taken by the map to a path of equal length. Typically one takes each r_{k} to be a pleated surface, see [10] or [34]. In particular, note that neighborhoods of the missing boundaries of w′_{l} are cusps, which map into cusps in N_{l}.)

Then (w′_{l}, τ_{k}) is isometric to the convex core of H^{2}/Θ_{k} for some discrete subgroup Θ_{k} of Isom(H^{2}) and there is an induced discrete faithful representation ψ_{k} : π_{1}(w_{l}) → Isom(H^{2}) with image Θ_{k} such that l_{ψ_{k}}(c) ≥ l_{ρ_{k}}(c) for any simple closed curve c in w_{l} and all k.

Thurston’s relative compactness theorem guarantees that there exists K such that l_{ψ_{k}}(∂w′_{l}) = l_{ρ_{k}}(∂w_{l}) ≤ K for all k. Then {ψ_{k}} satisfies the hypotheses of Proposition 5.6. Hence, after perhaps passing to another subsequence of {ρ_{k}}, again called {ρ_{k}}, we obtain a collection y of simple closed curves in w_{l} and a sequence of homeomorphisms h_{k} : w_{l} → w_{l} such that if R is a component of w_{l}−y, then {ψ_{k} ∘(h_{k}|_{R})_{∗}} converges in AH(π_{1}(R)). We then append y to x and we also add to x any component z of ∂w_{l} such that {l_{ρ_{k}}(z)} = {l_{ψ_{k}}(z)} converges to 0. Let the restriction of f_{k} to w_{l} agree with h_{k}. If R is a component of w_{l} − x, then the convergence of the Fuchsian representations {ψ_{k} ∘ (h_{k}|_{R})_{∗}} implies that any subsequence of {ρ_{k} ∘ (h_{k}|_{R})_{∗}} has a convergent subsequence in AH(π_{1}(R)). This is because, for an appropriate choice of basepoint in r_{k}(R), the translation distances of a generating set of elements in ρ_{k} ∘ (h_{k}|_{R})_{∗}(π_{1}(R)) are bounded by the translation distances in the Fuchsian representations (since r_{k} are pathwise isometries). Hence, we may pass to a further subsequence of {ρ_{k}}, again called {ρ_{k}}, such that if R is any component of w_{l} − x, then {ρ_{k} ∘ (h_{k}|_{R})_{∗}} converges in AH(π_{1}(R)).

Let {ρ_{k}} be the subsequence and {f_{k} : w → w} be the sequence of homeomorphisms obtained after applying the above process m times. Each f_{k} induces a homeomorphism F_{k} : W → W preserving ∂M −∂W, which is homotopic to the identity on each component of ∂W − ∂M. Hence, F_{k} extends to a homotopy equivalence F̂_{k} of M which is equal to the identity on the complement of a regular neighborhood of W.

Let ϕ′_{k} = (F̂_{k})_{∗} and let Z be the I-bundle over ∂w − x. Thurston’s relative compactness theorem and our construction of Z ∪X imply that there exists a subsequence of {ρ_{k}}, again denoted {ρ_{k}}, such that if M′ is any component of M −(Z∪X) whose closure is not a thickened torus, then {ρ_{k} ∘ ϕ′_{k}|_{π_{1}(M′)}} converges in AH(π_{1}(M′)). Notice further that if c is the core curve of any component of Z, then {l_{ρ_{k}∘ϕ′_{k}}(c)} converges to a positive number δ(c). On the other hand, if c is the core curve of a component of X, then {l_{ρ_{k}∘ϕ′_{k}}(c)} converges to 0.

In order to complete the proof, it is necessary to precompose by an additional sequence of automorphisms (and pass to an additional subsequence) to guarantee that the new sequence of representations converges on every component of M − X which is not a thickened torus. We will make repeated use of the following elementary lemma, which we state without proof.

Lemma 5.7. Let Q be a compact, hyperbolizable 3-manifold and let A be an essential annulus in Q with core curve a. Suppose that {ϕ_{i}} is a sequence in AH(π_{1}(Q)) such that if Q′ is any component of Q−A, then {ϕ_{i}|_{π_{1}(Q′)}} converges in AH(π_{1}(Q′)) and such that {l_{ϕ_{i}}(a)} converges to a positive number δ. Then, there exists a subsequence {ϕ_{k}} of {ϕ_{i}} and a sequence {g_{k}} of homeomorphisms of Q (each of which is a power of a Dehn twist about the annulus A) such that {ϕ_{k} ∘ (g_{k})_{∗}} converges in AH(π_{1}(Q)).

Let Z_{1}, …, Z_{n} be the components of Z. We inductively assume that we have chosen a subsequence {ρ_{j}} of {ρ_{k}} and a sequence {ϕ_{j}} of automorphisms of π_{1}(M) such that if M′ is a component of M − (X ∪ Z_{l} ∪ ⋯ ∪ Z_{n}) which is not a thickened torus, then {ρ_{j} ∘ ϕ_{j}|_{π_{1}(M′)}} converges in AH(π_{1}(M′)). (If l = 1, then {ρ_{j}} is the sequence {ρ_{k}} obtained above and {ϕ_{j}} = {ϕ′_{k}}.)

Let Q_{l} be the component of M −(X ∪Z_{l+1} ∪⋯∪Z_{n}) which contains Z_{l}. Then {ρ_{j} ∘ ϕ_{j}|_{π_{1}(Q_{l})}} and Z_{l} satisfy the hypotheses of Lemma 5.7. We may thus assume, perhaps after passing to a further subsequence, that there is a sequence {g_{j}} of homeomorphisms of Q_{l} such that {ρ_{j} ∘ ϕ_{j} ∘(g_{j})_{∗}} converges in AH(π_{1}(Q_{l})). Since each g_{j} is a power of a Dehn twist about the annulus Z_{l}, we may extend {g_{j}} to a sequence, again called {g_{j}}, of homeomorphisms of M, such that each g_{j} is the identity map on M − Q_{l}. Then, after replacing ϕ_{j} by ϕ_{j} ∘ (g_{j})_{∗}, we see that if M′ is any component of M − (X ∪ Z_{l+1} ∪ ⋯ ∪ Z_{n}) which is not a thickened torus, then {ρ_{j} ∘ ϕ_{j}|_{π_{1}(M′)}} converges in AH(π_{1}(M′)).

Applying the above process n times gives the desired subsequence and sequence of automorphisms. This completes the proof of Theorem 5.5.

6. 3-manifolds which are not generalized books of I-bundles

The main result of this section asserts that hyperbolizable 3-manifolds with incompressible boundary which are not generalized books of Ibundles have Λ-invariant strictly less than 1.

Theorem 2.9. If M is a compact, orientable, hyperbolizable 3-manifold with incompressible boundary which is not a generalized book of Ibundles, then Λ(M) < 1.

Proof of Theorem 2.9. Let M be a compact, orientable, hyperbolizable 3-manifold with incompressible boundary which is not a generalized book of I-bundles. We will assume that Λ(M) = 1 and arrive at a contradiction.

If Λ(M) = 1, there exists a sequence {N_{i}} in T T (M) such that {λ_{0}(N_{i})} converges to 1 where N_{i} = H^{3}/Γ_{i}. Let ρ_{i} : π_{1}(M) → PSL_{2}(C) be a discrete faithful representation with image Γ_{i}. Let {ρ_{j}}, {ϕ_{j}}, X, and x be as in Theorem 5.5.

Let M_{0} be a component of M − X which contains a component V of M − Σ(M) whose closure is not a solid torus or a thickened torus. Note that the fundamental group of V must be non-abelian, since any compact hyperbolizable 3-manifold with abelian fundamental group is a solid torus or a thickened torus. Hence, there exists a curve in V which is not homotopic into Σ(M) and the closure of M_{0} cannot be a thickened torus. Thus, Theorem 5.5 implies that {ρ_{j} ∘ ϕ_{j}|_{π_{1}(M_{0})}} converges to a discrete faithful representation ρ : π_{1}(M_{0}) → PSL_{2}(C) such that ρ(π_{1}(M_{0} ∩ X)) is parabolic. Let Γ^{0} = ρ(π_{1}(M_{0})) and N^{0} = H^{3}/Γ^{0}. Let Γ̂ be a geometric limit of some subsequence of {Γ_{j}^{0} = ρ_{j}(ϕ_{j}(π_{1}(M_{0})))} and let N_{j}^{0} = H^{3}/Γ_{j}^{0} and N̂ = H^{3}/Γ̂. Since 1 ≥ λ_{0}(N_{j}^{0}) ≥ λ_{0}(N_{j}) and lim_{j→∞} λ_{0}(N_{j}) = 1, we see that lim_{j→∞} λ_{0}(N_{j}^{0}) = 1. Lemma 5.2 then implies that λ_{0}(N̂) = 1 and hence that λ_{0}(N^{0}) = 1.

Let M′_{0} denote M_{0} − N (X) where N (X) is a regular neighborhood of X and let Y denote the intersection of the closure of N (X) with M′_{0}. Since ∂M′_{0} − Y is incompressible and the elements of Γ^{0} corresponding to π_{1}(Y) are parabolic, Lemma 5.1 implies that Γ^{0} satisfies Bonahon’s condition (B). Therefore, N^{0} is topologically tame. Since, λ_{0}(N^{0}) = 1, the results of [13] imply that Γ^{0} is either a Fuchsian group or an extended Fuchsian group. Thus N^{0} contains a compact submanifold R which is a strong deformation retract of N^{0} and is an I-bundle with base surface B, such that an element of Γ^{0} is parabolic if and only if it is conjugate to an element of π_{1}(∂B). (If Z is the totally geodesic hyperplane preserved by Γ^{0}, then we can take B to be a compact core for Z/Γ^{0} and R to be a regular neighborhood of B.)

Since Γ^{0} = ρ(π_{1}(M_{0})) and every element of Γ^{0} corresponding to an element of π_{1}(Y) is parabolic, there is a homotopy equivalence h : M′_{0} → R such that every element of Y maps into a component of S the sub-bundle over ∂B. If we let S_{0} denote the set of components of S which contain images of elements of Y, then h : (M′_{0}, Y) → (R, S_{0}) is a homotopy equivalence of pairs. Since every component of ∂M′_{0} −Y is incompressible and h is a homotopy equivalence of pairs, every component of ∂R − S_{0} is incompressible (see Proposition 1.2 in [6] or section 2 of Canary-McCullough [12]), which implies that S_{0} = S. Thus, (M′_{0}, Y) is homotopy equivalent to the I-pair (R, S) which implies that (M′_{0}, Y) is homeomorphic to (R, S) (see corollary 5.8 in Johannson [18]). This implies that M′_{0} is an admissibly embedded essential Ibundle and hence may be properly homotoped into Σ(M). This however contradicts the fact that there exist curves in V which are not homotopic into Σ(M). This contradiction completes the proof of Theorem 2.9.

7. Remarks and Conjectures

1. The most natural analogue of the Gromov norm is the invariant V (M) which is defined to be the infimum of the volumes of the convex cores of hyperbolic manifolds homeomorphic to the interior of M. One would make the following conjecture in the spirit of the paper.

Conjecture: V (M) = 0 if and only if every component of M’s incompressible core is a generalized book of I-bundles.

It seems likely that an argument similar to that in section 4 would give that V (M) = 0 whenever M is a generalized book of I-bundles. Whereas an argument similar to that in section 6, along with work of Taylor [31], should give that V (M) > 0 if M has incompressible boundary but is not a generalized book of I-bundles. If M is obtained from generalized books of I-bundles by adding 1-handles, then one needs to show that V (M) = 0. However, as the direct analogues for volumes of the results of Patterson for λ_{0}, used in Section 3, are false, one would need to find a more explicit proof for this case.

2. The work of Canary [11] and Burger-Canary [9] exhibits relationships between V (M) and Λ(M). The work of [11] gives the following upper bound for all M,

Λ(M) ≤ \frac{4π|χ(∂M)|}{V (M)}

(where χ(∂M) denotes the Euler characteristic of ∂M), while the work of [9] shows that there exist constants A > 0 and B > 0 such that if M has incompressible boundary then

Λ(M) ≥ \frac{A}{(V (M) + B|χ(∂M)|)^{2}} .

