Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 51–72

COLLARS IN PSL(2,H)

Ruth Kellerhals

Université Bordeaux I, Laboratoire MPB 351, cours de la Libération, F-33405 Talence cédex, France; ruth@math.u-bordeaux.fr

Abstract. Let M = H^{4}/Γ denote a 4-dimensional oriented hyperbolic manifold of finite volume. By identifying Iso^{+}(H^{4}) with PSL(2, H), we construct embedded tubular neighborhoods around short simple closed geodesics in M whose collar width depends on the length of g, only. We show that two non-intersecting short geodesics have disjoint collars. Moreover, the constructed collars do not intersect the canonical cusps associated to parabolic elements in Γ. Finally, we discuss some applications and provide bounds for the injectivity radius and the number of short simple closed geodesics in M.

0. Introduction

Consider the group of orientation preserving Möbius transformations of Ê^{3} = E^{3} ∪ {∞} which acts by direct isometries on hyperbolic 4-space realized in the upper half space model. As such, the group Iso^{+}(H^{4}) can be identified with the group PSL(2, H) of Clifford matrices with quaternion coefficients (cf. Section 1). In this setting, we study the geometry of discrete groups in Iso^{+}(H^{4}) and quotient manifolds of finite volume.

The main result of this paper is a collar theorem for hyperbolic 4-manifolds M providing around each simple closed geodesic g in M of length l ≤ l_{0} = (\sqrt{3} /4π) log^{2} 2 ≃ 0.06622 an embedded tubular neighborhood whose collar width depends on the length l of g, only (cf. Section 2.1). One important tool in the proof (cf. Section 2.2) is the inequality for discrete non-elementary two generator groups of Clifford matrices due to P. Waterman [Wat] which generalizes the wellknown trace inequality for PSL(2, C). Moreover, we make use of properties of the generalized cross ratio for an ordered quadruple of vectors in a Clifford algebra as developed by C. Cao and P. Waterman [CW, Section 6].

In the remaining part of Section 2, we prove some results about the size and the position of the collars in the thin part of M. In Section 2.3, we show that distinct simple closed geodesics have disjoint collars. In Section 2.4, we investigate parabolic group elements and cusps in a non-compact manifold M of finite volume (cf. also [K2]). We show that the canonical cusps and the collars do not intersect in M. Moreover, for a sequence of loxodromic elements converging to a parabolic one, the collars tend to the canonical cusp. In this way, we obtain a fairly good picture about the thin part of a hyperbolic 4-manifold M.

1991 Mathematics Subject Classification: Primary 53C22, 53C25, 57S30, 51N25, 15A45.

Partially supported by Schweizerischer Nationalfonds No. 2000-050579.97.

Finally, in Section 3, we discuss some applications concerning the geometry and topology of hyperbolic 4-manifolds. For example, the injectivity radius i(M) of a compact hyperbolic 4-manifold M satisfies the inequality i(M) ≥ const · vol_{4}(M)^{−2} which improves a result of A. Reznikov [Re].

Our results extend the well-known collar theorems for Riemannian surfaces of genus > 1 and hyperbolic 3-manifolds (cf. [Bu, p. 94] for relevant references) to discrete subgroups of Iso^{+}(H^{4}) and their quotient manifolds. In [CW, Section 9], Cao and Waterman derived a collar theorem for hyperbolic manifolds of arbitrary dimensions n ≥ 2. They approach isometries of hyperbolic n-space by means of Clifford matrices as well and utilize similar inequalities for discrete two generator groups as mentioned above. The methods differ where they make use of certain extremal values associated to the rotational part of loxodromic elements. Therefore, by specializing to the thin part of hyperbolic manifolds of dimension n = 4, it is not a surprise that our results are stronger when compared to theirs.

Acknowledgement. This work arose during research stays at the Forschungsinstitut FIM of the ETH Zürich (Switzerland) and at the Mathematical Institute of Fribourg (Switzerland). The author expresses her thanks to the Director of FIM, Professor A.-S. Sznitman, and to Professor E. Ruh for the invitation and the hospitality.

1. Hyperbolic isometries and Clifford matrices

1.1. Some hyperbolic geometry. Let H^{n} denote the hyperbolic n-space realized in the Poincaré conformal model of the upper half space En_{+} = {x = (x_{1}, …, x_{n}) ∈ E^{n} | x_{n} > 0} equipped with the metric

cosh dist(x, y) = 1 + \frac{|x − y|^{2}}{2x_{n}y_{n}} .

The compactification \overline{H^{n}} = H^{n} ∪ ∂H^{n} consists of H^{n} together with the set ∂H^{n} = Ê^{n−1} := E^{n−1} ∪ {∞} of its points at infinity.

The group I(H^{n}) of isometries of H^{n} is isomorphic to the subgroup M(En_{+}) ⊂ M(Ê^{n}) of Möbius transformations of Ê^{n} that leave En_{+} invariant. By means of Poincaré extension, there are the isomorphisms

I(H^{n}) ≅ M(En_{+}) ≅ M(Ê^{n−1}).

According to the fixed point behavior a Möbius transformation is either elliptic, parabolic, or loxodromic. For example, if φ ∈ M(En_{+}) has precisely one, respectively two, fixed points in Ê^{n−1} and none in En_{+} , then φ is parabolic, respectively loxodromic.

Collars in PSL(2, H) 53

Γ is elementary if Γ has a finite orbit Γp for some point p ∈ \overline{H^{n}}. Moreover, Γ is said to be of parabolic, respectively hyperbolic, type if Γ has one, respectively two, different fixed points in ∂H^{n} and no further finite orbits in \overline{H^{n}}. There are the following characterizations (cf. [Ra]). Γ is discrete, elementary and of parabolic type if and only if Γ is conjugate to an infinite discrete subgroup of the isometry group I(E^{n−1}) of E^{n−1} . Let S(E^{n−1})_{∗} denote the group of all Möbius transformations in M(En_{+}) leaving invariant the set {0, ∞}. Then, Γ is discrete, elementary and of hyperbolic type if and only if Γ is conjugate in M(En_{+}) to an infinite discrete subgroup of S(E^{n−1})_{∗}.

Finally, if φ, ψ ∈ M(En_{+}) are such that ψ is loxodromic with one fixed point in common with φ, then the subgroup ⟨φ, ψ⟩ generated by φ, ψ is not discrete (cf. [Ra, Theorem 5, 5.4]). Therefore, a discrete elementary group Γ containing a loxodromic (parabolic) element, consists of loxodromic (parabolic) elements, only, and they all have the same fixed points.

1.2. Discrete groups of Clifford matrices. Let Γ ⊂ I^{+}(H^{n}) denote a discrete group of orientation preserving isometries of H^{n}. We are interested in the geometrical behavior of Γ such as the uniform isolation of id in Γ. To this end, following K.Th. Vahlen [V] and L.V. Ahlfors [A1], [A2], we make use of the very elegant and important description of Möbius transformations as Clifford matrices. In the following, we provide a summary about the Clifford calculus in n variables.

The Clifford algebra C_{n} is the associative algebra over R generated by n − 1 elements i_{1}, …, i_{n−1} subject to the relations i_{k}i_{l} = −i_{l}i_{k} (k ≠ l) and i_{k}^{2} = −1. Each element a ∈ C_{n} can be uniquely represented in the form

a = ∑_{I} a_{I}I, a_{I} ∈ R,

where I runs through all products i_{k_{1}} ⋯ i_{k_{r}} with 0 < k_{1} < ⋯ < k_{r} < n. Here, the empty product is included and identified with i_{0} = i_{k_{0}} := 1. The number r ≥ 0 is called the degree of I.

Examples are C_{1} = R, C_{2} = C and C_{3} = H.

C_{n} is a real vector space of dimension 2^{n−1} which can be turned into a normed space by imposing the Euclidean norm

|a|^{2} := ∑_{I} a_{I}^{2} for a = ∑_{I} a_{I}I.

Moreover, C_{n} admits a direct sum decomposition

C_{n} = ⨁_{r=0}^{n−1} C_{n}(r),

where C_{n}(r) ⊂ C_{n} is spanned by the products I of degree r. Therefore, we can write

a = a(0) + a(1) + ⋯ + a(n − 1) with a(r) ∈ C_{n}(r).

There are three involutions on C_{n}. The mapping a ↦ a^{∗} is defined by sending each I = i_{k_{1}} ⋯ i_{k_{r}} to I^{∗} := i_{k_{r}} ⋯ i_{k_{1}} , while a ↦ a′ is given by replacing each factor i_{k} by −i_{k}. The conjugation a ↦ \overline{a} is the composition \overline{a} := a^{′∗}. We obtain (cf. [Wad, p. 126])

(1.1)  \overline{a} = a(0) − a(1) − a(2) + a(3) + ⋯ + (−1)^{(n−1)n/2}a(n − 1).

Clifford numbers of the form x = x_{0} +x_{1}i_{1} +⋯+x_{n−1}i_{n−1} are called vectors. They satisfy x = x^{∗}. Vectors form an n-dimensional linear subspace of C_{n} which is usually identified with E^{n}. Non-zero vectors are invertible with inverse

x^{−1} = \frac{1}{|x|^{2}} (x_{0} − x_{1}i_{1} − ⋯ − x_{n−1}i_{n−1}).

Hence, products of non-zero vectors are invertible and give rise to a multiplicative subgroup, the Clifford group G_{n}. For a, b ∈ G_{n}, one has |ab| = |a|·|b|. Embedding E^{n} ⊂ E^{n+1} in the natural way, one obtains G_{n} < G_{n+1} .

A Clifford matrix is an element of the set

\begin{aligned}SL(2, C_{n}) := \{T = ( \begin{matrix}a & b \\ c & d \end{matrix} ) | a, b, c, d ∈ G_{n} ∪ \{0\}; & ab^{∗}, cd^{∗}, c^{∗}a, d^{∗}b ∈ E^{n}; \\ & ad^{∗} − bc^{∗} = 1\}. \end{aligned}

By a result of Vahlen and H. Maass (cf. [A2, p. 221]), SL(2, C_{n}) is a group under matrix multiplication with

(1.2)  T^{−1} = ( \begin{matrix}d^{∗} & −b^{∗} \\ −c^{∗} & a^{∗} \end{matrix} ) for T = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, C_{n}).

There are different ways to introduce the notion of trace for a matrix

T = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, C_{n}).

The quantity tr(T) = a + d^{∗} is often called the trace of T. One checks that the scalar part of tr(T) is a conjugacy invariant (cf. [Wat, p. 99]) which we denote by

(1.3)  Tr(T) := tr(T)(0) = (a + d^{∗})(0).

Consider the subgroup

PSL(2, C_{n}) := SL(2, C_{n})/{λE | λ ∈ R − {0}},

which satisfies PSL(2, C_{n}) < PSL(2, C_{n+1}). The group PSL(2, C_{n}) acts bijectively on Ê^{n} by

T (x) = (ax + b)(cx + d)^{−1}, T (0) = bd^{−1}, T (∞) = ac^{−1},

and this action can be extended to Ê^{n+1} = {x_{0}+x_{1}i_{1}+⋯+x_{n}i_{n} | x_{k} ∈ R}∪{∞}, which we denote by the same symbol. Furthermore, PSL(2, C_{n}) is isomorphic to the group M^{+}(Ê^{n}) of orientation preserving Möbius transformations of Ê^{n} where matrix multiplication corresponds to composition of mappings. Each T ∈ PSL(2, C_{n}) preserves the upper half space (cf. Section 1.1) since

(1.4)  [T (x)]_{n} = \frac{x_{n}}{|cx + d|^{2}} ; \frac{|T′(x)|}{[T (x)]_{n}} = \frac{1}{x_{n}} .

Therefore, we have

(1.5)  I^{+}(H^{n+1}) ≅ PSL(2, C_{n}) for n ≥ 1.

Among the generators of PSL(2, C_{n}) one has the

(1.6)  dilation ( \begin{matrix}ρ & 0 \\ 0 & 1/ρ \end{matrix} ) , x ↦ ρ^{2}x, ρ ≠ 0;
(1.7)  direct orthogonal  ( \begin{matrix}a & 0 \\ 0 & a′ \end{matrix} ) , x ↦ axa^{∗}, |a| = 1;
(1.8)  inversion  ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , x ↦ −x^{−1};
(1.9)  translation  ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) , x ↦ x + μ, μ ∈ E^{n}.

By [CW, Section 6], a notion of generalized cross ratio for an ordered tuple {x_{1}, x_{2}, x_{3}, x_{4}} of four points in E^{n}, no three of which coincide, is defined by

(1.10)  [x_{1}, x_{2}, x_{3}, x_{4}] := (x_{1} − x_{3})(x_{1} − x_{2})^{−1}(x_{2} − x_{4})(x_{3} − x_{4})^{−1}

and extended to allow x_{1} , x_{2} or x_{4} to be infinite. This cross ratio satisfies several properties such as

(1.11)  \begin{aligned}[x_{1}, x_{2}, x_{3}, x_{4}] & = [x_{4}, x_{2}, x_{3}, x_{1}]^{−1}; \\ [∞, x, 0, y] & = (y − x)y^{−1}. \end{aligned}

Finally, for a Möbius transformation

T = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ PSL(2, C_{n}),

there is the following pseudo-invariance

(1.12) [T (x_{1}), T (x_{2}), T (x_{3}), T (x_{4})] = (cx_{3} + d)^{∗}^{−1}[x_{1}, x_{2}, x_{3}, x_{4}](cx_{3} + d)^{∗}.

We are interested in the geometry of a finitely generated discrete group of Möbius transformations, or more precisely, of its two generator subgroups. For PSL(2, C), Jørgensen’s trace inequality (cf. [J], [Be, Section 5.4]) provides a very satisfactory picture. Using similar techniques, P. Waterman [Wat] generalized the result in various ways.

Theorem 1.1 ([Wat, Theorem 9]). Let

S = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and T = ( \begin{matrix}λ & 0 \\ 0 & λ^{∗−1} \end{matrix} )

generate a discrete and non-elementary subgroup in PSL(2, C_{n}). Let λ̃ = 2λ(0) − λ. Then,

|\overline{λ̃} − λ^{−1}|^{2} · (1 + |bc|) ≥ 1 if |λ| ≠ 1.

Theorem 1.2 ([Wat, Theorem 8]). Let

S = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and T = ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) , μ ∈ E^{n},

generate a discrete and non-elementary subgroup in PSL(2, C_{n}). Then,

|c| · |μ| ≥ 1.

Proof. We present a proof different in parts from [Wat] by adapting ideas of [Sh, p. 42]. Define recursively the sequence

S_{0} = ( \begin{matrix}a_{0} & b_{0} \\ c_{0} & d_{0} \end{matrix} ) := S, S_{n} = ( \begin{matrix}a_{n} & b_{n} \\ c_{n} & d_{n} \end{matrix} ) := S_{n−1}T S_{n−1}^{−1} for n ≥ 1.

By (1.2), one computes

(1.13)  \begin{aligned}a_{n} & = −a_{n−1}μc_{n−1}^{∗} + a_{n−1}d_{n−1}^{∗} − b_{n−1}c_{n−1}^{∗}, \\ b_{n} & = a_{n−1}μa_{n−1}^{∗} − a_{n−1}b_{n−1}^{∗} + b_{n−1}a_{n−1}^{∗}, \\ c_{n} & = −c_{n−1}μc_{n−1}^{∗} + c_{n−1}d_{n−1}^{∗} − d_{n−1}c_{n−1}^{∗}, \\ d_{n} & = c_{n−1}μa_{n−1}^{∗} + (a_{n−1}d_{n−1}^{∗} − b_{n−1}c_{n−1}^{∗})^{∗}. \end{aligned}

Since a_{k}d_{k}^{∗} − b_{k}c_{k}^{∗} = 1, k ≥ 0, and since x = x^{∗} for the vectors x = a_{n−1}b_{n−1}^{∗}, c_{n−1}d_{n−1}^{∗}, we deduce

\begin{aligned}a_{n} & = 1 − a_{n−1}μc_{n−1}^{∗}, \\ b_{n} & = a_{n−1}μa_{n−1}^{∗}, \\ c_{n} & = −c_{n−1}μc_{n−1}^{∗}, \\ d_{n} & = 1 + c_{n−1}μa_{n−1}^{∗}. \end{aligned}

By induction, one obtains

|c_{n}μ| = |cμ|^{2^{n}},

and, by means of the triangle inequality,

|a_{n} − 1| = |d_{n} − 1| ≤ |cμ|^{2^{n−1}} ∑_{k=0}^{∞} |cμ|^{k} + |a_{0} − 1| · |cμ|^{2^{n−1}+···+1}.

Suppose that |cμ| < 1. Then,

|a_{n} − 1| = |d_{n} − 1| ≤ \frac{|cμ|^{2^{n−1}}}{1 − |cμ|} + |a − 1| · |cμ|^{2^{n}−1}.

Therefore, c_{n} → 0, a_{n}, d_{n} → 1 and b_{n} → μ, that is, the sequence S_{n} converges to T. But ⟨S, T ⟩ is discrete, so that S_{n} = T —and especially c_{n} = 0—for all n sufficiently large. Hence, c = 0, and the group ⟨S, T ⟩ is elementary of parabolic type which is a contradiction to the assumption.

1.3. Loxodromic elements in PSL(2, H). Consider hyperbolic space H^{4} as subset of {x = x_{0} + x_{1}i_{1} + x_{2}i_{2} + yi_{3} ∈ C_{3} | y > 0}. By (1.5), we have I^{+}(H^{4}) ≅ PSL(2, H). The coefficient algebra H of quaternions is a real vector space with basis

1, i := i_{1}, j := i_{2}, k := ij satisfying i^{2} = j^{2} = k^{2} = −1.

Accordingly, each element a = a_{0} + a_{1}i + a_{2}j + a_{3}k ∈ H can be written as a = Sa + V a with scalar part Sa := a_{0} and, by abuse of language, vector part V a := a_{1}i + a_{2}j + a_{3}k. Since

\begin{aligned}\overline{a} & = Sa − V a, \\ |a|^{2} & = a\overline{a} = (Sa)^{2} − (V a)^{2}, \end{aligned}

a unit quaternion a ∈ H, i.e., |a|^{2} = a_{0}^{2} + (a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) = 1, may be expressed as

(1.14)  a = cos α + q sin α for some α ∈ [0, 2π),

where q is a pure unit quaternion, i.e., q satisfies Sq = 0 and therefore q = −\overline{q}. Actually, q = V a/|V a|. Since q^{2} = −1, de Moivre’s theorem yields (cf. [Co])

(1.15)  a^{r} = cos rα + q sin rα, for all r ∈ R.

The above motivates the notation exp(qα) := cos α + q sin α. For a fixed q, this function satisfies all well-known properties such as

exp(qα) · exp(qβ) = exp(q(α + β)) = exp(qβ) · exp(qα).

Notice that by (1.1) this picture fails for Clifford numbers a ∈ C_{n} for n > 3.

Now, consider a loxodromic element φ ∈ I^{+}(H^{4}) with fixed points u, v ∈ ∂H^{4} . The geodesic joining u and v is the unique line left invariant by φ, the so-called axis a_{φ} of φ, restricted to which φ acts as a translation with the translational length τ_{1} = 2 log ρ, say. Globally, φ acts as a dilation φ_{1} with multiplier e^{τ_{1}} = ρ^{2} ≠ 1 followed by a special orthogonal transformation φ_{2} , rotating in some 2-plane by an angle τ_{2} = 2ω, say, and having one rotational axis (cf. (1.6), (1.7)).

In order to represent φ by a matrix in PSL(2, H), we simplify first. Conjugate φ ∈ M^{+}(Ê^{3}) to obtain the Möbius transformation ψ̃(x) = ρ^{2}Ax in M^{+}(Ê^{3}) rotating in the plane spanned by 1, i with fixed j -axis, i.e.,

A = ( \begin{matrix}cos 2ω & − sin 2ω & 0 \\ sin 2ω & cos 2ω & 0 \\ 0 & 0 & 1 \end{matrix} ) ∈ SO(3).

Then, the loxodromic transformation ψ̃(x) = ρ^{2}Ax can be represented by

(1.16)  T = Tψ̃ = ( \begin{matrix}ρ exp(iω) & 0 \\ 0 & ρ^{−1} exp(−iω) \end{matrix} ) ∈ PSL(2, H).

By (1.3), Tr(T) = 2 cosh(τ_{1}/2) · cos(τ_{2}/2). In analogy to the complex case (cf. [M, p. 1040]), we call the conjugacy invariant pair τ = (τ_{1}, τ_{2}) the (quaternionic) length vector of the loxodromic transformation φ ∈ I^{+}(H^{4}) (and of T ∈ PSL(2, H)).

Later, we shall need the following property of loxodromic elements (cf. Proposition 2.6). Consider an arbitrary loxodromic element T ∈ PSL(2, H) with quaternionic length vector τ = (τ_{1}, τ_{2}).

Lemma 1.3. Let T ∈ PSL(2, H) be a loxodromic element with axis a_{T} and with quaternionic length vector (τ_{1}, τ_{2}). Let p ∈ H^{4} such that p ∉ a_{T} and assume that the perpendicular from p to a_{T} meets a_{T} at p̂. Denote by α = α(p) the angle at p̂ in the triangle (p, p̂, T_{2}(p)). Let d = dist(p, T (p)) and δ = dist(p, a_{T}). Then,

(1.17)  cosh d = cosh τ_{1} + sinh^{2} δ · (cosh τ_{1} − cos α).

Proof. Assume without loss of generality that

T = ( \begin{matrix}e^{τ_{1}/2} exp(iτ_{2}/2) & 0 \\ 0 & e^{−τ_{1}/2} exp(−iτ_{2}/2) \end{matrix} ) ,

that is, a_{T} equals the positive i_{3} -axis, and T_{2} is a rotation in the (1, i)-plane of {x = x_{0} + x_{1}i + x_{2}j + yi_{3} ∈ C_{3}}.

Observe that δ = dist(p, p̂) = dist(|p|i_{3}, p) = dist(|p|i_{3}, T_{2}(p)) satisfying cosh δ = |p|/p_{3} . Now, project the hyperbolic triangle (p, |p|i_{3}, T_{2}(p)) orthogonally down to {y = 0} ≃ E^{3} and use Euclidean trigonometry to verify that

(1.18)  cos α = \frac{(p_{0}^{2} + p_{1}^{2}) cos τ_{2} + p_{2}^{2}}{p_{0}^{2} + p_{1}^{2} + p_{2}^{2}} ≥ cos τ_{2}.

Write p = z + p_{2}j + p_{3}i_{3} for some z = r exp(iψ) with r^{2} = p_{0}^{2} + p_{1}^{2} . Then, we obtain

cosh d = 1 + \frac{|p − T (p)|^{2}}{2p_{3}^{2}e^{τ_{1}}} = 1 + \frac{|p − e^{τ_{1}} exp(iτ_{2}/2)p exp(iτ_{2}/2)|^{2}}{2p_{3}^{2}e^{τ_{1}}}

= 1 + \frac{|(r exp(iψ), p_{2}, p_{3}) − e^{τ_{1}}(r exp(i(ψ + τ_{2})), p_{2}, p_{3})|^{2}}{2p_{3}^{2}e^{τ_{1}}}

= 1 + \frac{|r exp(iψ) − re^{τ_{1}} exp(i(ψ + τ_{2}))|^{2} + (p_{2}^{2} + p_{3}^{2})(e^{τ_{1}} − 1)^{2}}{2p_{3}^{2}e^{τ_{1}}}

= cosh τ_{1} + 2\frac{p_{2}^{2}}{p_{3}^{2}} sinh^{2}(τ_{1}/2) + \frac{p_{0}^{2} + p_{1}^{2}}{p_{3}^{2}} (cosh τ_{1} − cos τ_{2})

= cosh τ_{1} + 2 sinh^{2}(τ_{1}/2)\frac{p_{0}^{2} + p_{1}^{2} + p_{2}^{2}}{p_{3}^{2}} + \frac{p_{0}^{2} + p_{1}^{2} − (p_{0}^{2} + p_{1}^{2}) cos τ_{2}}{p_{3}^{2}}

= cosh τ_{1} + sinh^{2} δ · (cosh τ_{1} − cos α),

since sinh^{2} δ = (p_{0}^{2} + p_{1}^{2} + p_{2}^{2})/p_{3}^{2} . This proves (1.17).

2. The collar theorem

2.1. Statement of the result. Let M denote an oriented hyperbolic 4manifold of finite volume. Then M can be written as a quotient H^{4}/Γ by a discrete torsion-free subgroup Γ < PSL(2, H) which is non-elementary.

Let g be a simple (i.e., with no self-intersection) closed geodesic in M of length l(g). Consider a collar or tubular neighborhood

T_{g}(r) = {p ∈ M | dist(p, g) < r}

around g embedded in M of collar width or radius r = r(g) ≥ 0. We present a non-trivial lower bound for r depending on l(g), only, which generalizes results of Brooks–Matelski [BM] and Meyerhoff [M].

Theorem 2.1. Let l_{0} = (\sqrt{3} /4π) log^{2} 2 ≃ 0.06622. Then, each simple closed geodesic g in M of length l(g) ≤ l_{0} has a collar of radius r satisfying

cosh(2r) = \frac{1 − 3k}{k} , where k = cosh \sqrt{\frac{4πl(g)}{\sqrt{3}}} − 1.

2.2. The proof. Each lift g̃ of g in H^{4} is the axis of a loxodromic transformation in Γ given by some matrix T ∈ PSL(2, H) having length vector τ = (τ_{1}, τ_{2}). Here, τ_{1} = l(g) and τ_{2} encodes the twisting of T. We call τ also the length vector of g.

Two different lifts g̃_{1}, g̃_{2} of g give rise to Γ-conjugate loxodromic elements T_{1} , T_{2} with disjoint axes a_{T_{1}} , a_{T_{2}} but equal length vector τ. Denote by p the common perpendicular line of a_{T_{1}} , a_{T_{2}} . We have to estimate the length 2r of the segment of p between a_{T_{1}} and a_{T_{2}} in terms of the translational length τ_{1} . This will be done in two steps.

Since τ is a conjugacy invariant, we conjugate Γ such that we may assume that (cf. Section 1.3)

T_{1} = ( \begin{matrix}e^{τ_{1}/2} exp(jτ_{2}/2) & 0 \\ 0 & e^{−τ_{1}/2} exp(−jτ_{2}/2) \end{matrix} ) ,
T_{2} =UT_{1}U^{−1} with U = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

Since ⟨T_{1}, T_{2}⟩ is non-elementary, ⟨T_{1}, U⟩ is non-elementary as well (cf. Section 1.1). Now, Theorem 1.1 applied to ⟨T_{1}, U⟩ yields

(2.1) |e^{τ_{1}/2} exp(jτ_{2}/2) − e^{−τ_{1}/2} exp(−jτ_{2}/2)|^{2} · (1 + |bc|) = 2k · (1 + |bc|) ≥ 1,

where k = cosh τ_{1} − cos τ_{2} .

Next, we have to relate |bc| to 2r (cf. [Be, p. 112] for the case of dimension 3). For this, we take a Möbius transformation

V = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ PSL(2, H)

which maps 0, ∞, U(0), U(∞), the fixed points of T_{1} , T_{2} , to −w, w, −1, 1 with |w| > 1, say. Hence, the common perpendicular p is mapped to the positive i_{3} -axis, and

2r = dist(a_{T_{1}}, a_{T_{2}}) = dist(a_{UT_{1}}, a_{UT_{2}}) = log |w|.

Furthermore, by Section 1.2 and (1.12), we have

[−1, 1, −w, w] = δ^{∗}^{−1}[bd^{−1}, ac^{−1}, 0, ∞]δ^{∗}.

A short computation, using ad^{∗} − bc^{∗} = 1 and (1.11), yields

(2.2)  \frac{1}{4}(1 − w)^{2}w^{−1} = δ^{∗}^{−1}bc^{∗}δ^{∗}.

Now, write w = ρ exp(Iω) in E^{3} for some ω ∈ [0, 2π) and a unit pure element I ∈ H(1). Then,

2r = log ρ.

Putting z := (2r + Iω)/2, we obtain

w = e^{2r} exp(Iω) =: exp(2r + Iω) = exp(2z).

Next, define

sinh z := \frac{1}{2}\{exp(z) − exp(−z)\}.

Then, by (1.15),

sinh^{2} z = \frac{1}{4}\{w + w^{−1} − 2\} = \frac{1}{4}(1 − w)^{2}w^{−1}.

Moreover, one computes

(2.3)  | sinh z|^{2} = \frac{1}{2}(cosh(2r) − cos ω) ≤ \frac{1}{2}(cosh(2r) + 1).

Thus, (2.1)–(2.3) yield

\frac{1}{2}(cosh(2r) + 1) ≥ | sinh z|^{2} = |δ^{∗}^{−1}bc^{∗}δ^{∗}| = |bc| ≥ \frac{1}{2k} − 1.

Hence, we proved the following (cf. [BM, Theorem 1, p. 166] and [M, Theorem, p. 1042] in the case of dimension 3 and [CW, Lemma 9.1, p. 133] in the case of arbitrary dimension).

Lemma 2.2. Let g be a simple closed geodesic in M with quaternionic length vector τ such that k = k(τ) = cosh τ_{1} − cos τ_{2} < \frac{1}{4} . Then, there is a collar T_{g}(r) around g in M of radius r satisfying

cosh(2r) = \frac{1 − 3k}{k} .

The second step consists in estimating cosh(2r) in terms of l(g) = τ_{1} alone. This can be achieved by following the lines of proof in [M, pp. 1044–1046]. That is, observe first that Lemma 2.2 holds also for

k(T^{n}) = k(nτ) = cosh(nτ_{1}) − cos(nτ_{2}) < \frac{1}{4}, for all n ∈ N.

Then, use the following lemma which is due to D. Zagier.

Lemma 2.3. ([M, Lemma, p. 1045]). For 0 < τ_{1} < π\sqrt{3} and τ_{2} ∈ [0, 2π), there exists a number n ∈ N such that

cosh(nτ_{1}) − cos(nτ_{2}) ≤ cosh \sqrt{\frac{4πτ_{1}}{\sqrt{3}}} − 1.

By choosing τ_{1} = (\sqrt{3} /4π) log^{2} 2, Lemma 2.3 implies that

k(nτ) ≤ cosh(log 2) − 1 = \frac{1}{4}.

2.3. The relative size of a collar in M. In this paragraph we investigate some properties of the collars just constructed around sufficiently short closed geodesics in M such as the growth behavior of a collar in terms of the collar width and the mutual position of the collars around disjoint loops.

Let g be a simple closed geodesic in M of length l = l(g). If l ≤ l_{0} = (\sqrt{3} /4π) log^{2} 2, then Theorem 2.1 yields a collar T_{g}(r) around g of radius r given by

cosh(2r) = \frac{1 − 3k}{k} , where k = cosh \sqrt{\frac{4πl}{\sqrt{3}}} − 1.

Remark. It is easy to see that r = r(l) is strictly decreasing.

We show that the volume of T_{g}(r) as a function of l is strictly decreasing as well.

Lemma 2.4. Let Cyl(r, l) ⊂ H^{n} denote a hyperbolic n-cylinder of radius r with axis of length l. Then, the volume vol_{n}(Cyl(r, l)) of Cyl(l, r) is given by

vol_{n}(Cyl(r, l)) = \frac{2π}{n − 1} · l · sinh^{n−1} r.

Proof. Consider the line h ⊂ H^{n} determined by the axis of Cyl(l, r), and let S be the unit sphere in its orthogonal complement. A point x ∈ H^{n} can be represented in the coordinates (t, y, s) where t = dist(x, h), y is the projection from x to h, and s ∈ S is the tangent vector of the segment yx at y. With respect to these spherical-equidistant (or Fermi-) coordinates, the volume element of H^{n} turns into (cf. [G, (20), p. 91])

dvol_{n} = cosh t sinh^{n−2} t dt dy ds.

Therefore, we can write

vol_{n}(Cyl(r, l)) = ∫_{0}^{r} ∫_{0}^{l} ∫_{0}^{2π} cosh t sinh^{n−2} t dt dy ds

= 2π · l · ∫_{0}^{sinh r} u^{n−2} du

= \frac{2π}{n − 1} · l · sinh^{n−1} r.

Proposition 2.5. Let g denote a simple closed geodesic in M of length l ≤ l_{0} . Then, the volume vol_{4}(T_{g}(r)) of the collar T_{g}(r) of radius r = r(l) is a strictly decreasing function of l.

Proof. We have to investigate the growth of vol_{4}(T_{g}(r)) = vol_{4}(Cyl(r, l)). By Lemma 2.4, we can write

(2.4)  vol_{4}(Cyl(r, l)) = 2π/3 · l · sinh^{3} r = 2/3 · sinh r · vol_{3}(Cyl(r, l)).

Now, sinh r is strictly monotonely decreasing in terms of l by the above remark. Moreover, the same is true for vol_{3}(Cyl(r, l)) = π · l · sinh^{2} r(l). Namely, since l · k′(l) > k(l) by [M, p. 1047], one sees that

\frac{d}{dl} (l · sinh^{2} r(l)) = \frac{d}{dl} (l · \frac{1 − 4k(l)}{2k(l)}) < 0, where k(l) = cosh \sqrt{\frac{4πl}{\sqrt{3}}} − 1.

Hence, by (2.4), the assertion is proved.

Proposition 2.6. Let g, g′ be two simple closed geodesics in M of lengths l, l′ ≤ l_{1} := (\sqrt{3}/4π) log^{2} \frac{17}{9} (< l_{0}) which do not intersect. Then, the collars T_{g}, T_{g′} of radius r(l), r(l′) are disjoint.

Proof. Let g̃, g̃′ be lifts to H^{4} of g, g′ in M = H^{4}/Γ, and denote by T, T′ ∈ Γ loxodromic elements with axes g̃, g̃′ and quaternionic length vectors τ, τ′. Put δ = dist(g̃, g̃′). We have to show that δ ≥ r + r′, where r = r(l), r′ = r(l′).

For this, conjugate T, T′ in PSL(2, H) in order to obtain the elements

X = ( \begin{matrix}e^{τ_{1}/2} exp(jτ_{2}/2) & 0 \\ 0 & e^{−τ_{1}/2} exp(−jτ_{2}/2) \end{matrix} ) , Y = ( \begin{matrix}a & b \\ c & d \end{matrix} )

of quaternionic length vectors equal to τ, τ′. Consider the element Y XY^{−1} conjugate to X whose axis a_{Y XY^{−1}} = Y (a_{X}) is disjoint from a_{X} and a_{Y} . Let p ∈ a_{X} denote the point such that δ = dist(a_{X}, a_{Y}) = dist(p, a_{Y}); p is the foot point on a_{X} of the common perpendicular of a_{X}, a_{Y} . By construction, d := dist(p, Y (p)) ≥ 2r. Now, using the notation k′ := k(Y) = cosh τ′_{1} − cos τ′_{2} , Lemma 1.3 and (1.18) imply that

cosh(2r) ≤ cosh d = cosh τ′_{1} + sinh^{2} δ(cosh τ′_{1} − cos α)
≤ k′ + 1 + sinh^{2} δ · k′ = cosh^{2} δ · k′ + 1.

By Theorem 2.1, we deduce that

cosh(2δ) = 2 cosh^{2} δ − 1 ≥ 2 · \frac{cosh(2r) − 1}{k′} − 1 = 2 · \frac{1 − 4k}{kk′} − 1

= \frac{1 − 4k}{kk′} + \frac{1 − 4k − kk′}{kk′} .

Suppose that k′ ≥ k (otherwise, exchange the role of X and Y ). Then, we obtain

cosh(2δ) ≥ \frac{\sqrt{1 − 4k}}{k} · \frac{\sqrt{1 − 4k′}}{k′} + \frac{1 − 4k − kk′}{kk′} .

By assumption, l ≤ l_{1} = (\sqrt{3}/4π) log^{2} 17/9 < (\sqrt{3}/4π) log^{2}(11 + \sqrt{40} )/9. Therefore, by Theorem 2.1,

k = cosh \sqrt{\frac{4πl}{\sqrt{3}}} − 1 < \frac{2}{9}.

Hence,

cosh(2r) = \frac{1 − 3k}{k} < \frac{\sqrt{1 − 4k}}{k} ,

and similarly for cosh(2r′). In order to conclude that cosh(2δ) ≥ cosh(2r + 2r′) = cosh(2r) · cosh(2r′) + sinh(2r) · sinh(2r′), we will show that

\frac{1 − 4k − kk′}{kk′} ≥ \frac{\sqrt{1 − 4k − k^{2}}}{k} · \frac{\sqrt{1 − 4k′ − k^{′2}}}{k′} ≥ sinh(2r) · sinh(2r′).

First, it is very easy to see that

(1 − 4k − k^{2}) · (1 − 4k′ − k^{′2}) ≤ (1 − 4k − kk′) · (1 − 4k′ − kk′).

Therefore,

\begin{aligned}\frac{1 − 4k − kk′}{kk′} & ≥ \frac{\sqrt{1 − 4k − kk′}}{k} · \frac{\sqrt{1 − 4k′ − kk′}}{k′} \\ & ≥ \frac{\sqrt{1 − 4k − k^{2}}}{k} · \frac{\sqrt{1 − 4k′ − k^{′2}}}{k′} . \end{aligned}

Secondly, since k < \frac{2}{9} , it can be checked that

sinh(2r) = \frac{\sqrt{1 − 6k + 8k^{2}}}{k} < \frac{\sqrt{1 − 4k − k^{2}}}{k} ,

and similarly for sinh(2r′). This finishes the proof.

2.4. Cusps and collars. In this paragraph, assume that M = H^{4}/Γ is a non-compact oriented manifold of finite volume. That is, Γ < PSL(2, H) is a discrete non-elementary group without torsion containing parabolic elements which give rise to cusps in M.

In general, a cusp C ⊂ H^{n}/Γ can be written as C = C_{q} = V_{q}/Γ_{q} for some point q ∈ ∂H^{n}, where Γ_{q} < Γ is of parabolic type with fixed point q (cf. Section 1.1), and where V_{q} ⊂ H^{n} is some precisely invariant horoball based at q. Actually, one can associate to Γ_{q} a particular horoball B_{q} based at q such that B_{q}/Γ_{q} embeds in H^{n}/Γ. Assume for simplicity that q = ∞. By a theorem of Bieberbach, the subgroup Λ = Λ(∞) ⊂ Γ_{∞} consisting of all translations (cf. (1.9)) is a lattice of finite index and rank n − 1. Let μ ∈ Λ ≅ E^{n−1} denote a shortest nontrivial vector with associated translation t_{μ} (cf. (1.9)). Then,

B(μ) = B_{∞}(μ) = \{x ∈ H^{n} | sinh \frac{1}{2}dist(x, t_{μ}(x)) < \frac{1}{2}\} = \{x ∈ H^{n} | x_{n} > |μ|\}

is called the canonical horoball of Γ_{∞}. Generalizing [Be, Theorem 5.4.4]), S. Hersonsky [H] showed that B_{∞}(μ)/Γ_{∞} embeds in M and that the canonical horoballs associated to inequivalent parabolic elements in Γ are disjoint. The following two lemmata provide a different proof; this one is inspired by [Sh].

Lemma 2.7. Let Γ ⊂ PSL(2, C_{n−1}) be a discrete non-elementary subgroup

containing the translation ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ). Define the horoball

B(μ) := {x ∈ H^{n} | x_{n} > |μ|}.

Then, for all S ∈ Γ − Γ_{∞},

S(B(μ)) ∩ B(μ) = ∅.

Proof. Suppose that there is an S ∈ Γ − Γ_{∞} such that S(B(μ)) ∩ B(μ) ≠ ∅.

Hence, there are points x, y ∈ B(μ) such that S(x) = y. Write S = ( \begin{matrix}a & b \\ c & d \end{matrix} ).

Since S ∉ Γ_{∞}, one has c ≠ 0 and S^{−1}(∞) = −c^{−1}d ∈ E^{n−1} with [c^{−1}d]_{n} = 0. By (1.4), we deduce

(2.8)  |μ| < y_{n} = [S(x)]_{n} = \frac{x_{n}}{|cx + d|^{2}} ≤ \frac{1}{|c|^{2} · x_{n}} ,

since |cx + d|^{2} = |c|^{2} · |x + c^{−1}d|^{2} ≥ |c|^{2} · x_{n}^{2} . Therefore, |c|^{2} · |μ|^{2} < 1, which contradicts Theorem 1.2.

Lemma 2.8. Let Γ ⊂ PSL(2, C_{n−1}) be a discrete non-elementary subgroup. For i = 1, 2, let T_{i} ∈ Γ be a parabolic element with fixed point q_{i} such that

S_{i}^{−1}T_{i}S_{i} = ( \begin{matrix}1 & μ_{i} \\ 0 & 1 \end{matrix} ) for some S_{i} ∈ PSL(2, C_{n−1}), and put K_{i} := S_{i}(B(μ_{i})) =

{S_{i}(x) | x_{n} > |μ_{i}|}. Then,

U(K_{1}) ∩ K_{2} ≠ ∅ for some U ∈ Γ ⟹ U(q_{1}) = q_{2}.

Proof. Consider the matrix V = ( \begin{matrix}a & b \\ c & d \end{matrix} ) := S_{2}^{−1}US_{1} ∈ PSL(2, C_{n−1}).

By assumption, there are y_{i} ∈ B(μ_{i}), i = 1, 2, such that V (y_{1}) = y_{2} . Since S_{i}(∞) = q_{i}, i = 1, 2, we have to show that V (∞) = ∞, or equivalently, c = 0.

Let W = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) := V S_{1}^{−1}T_{1}S_{1}V^{−1} . Then, W = S_{2}^{−1}UT_{1}U^{−1}S_{2} implying

that W ∈ S_{2}^{−1}ΓS_{2} , and by (1.13),

γ = −cμ_{1}c^{∗} + cd^{∗} − dc^{∗} = −cμ_{1}c^{∗}.

Since V (y_{1}) = y_{2} , we obtain in analogy to (2.8) that |c|^{2}|μ_{1}μ_{2}| < 1, whence

(2.9)  |γ| · |μ_{2}| = |c|^{2}|μ_{1}μ_{2}| < 1.

Consider now the elements

W = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) , S_{2}^{−1}T_{2}S_{2} = ( \begin{matrix}1 & μ_{2} \\ 0 & 1 \end{matrix} )

in the discrete group S_{2}^{−1}ΓS_{2} . They generate a discrete subgroup which, by (2.9) and Theorem 1.2, is elementary. This implies γ = c = 0.

Consider an oriented hyperbolic 4-manifold M with cusps. We show now that the canonical cusps, i.e., those covered by canonical horoballs, and the collars around short closed geodesics, i.e., simple closed geodesics g of length l(g) ≤ l_{0} = (\sqrt{3} /4π) log^{2} 2 ≃ 0.06622, are disjoint.

Theorem 2.9. Let M denote a non-compact oriented hyperbolic 4-manifold of finite volume. Then, the canonical cusps and the collars around short closed geodesics in M do not intersect.

Proof. Let M = H^{4}/Γ, and assume without loss of generality that Γ contains a parabolic element fixing ∞. Denote by

X = ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) ∈ Γ_{∞}

the translation by a vector μ ≠ 0 of minimal length in Λ(Γ_{∞}). Hence, B(μ)/Γ_{∞} is a cusp in M.

Let g be a short closed geodesic in M. Denote by Y ∈ Γ a loxodromic element with quaternionic length vector τ = (τ_{1}, τ_{2}) (cf. Section 1.3) whose axis projects to g. Consider an embedded collar T_{g}(r) around g of radius r satisfying (cf. Lemma 2.2)

cosh(2r) = \frac{1 − 3k}{k} , where k = cosh τ_{1} − cos τ_{2} < \frac{1}{4}.

We have to show that T_{g}(r) is disjoint from B(μ)/Γ_{∞}.

Since ⟨X, Y ⟩ ⊂ Γ is discrete, by Section 1.1, the fixed points of Y lie in {y = 0} ≃ E^{3} . Denote by z ∈ E^{3} the center of the circle which contains the axis of Y . Conjugate Γ suitably by a translation which maps z to 0 and the fixed points of Y to ±ρ exp(Jω). Conjugating further by a suitable rotary dilation, we may assume that this new Y has fixed points ±1. That is, we may work with

X = ( \begin{matrix}1 & μ/ρ \\ 0 & 1 \end{matrix} ) , Y = ( \begin{matrix}a & b \\ b & a \end{matrix} ) for some a, b ∈ H,

generating a discrete and non-elementary subgroup of Γ, and we have to show that

|μ|/ρ ≥ e^{r}.

By Theorem 1.2, we know that

|b| · |μ|/ρ ≥ 1.

Therefore, it is enough to prove that

(2.10)  |b|^{2} · e^{2r} ≤ 1.

First, observe that

(2.11)  e^{2r} = cosh(2r) + sinh(2r) ≤ 2 cosh(2r) = \frac{2}{k} (1 − 3k) < \frac{2}{k} .

Next, we estimate |b|^{2} proceeding as in the proof of Theorem 2.1. Let

V = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ PSL(2, H)

be a Möbius transformation mapping 0, ∞, Y (0),Y (∞) to −w, w, −1, 1 (with w ∉ R). By Section 1.2 and (1.12), we deduce that

[−1, 1, −w, w] = δ^{∗}^{−1}[ba^{−1}, ab^{−1}, 0, ∞]δ^{∗},

and therefore

\frac{1}{4}(1 − w)^{2}w^{−1} = δ^{∗}^{−1}bb^{∗}δ^{∗}.

We can write w = e^{τ_{1}} exp(Iτ_{2}) =: exp(τ_{1} + Iτ_{2}) = exp(2z) for some unit pure element I ∈ H(1). Since sinh^{2} z = \frac{1}{4}(1 − w)^{2}w^{−1} , we obtain

(2.12)  |b|^{2} = | sinh z|^{2} = \frac{1}{2}(cosh τ_{1} − cos τ_{2}) = \frac{1}{2}k.

Now, (2.12) combined with (2.11) yields (2.10).

Proposition 2.10. Consider a sequence of discrete torsion-free groups Γ_{n} < PSL(2, H) containing loxodromic elements S_{n} ∈ Γ_{n} with axes a_{S_{n}} and length

vectors τ_{n}. For τ_{n} → 0, let S_{n} → ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) for some μ ∈ E^{3} − \{0\}. Then, the

collars T_{n} around a_{S_{n}} of radius r(τ_{n}) tend to the horoball B(μ) = {x ∈ H^{4} | x_{n} > |μ|} in H^{4} .

Proof. We adopt ideas of [BM, pp. 167–168]. The translation ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) has

fixed point ∞. As usually, let μ = |μ| exp(Iω).

We may conjugate Γ_{n} by

U := ( \begin{matrix}0 & |μ|^{1/2} exp(Iω/2) \\ |μ|^{−1/2} exp(−Iω/2) & 0 \end{matrix} ) ∈ PSL(2, H),

so that

U^{−1} ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) U = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) =: P,

which fixes 0, and assume that S_{n} ∈ Γ_{n} are loxodromic elements with limit P for τ_{n} → 0. That is, S_{n}(∞) ≠ ∞ for n ≥ n_{0} , and the axis a_{S_{n}} of S_{n}, n ≥ n_{0} , is a semicircle in E4_{+} perpendicular to {y = 0}. This semicircle can be shifted by a translation t_{μ_{n}} to a semicircle centered at 0 and by a direct rotation with matrix A_{n} such that the fixed points are ±ρ_{n}^{−1} ∈ R. This, together with the fixed point condition, yields the representation

(2.13)  S_{n} = ( \begin{matrix}1 & −μ_{n} \\ 0 & 1 \end{matrix} ) A_{n}^{−1} ( \begin{matrix}a_{n} & ρ_{n}^{−1}b_{n} \\ ρ_{n}b_{n} & a_{n} \end{matrix} ) A_{n} ( \begin{matrix}1 & μ_{n} \\ 0 & 1 \end{matrix} ) .

The axis a_{S_{n}} is at hyperbolic distance | log ρ_{n}| from the Euclidean hyperplane {y = 1}. For simplicity, assume that ρ_{n} > 1.

Since S_{n} → P, the endpoints of a_{n} tend to 0 ∈ E^{4} , and we also have A_{n}t_{μ_{n}} → E. Moreover, by looking at the non-diagonal entries of S_{n} in (2.13), the same reasoning as in the proof of Theorem 2.9 (cf. (2.12)) yields

|ρ_{n}b_{n}| = ρ_{n}| sinh(τ_{n}/2)| → 1,
|ρ_{n}^{−1}b_{n}| = ρ_{n}^{−1}| sinh(τ_{n}/2)| → 0.

In the difference, this gives the asymptotic behavior

(2.14)  sinh(log ρ_{n}) ∼ \frac{1}{2| sinh(τ_{n}/2)|} for n ≥ n_{0}.

Now, associate to each S_{n}, n ≥ n_{0} , its tubular neighborhood T_{n} of radius r_{n} = r(τ_{n}) given by (cf. Lemma 2.2)

(2.15)  sinh(r_{n}) = \sqrt{\frac{1 − 2k(τ_{n})}{2k(τ_{n})}} = \sqrt{\frac{1}{4| sinh(τ_{n}/2)|^{2}} − 1}.

The hyperbolic distance from the top of T_{n} to the Euclidean hyperplane {y = 1} equals log ρ_{n}−r_{n}. Hence, the corresponding Euclidean distance behaves according to

exp(log ρ_{n} − r_{n}) − 1 ∼ \frac{sinh(log ρ_{n})}{sinh r_{n}} − 1,

and tends to 0 by (2.4) and (2.15).

3. Some applications

Let M be a compact oriented hyperbolic 4-manifold with Euler-characteristic χ(M). Closed geodesics in M come along with embedded tubular neighborhoods whose volumes increase the shorter the geodesics are. This property allows to derive some global assertions about the geometry and topology of M.

Consider the injectivity radius i(M) of M which is related to the length of a shortest (and necessarily simple) closed geodesic g_{0} in M by i(M) = \frac{1}{2}l(g_{0}). Since M is 4-dimensional, by [Re, Theorem], we have the inequality

(3.1)  i(M) ≥ const · vol_{4}(M)^{−5}.

The following result improves the exponent of vol_{4}(M) in (3.1).

Proposition 3.1. Let M denote a compact oriented hyperbolic 4-manifold. Then, the injectivity radius and the volume of M are related by

(3.2)  i(M) ≥ const · vol_{4}(M)^{−2}.

Proof. Assume that there is a short simple closed geodesic g of length l in M. By Theorem 2.1, there is a tubular neighborhood T_{g}(r) ⊂ M around g of radius r satisfying

sinh^{2} r = \frac{1}{2k} − 2, where k = cosh \sqrt{\frac{4πl}{\sqrt{3}}} − 1.

By Lemma 2.4, we get

vol_{4}(M) ≥ vol_{4}(T_{g}(r)) = \frac{2π}{3} · l · sinh^{3} r.

Since sinh^{3} r ∼ const · l^{−3/2} , for small l, we obtain

l ≥ const · vol_{4}(M)^{−2}

as desired.

Remark. By [S, Theorem], the first positive element λ_{1}(M) in the discrete spectrum associated to the eigenvalue problem △f + λf = 0 shows the same behavior (3.2) with respect to the volume vol_{4}(M) like the injectivity radius i(M), namely, λ_{1}(M) ≥ const · vol_{4}(M)^{−2} . More generally, there is a close relationship between the length (or geometric) spectrum and the eigenvalue spectrum of a compact hyperbolic n-manifold M (cf. [BB] and [Ri]). Well understood are their asymptotic distributions. For example, if Π_{M}(t) denotes the number of all free homotopy classes of length ≤ t on M, then Π_{M}(t) < ∞, and

Π_{M}(t) ∼ \frac{e^{(n−1)t}}{(n − 1)t} for t → ∞.

Moreover, two compact hyperbolic n-manifolds have the same eigenvalue spectrum if and only if they have the same length spectrum.

Next, we present a bound for the number of different short closed geodesics in M. Two closed geodesics g, g′ in M are called inequivalent if they do not belong to the same free homotopy class of closed curves in M (cf. [Bu, Section 1.6]).

Proposition 3.2. Let M denote a compact oriented hyperbolic 4-manifold with Euler-characteristic χ(M). Let g_{1}, …, g_{m} be pairwise disjoint inequivalent simple closed geodesics in M of length < 0.04. Then, m < 100χ(M).

Proof. As above, each of the simple closed geodesics g_{i}, i = 1, …, m, of length l(g_{i}) < l_{1} := 0.04 < l_{0} = (\sqrt{3}/4π) log^{2} 2 has a collar T_{i} = T_{i}(r_{i}) embedded in M of radius r_{i} = r(l(g_{i})). By Proposition 2.6, the collars T_{1}, …, T_{m} are pairwise disjoint. Therefore, vol_{4}(M) ≥ ∑_{i=1}^{m} vol_{4}(T_{i}). By Lemma 2.4 and by Proposition 2.5, we obtain

∑_{i=1}^{m} vol_{4}(T_{i}) ≥ m · vol_{4} (T (r(l_{1}))) = \frac{2π}{3} · m · l_{1} · sinh^{3}(r(l_{1})).

On the other hand, the theorem of Gauss–Bonnet–Chern (cf. [Ch]) gives

vol_{4}(M) = \frac{4}{3}π^{2}χ(M),

which implies the bound m < 100χ(M).

Remark. The proof shows that the data in the assertion of Proposition 3.2 can be varied. In general, the number m of pairwise disjoint inequivalent simple closed geodesics in M decreases the smaller the upper bound on their lengths is.

Example. There is one explicit geometrical construction of a compact oriented hyperbolic 4-manifold M^{∗} which is due to M. Davis [D]. His construction is based on identifying through transvections opposite dodecahedral facets in the regular polytope P = {5, 3, 3} ⊂ H^{4} , the so-called 120-cell, with dihedral angles equal to \frac{2}{5}π. The polytope P can be decomposed into 14,400 congruent orthoschemes each of volume 13π^{2}/5,400. This implies that the manifold M^{∗} has Euler-characteristic 26 (cf. [D] and also [K1, p. 92]). The trigonometry of the dissecting orthoschemes can be well controlled. In particular, the edge length L of P can be computed by

cosh L = \frac{3τ}{2 − τ} , where τ = 2 cos \frac{π}{5} ,

which yields L ≃ 3.23384. Therefore, shortest closed geodesics in M^{∗} are of lengths ≤ 3.23384. On the other side, proceeding as in the proof of Proposition 3.2, an arbitrary compact oriented hyperbolic 4-manifold M with χ(M) = 26 has closed geodesics of lengths > 9.82 · 10^{−8} .

References

[A1] Ahlfors, L.V.: Möbius transformations and Clifford numbers. - In: Differential Geometry and Complex Analysis—in Memory of H.E. Rauch, edited by I. Chavel and H. Farkas. Springer-Verlag, 1985.

[A2] Ahlfors, L.V.: Möbius transformations in R^{n} expressed through 2 × 2 matrices of Clifford numbers. - Complex Variables Theory Appl. 5, 1986, 215–224.

[Be] Beardon, A.F.: The Geometry of Discrete Groups. - Springer-Verlag, 1983.

[BB] Bérard Bergery, L.: Sur les longueurs des géodésiques périodiques et le spectre des formes d’espace hyperbolique compactes. - In: Variétés à courbure négative: Séminaire de géométrie riemannienne 1970–1971, edited by L. Bérard Bergery, J. P. Bourguignon, E. Mazet. Publications mathématiques de l’Université Paris VII, 1979.

[BM] Brooks, R., and J. Matelski: Collars in Kleinian groups. - Duke Math. J. 49, 1982, 163–182.

[Bu] Buser, P.: Geometry and spectra of compact Riemann surfaces. - Birkhäuser, 1992.

[CW] Cao, C., and P.L. Waterman: Conjugacy invariants of Möbius groups. - In: Quasiconformal Mappings and Analysis, A Collection of Papers Honoring F.W. Gehring, edited by P.L. Duren et al. Springer-Verlag, 1998.

[Ch] Chern, S.S.: On the curvature and characteristic classes of a Riemannain manifold. - Abh. Math. Sem. Univ. Hamburg 20, 1956, 117–126.

[Co] Coxeter, H.S.M.: Quaternions and reflections. - Amer. Math. Monthly 53, 1946, 136– 146.

[D] Davis, M.W.: A hyperbolic 4-manifold. - Proc. Amer. Math. Soc. 93, 1985, 325–328.

[G] Geometry II: In: Encyclopaedia of Math. Sciences, vol. 29, edited by E.B. Vinberg, Springer-Verlag, 1993.

[H] Hersonsky, S.: Covolume estimates for discrete groups of hyperbolic isometries having parabolic elements. - Michigan Math. J. 40, 1993, 467–475.

[J] Jørgensen, T.: On discrete groups of Möbius transformations. - Amer. J. Math. 98, 1976, 739–749.

72 Ruth Kellerhals

[K1] Kellerhals, R.: Volumina von hyperbolischen Raumformen. - Preprint MPI/95–110, 1995.

[K2] Kellerhals, R.: Volumes of cusped hyperbolic manifolds. - Topology 37, 1998, 719–734.

[M] Meyerhoff, R.: A lower bound for the volume of hyperbolic 3-manifolds. - Canad. J. Math. 39, 1987, 1038–1056.

[Ra] Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. - Springer-Verlag, 1994.

[Re] Reznikov, A.: The volume and the injectivity radius of a hyperbolic manifold. - Topology 34, 1995, 477–479.

[Ri] Riggenbach, H.: Freie Homotopieklassen und das Eigenwertspektrum des Laplace-Ope-rators. - PhD thesis, Basel, 1975.

[S] Schoen, R.: A lower bound for the first eigenvalue of a negatively curved manifold. - J. Differential Geom. 17, 1982, 233–238.

[Sh] Shimizu, H.: On discontinuous groups operating on the product of the upper half planes. - Ann. Math. 77, 1963, 33–71.

[V] Vahlen, K.Th.: Über Bewegungen und komplexe Zahlen. - Math. Ann. 55, 1902, 585–593.

[Wad] Wada, M.: Conjugacy invariants of Möbius transformations. - Complex Variables Theory Appl. 15, 1990, 125–133.

[Wat] Waterman, P.L.: Möbius transformations in several dimensions. - Adv. Math. 101, 1993, 87–113.

Received 16 December 1998 Received in revised form 20 January 2000