APPEARED IN BULLETIN OF THE

arXiv:math/9207211v1 [math.DG] 1 Jul 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 1, July 1992, Pages 154-158GENERALIZING THE HYPERBOLIC COLLAR LEMMAAra BasmajianAbstract. We discuss two generalizations of the collar lemma.

The first is the stableneighborhood theorem which says that a (not necessarily simple) closed geodesic ina hyperbolic surface has a “stable neighborhood”whose width only depends on thelength of the geodesic. As an application, we show that there is a lower bound forthe length of a closed geodesic having crossing number k on a hyperbolic surface.This lower bound tends to infinity with k.Our second generalization is to totally geodesic hypersurfaces of hyperbolic man-ifolds.Namely, we construct a tubular neighborhood function and show that anembedded closed totally geodesic hypersurface in a hyperbolic manifold has a tubu-lar neighborhood whose width only depends on the area of the hypersurface (andhence not on the geometry of the ambient manifold).

The implications of this re-sult for volumes of hyperbolic manifolds is discussed. We also derive a (hyperbolic)quantitative version of the Klein-Maskit combination theorem (in all dimensions) forfree products of fuchsian groups.

Using this last theorem, we construct examples toillustrate the qualitative sharpness of the tubular neighborhood function.1. IntroductionIt is natural to look for universal properties of discrete subgroups of a fixed liegroup.

The lie group we are concerned with is the orientation preserving isometriesof real hyperbolic space Isom+(Hn). A fundamental universal property in dimensionthree is the Jorgensen inequality [Jo].

Another celebrated property is the Margulislemma [Th] (actually, this lemma holds for a bigger class of lie groups). We wouldlike to focus on the well-known collar lemma due to Keen [Ke]; the sharp form isdue to Buser [Bu].

There are many different versions of this lemma in the literature(cf. [Ba1, Be, Ha, M1, Ma, R]).Let r : R+ →R+ be the function r(x) = log coth(x/2).Observe that thisfunction monotonically decreases to zero and satisfies r2 = 1.

The collar lemmasays that any simple closed geodesic of length ℓon a hyperbolic surface (a completesurface of constant curvature negative one) has a collar neighborhood of width r( ℓ2).Furthermore, a collection of disjoint simple closed geodesics have disjoint collars.This result is universal in the sense that the collar width does not depend on theunderlying hyperbolic structure of the surface, only on the length of the geodesic.We would like to discuss two generalizations of the collar lemma. In the firstgeneralization (the stable neighborhood theorem), we remove the hypothesis inthe collar lemma that the closed geodesic be simple.

The concept of a collar is1991 Mathematics Subject Classification. Primary 30F40, 53C20.Key words and phrases.

Collar, geodesic, hyperbolic manifold, self-intersection, totally geo-desic hypersurface, tubular neighborhood, volume.Received by the editors October 25, 1991 and, in revised form, January 28, 1992c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2ARA BASMAJIANthen replaced by the notion of a stable neighborhood. The second generalizationof the collar lemma involves totally geodesic hypersurfaces in hyperbolic manifolds(complete riemannian manifolds of constant curvature negative one).

Simple closedgeodesics are replaced by totally geodesic embedded hypersurfaces and the notionof a collar is then replaced by a tubular neighborhood.2. The stable neighborhood theoremThe neighborhood of width d, Ud(ω), about a geodesic ω in a hyperbolic surfaceis the set of all points within a distance d from the geodesic.

Suppose ω is a closedgeodesic. Then the neighborhood Ud(ω) is said to be stable if for any two connected(smooth) lifts of ω, say ω1 and ω2, we haveω1 ∩ω2 ̸= ∅if and only ifUd(ω1) ∩Ud(ω2) ̸= ∅.Theorem 1 (The Stable Neighborhood Theorem).

A closed geodesic of length ℓon a hyperbolic surface has a stable neighborhood of width r( ℓ2). Furthermore, twodisjoint closed geodesics ω1 and ω2 on the surface having lengths ℓ1 and ℓ2 havedisjoint stable neighborhoods of widths r( ℓ12 ) and r( ℓ22 ) respectively, if ω1 and ω2are separated by a disjoint union of simple closed geodesics.The above separation condition is necessary, since there are examples of closedgeodesics that get arbitrarily close to any fixed boundary geodesic on a pair ofpants.Hempel, Nakanishi, and Yamada [He, N, Y1, Y2]) have independently shownthat there is a universal lower bound for the length of a nonsimple closed geodesicon a hyperbolic surface.

As a consequence of the stable neighborhood theorem wehaveCorollary 2. There exists an increasing sequence Mk (for k = 1, 2, 3, .

. .

), tendingto infinity so that if ω is a closed geodesic with self-intersection number k, thenℓ(ω) > Mk. Thus the length of a closed geodesic gets arbitrarily large as its self-intersection gets large.3.

The tubular neighborhood theoremIn [Ba2], it was shown that associated to any totally geodesic hypersurface ina hyperbolic manifold there exists a spectrum of numbers called the orthogonalspectrum. This spectrum is essentially the lengths of orthogonals that start andend in the hypersurface.

The following can be thought of as a geometric study ofthe first element in that spectrum.Let Vn(r) be the volume of the n-dimensional hyperbolic ball of radius r. Thefunctioncn(A) = 12(Vn−1 ◦r)−1(A)is called the n-dimensional tubular neighborhood function. Observe that this func-tion is monotone decreasing and tends to zero as A goes to infinity.

The area ofa hypersurface is with respect to the induced metric from the ambient hyperbolicmanifold.The following theorem shows that a closed embedded totally geodesic hypersur-face in a hyperbolic manifold has a tubular neighborhood whose width only dependson the area of the hypersurface.

GENERALIZING THE HYPERBOLIC COLLAR LEMMA3Theorem 3 (The Tubular Neighborhood Theorem). Suppose M n is a hyperbolicmanifold containing Σ, an embedded closed totally geodesic hypersurface of area A.Then Σ has a tubular neighborhood of width cn(A).

That is, the set of points{x ∈M : d(x, Σ) < cn(A)}is isometric to the product Σ × (−cn(A), cn(A)).Furthermore, any disjoint set of such hypersurfaces has disjoint tubular neigh-borhoods.The main idea in the proof of the tubular neighborhood theorem is that thehypersurface in question must contain an embedded disc of radius r(d), where d isthe length of the shortest common orthogonal from Σ to itself.It is well known that there exists a constant a, so that if M is a hyperbolic threemanifold containing K rank two cusps then the volume of M is bigger than Ka (see[Th]). Analyzing volumes of tubular neighborhoods, we can prove a version of thisresult for totally geodesic hypersurfaces in hyperbolic manifolds of all dimensions.Specifically, we haveTheorem 4.

There exist positive constants an, for each n ≥3, depending onlyon the dimension n, so that if M n is a hyperbolic manifold containing K closedembedded disjoint totally geodesic hypersurfaces then,Vol(M) > Kan.For hyperbolic three manifolds, a3 can be taken to be πlog 2 +√22, which is ap-proximately 4.4.In dimensions n ≥4, if M ni is a sequence of (not necessarily distinct) hyperbolicn-manifolds each containing an embedded totally geodesic closed hypersurface ofarea Ai, where Ai →∞, then the volumes of the tubular neighborhoods of widthcn(Ai) tend to infinity. In particular, the volumes Vol(Mi) →∞.In dimension three, there exist examples of totally geodesic surfaces in hyperbolicthree manifolds whose areas get arbitrarily large, but whose best possible tubularneighborhoods have bounded volume.The main lemma needed to prove the lower volume bound in Theorem 4 is therate of growth lemma.Lemma (Rate of growth lemma).

Vn, the volume of a (one-sided) tubular neighbor-hood of width cn(x) about a hypersurface having area x, has the following behaviorat infinity,limx→∞Vn(x) =∞for n ≥4,πfor n = 3,0for n = 2.V3 is monotone increasing and V2 is monotone decreasing.We remark that Kojima and Miyamoto [KM] have found the smallest volumehyperbolic three manifolds with totally geodesic boundary. This volume is about6.45.In dimension three, a rank two cusp has a cusp neighborhood whose boundaryis a flat torus having shortest closed geodesic of length one.

Furthermore, if the

4ARA BASMAJIANhyperbolic three manifold has more than one cusp, these regions will be disjoint.In fact, this is the basis of Meyerhoff’s lower volume bound for a hyperbolic threemanifold containing rank two cusps [Me].The next corollary extends this to our setting:Corollary 5. The tubular neighborhood of width c3(A) about a totally geodesicembedded closed surface of area A in a hyperbolic three manifold is disjoint fromthe tori on the boundary of a rank two cusp having shortest closed geodesic of lengthone.

In particular, if the three manifold M has n rank two cusps and k disjointtotally geodesic closed embedded surfaces, thenVol(M) > (√3/4)n + (4.4)k.This corollary follows easily from the nextLemma. Suppose M is a hyperbolic three manifold containing a (rank one or two)cusp and an embedded closed totally geodesic surface Σ.

Let T2 be the boundarytorus or annulus having its shortest closed geodesic of length one. Then the distanced(T2, Σ) > log 2.A fuchsian subgroup in the group of isometries Isom+(Hn) is a pair (F, X) whereX is a hyperbolic hyperplane and F is a discrete subgroup keeping X and the half-spaces it bounds in Hn invariant.

The injectivity radius of F at x ∈X, denotedinj(x), is the largest hyperbolic disc centered at x whose F-translates are all disjoint.If x is an elliptic fixed (orbifold) point, then its injectivity radius is zero. There isa unique common orthogonal between any two disjoint (in Hn) hyperplanes.The following is a hyperbolic (quantitative) version of the Klein-Maskit combi-nation theorem for fuchsian subgroups in all dimensions.

Its proof makes essentialuse of the combination theorem in all dimensions (see [Ma2, Ap]).Theorem 6. Suppose (F1, X1) and (F2, X2) are fuchsian subgroups of the fullisometry group Isom+(Hn) with X1 and X2 disjoint in Hn.Let x1 ∈X1 andx2 ∈X2 be the endpoints of the unique common perpendicular between the hyper-planes X1 and X2.

If the following inequality holds(*)r(inj(x1)) + r(inj(x2)) < d(X1, X2),then the group G =< F1, F2 > is a discrete group of the second kind that, abstractly,is the free product of F1 and F2. Furthermore, the hyperbolic (n −1)-hypersurfacesΣi = Xi/Fi ( for i = 1, 2) are totally geodesic boundary hypersurfaces for the hy-perbolic manifold N = Hn/G satisfying dN(Σ1, Σ2) = d(X1, X2).

The group G istorsion-free if and only if F1 and F2 are torsion-free.The above theorem with very few exceptions holds when there is equality in (∗).As a consequence of this, we haveCorollary 7. Suppose Σ1 and Σ2 are hyperbolic (n −1)-manifolds containing em-bedded balls of radii R1 and R2, respectively.Then there exists a hyperbolic n-manifold N having totally geodesic boundary hypersurfaces Σ1 and Σ2 satisfying,dN(Σ1, Σ2) = r(R1) + r(R2).□The dimension three examples in Theorem 4 are constructed using the abovecorollary.

GENERALIZING THE HYPERBOLIC COLLAR LEMMA5The proofs of the stable neighborhood theorem, its corollary, and other resultson short nonsimple closed geodesics are contained in the paper [Ba3]. The proofsof the theorems on totally geodesic hypersurfaces, a more general form of Theorem6, along with additional consequences of this approach are contained in [Ba4].AcknowledgementsThe author would like to thank Colin Adams, Boris Apanasov, and BernardMaskit for helpful conversations.References[Ap]Boris Apanasov, Discrete groups in space and uniformization problems, