Betti numbers and injectivity radii

We give lower bounds on the maximal injectivity radius for a closed orientable hyperbolic 3-manifold M with first Betti number 2, under some additional topological hypotheses. A corollary of the main result is that if M has first Betti number 2 and contains no fibroid surface then its maximal injectivity radius exceeds 0.32798. For comparison, Andrew Przeworski showed, with no topological restrictions, that the maximal injectivity radius exceeds arcsinh(1/4) = 0.247..., while the authors showed that if M has first Betti number at least 3 then the maximal injectivity exceeds log(3)/2 = 0.549.... The proof combines a result due to Przeworski with techniques developed by the authors in the 1990s.

• P is an I-bundle over a non-empty compact 2-manifold-with-boundary; • each component of B is homeomorphic to D 2 × S 1 ; • the set A = P ∩ B is the vertical boundary of the I-bundle P; and • each component of A is an annulus in ∂B which is homotopically non-trivial in B.
(Note that this terminology differs slightly from that of [1], where it is the triple (W, B, P) that is called a book of I-bundles.) As in [5], we define a fibroid in a closed, connected, orientable 3-manifold M to be a connected incompressible surface F with the property that each component of the compact manifold obtained by cutting M along F is a book of I-bundles. Note that in defining a fibroid to be connected, we are following the convention of [5] rather than that of [6].
We define a function R(x) for x > 0 by (1) R . As in [3, Section 10], we define a function V (x) for x > 0 by Thus, in a closed hyperbolic 3-manifold, if a geodesic of length is the core of an embedded tube of radius R( ), then this tube has volume V ( ).
The following result is implicit in [6], but we will supply a proof for the sake of comprehensibility.
Proposition 1. Let M be a closed hyperbolic 3-manifold. Suppose that there is an infinite subset N of H 2 (M ; Z) such that every element of N is represented by some connected, incompressible surface which is not a fibroid. Let λ be a positive number less than log 3. Then either the maximal injectivity radius of M is at least λ/2, or M contains a closed geodesic C of length at most λ such that the maximal tube about C has radius at least R(λ) and volume at least V (λ), where R(λ) and V (λ) are defined by (1) and (2).
Proof. The hypothesis implies in particular that H 2 (M ; Z) has infinitely many primitive elements, and so the first Betti number β 1 (M ) is at least 2. If β 1 (M ) ≥ 3, then according to [4,Corollary 10.4], the maximal injectivity radius is at least 1 2 log 3 > λ/2. We may therefore assume that β 1 (M ) = 2. Hence the quotient of H 1 (M, Z) by its torsion subgroup is a free abelian group L of rank 2. We let h : π 1 (M ) → L denote the natural homomorphism.
We distinguish two cases. First consider the case in which M contains a non-trivial closed geodesic C of some length < λ such that the conjugacy class represented by C is contained in the kernel of h. Let T denote the maximal embedded tube about C. According to [3,Corollary 10.5] we have Vol T ≥ V (λ). If ρ denotes the radius of T , this gives and hence ρ > R(λ).
Thus the second alternative of the proposition holds in this case.
We now turn to the case in which no non-trivial closed geodesic of length < λ represents a conjugacy class contained in the kernel of h. Since M is closed, there are only a finite number n ≥ 0 of conjugacy classes in π 1 (M ) that are represented by closed geodesics of length < l. Let γ 1 , . . . , γ n be elements belonging to these n conjugacy classes. Thenγ i = h(γ i ) is a non-trivial element of L for i = 1, . . . , n. Since L is a free abelian group of rank 2, there exists, for each i ∈ {1, . . . , n}, a homomorphism φ i of L onto Z such that φ i (γ i ) = 0. Becausē γ i = 0, the epimorphism φ i is unique up to sign.
The epimorphism φ i • h : π 1 (M ) → Z corresponds to a primitive element of H 1 (M ; Z), whose Poincaré dual in H 2 (M ; Z) we shall denote by c i . Since the set N ⊂ H 2 (M ; Z) given by the hypothesis of the theorem is infinite, there is an element c of N which is distinct from ±c i for i = 1, . . . , n. Since c ∈ N it follows from the hypothesis that there is a connected incompressible surface S ⊂ M which represents the homology class c and is not a fibroid.
We now apply Theorem A of [5], which asserts that if S is a connected non-fibroid incompressible surface in a closed, orientable hyperbolic 3-manifold M , and if λ is any positive number, then either (i) M contains a non-trivial closed geodesic of length < λ which is homotopic in M to a closed curve in M − S, or (ii) M contains a hyperbolic ball of radius λ/2. In the present situation, with λ chosen as above, we claim that alternative (i) of the conclusion of Theorem A of [5] cannot hold.
Indeed, suppose that C is a non-trivial closed geodesic of length < λ with the properties stated in (i). Since C has length < λ, the conjugacy class represented by C contains γ i for some i ∈ {1, . . . , n}. Since C is homotopic to a closed curve in M −S it follows that the image of γ i in H 1 (M ; Z) has homological intersection number 0 with c. Thus if ψ : π 1 (M ) → Z is the homomorphism corresponding to the Poincaré dual of c, we have ψ(γ i ) = 0. Now since L is the quotient of H 1 (M ) by its torsion subgroup, ψ factors as φ • h, where φ is some homomorphism from L to Z. Since c is primitive, ψ is surjective, and hence so is φ. But we have φ(γ i ) = ψ(γ i ) = 0. In view of the uniqueness that we observed above for φ i , it follows that φ = ±φ i , so that ψ = ±φ i • h and hence c = ±c i . This contradicts our choice of c.
Hence (ii) must hold. This means that the maximal injectivity radius of M is at least λ/2. Thus the first alternative of the proposition holds in this case.
Proposition 2. Let M be a closed hyperbolic 3-manifold. Suppose that there is an infinite subset N of H 2 (M ; Z) such that every element of N is represented by some connected, incompressible surface which is not a fibroid. Then the maximal injectivity radius of M exceeds 0.32798.
According to Proposition 1, either the maximal injectivity radius of M is at least λ/2so that the conclusion of the theorem holds -or M contains a closed geodesic C of length at most λ such that the maximal tube about C has volume at least V (λ) , where V (λ) is defined by (2). In the latter case, if R denotes the radius of T , we have R ≥ R(λ) = 0.806787 . . . . [7,Proposition 4.1], the maximal injectivity radius of M is bounded below by

Now according to
This gives the conclusion of the theorem in this case. Proof. If π 1 (M ) has a non-abelian free quotient, then by [6, Theorem 1.3], the maximal injectivity radius of M is at least 1 2 log 3 = 0.549 . . .. Now suppose that π 1 (M ) has no nonabelian free quotient. If N is the set given by the hypothesis of Theorem 3, it now follows from [6, Proposition 2.1] that every element of N is represented by a connected incompressible surface, which by hypothesis cannot be a fibroid. Thus N has the properties stated in the hypothesis of Proposition 2. The latter result therefore implies that the maximal injectivity radius of M exceeds 0.32798.
If M is a 3-manifold whose first Betti number is at least 2, then H 2 (M ; Z) has infinitely many primitive elements. If a non-trivial element of H 2 (M ; Z) is represented by a connected surface it must be primitive, since it has intersection number 1 with a class in H 1 (M ; Z). Hence Theorem 3 implies: Corollary 4. Let M be a closed hyperbolic 3-manifold. Suppose that the first Betti number of M is at least 2, and that M contains no non-separating fibroid. Then the maximal injectivity radius of M exceeds 0.32798.