3. If M is acylindrical then there is a unique hyperbolic manifold N whose convex core has totally geodesic boundary and is homeomorphic to M. One expects that D(M) = d(N), Λ(M) = λ_{0}(N), and that V (M) is the volume vol(C(N)) of the convex core C(N) of N. In fact, Bonahon [7] has shown that N is a local minimum for the volume (of the convex core) function on the space GF (M) of geometrically finite hyperbolic 3-manifolds homeomorphic to the interior of M. More generally, if M has incompressible boundary and M_{i} is a component of M − Σ(M) which is not homeomorphic to a solid torus, then there exists a unique hyperbolic 3-manifold N_{i} such that the convex core C(N_{i}) has totally geodesic boundary and C(N_{i}) is homeomorphic to M_{i}.

Conjecture: If M has incompressible boundary, {M_{1}, …, M_{n}} are the components of M − Σ(M) which aren’t homeomorphic to solid tori, and N_{i} are as above, then V (M) = ∑_{i=1}^{n} vol(C(N_{i})) and Λ(M) = min{λ_{0}(N_{i})}.

A positive solution to the conjecture above would imply that Λ is a homotopy invariant, since the incompressible cores of homotopyequivalent hyperbolic 3-manifolds are homotopy-equivalent, and Johannson’s theorem [18] implies that the complements of the characteristic submanifolds of homotopy-equivalent hyperbolizable 3-manifolds with incompressible boundary are homeomorphic.

We note that ∑_{i=1}^{n} vol(C(N_{i})) is equal to half the Gromov norm of the double of M. One can generalize this conjecture to obtain a similar conjecture for arbitrary compact hyperbolizable 3-manifolds.

4. It would be interesting to know more about the distribution of the set of values assumed by the invariant Λ. In this remark we show that the set of values accumulates at 0.

Let M_{i} be a compact hyperbolizable 3-manifold whose boundary is incompressible and has i toroidal boundary components and two genus two boundary components. One may obtain such manifolds by removing suitably chosen collections of boundary-parallel curves from a product S × [0, 1] where S is a surface of genus 2. We will show that V (M_{i}) → ∞, by observing that all but a bounded number of the cusps contribute a definite amount to the volume of the convex core of any hyperbolic structure on M_{i}.

If x ∈ N, then inj_{N}(x) denotes the injectivity radius of the N at the point x. The Margulis lemma asserts that there exists a constant M_{3} such that if ϵ < M_{3} and N is a complete hyperbolic 3-manifold, then every non-compact component of N_{thin(ϵ)} = {x ∈ N|inj_{N}(x) < ϵ} is the quotient of a horoball by a group of parabolic transformations fixing the horoball. Moreover, for all M_{3} > ϵ > 0, there exists D(ϵ) > 0 such that any non-compact component of N_{thin(ϵ)} has volume at least D(ϵ). There also exists K(ϵ) > 0 such that at most K(ϵ)|χ(∂C(N_{i}))| components of N_{thin(ϵ)} intersect ∂C(N). If N_{i} ∈ T T (M_{i}) then there are at least i non-compact components of (N_{i})_{thin(ϵ)} (one for each toroidal component of ∂M_{i}.) Since |χ(∂C(N_{i}))| ≤ 4 and every component of (N_{i})_{thin(ϵ)} intersects C(N_{i}), it follows that

vol(C(N_{i})) > (i − 4K(ϵ))D(ϵ).

Hence, V (M_{i}) ≥ (i − 4K(ϵ))D(ϵ), so V (M_{i}) converges to infinity and Λ(M_{i}) ≤ \frac{16π}{V (M_{i})} converges to 0. On the other hand, Λ(M_{i}) ≠ 0, by Proposition 2.5. Thus 0 is an accumulation point of the set of Λ values.

5. If Γ is a quasi-Fuchsian group, denote by K(Γ) the minimal K for which there exists a K-quasiconformal map conjugating Γ to a Fuchsian group. One may use the methods of section 4 to construct a sequence {Γ_{j}} of quasiconformally conjugate quasi-Fuchsian groups such that the Hausdorff dimensions of the limit sets of the Γ_{j} converge to 1, but {K(Γ_{j})} converges to ∞. We will discuss this more fully in a future note.

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109

Department of Mathematics, SUNY-Stony Brook, Stony Brook, NY 11794

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109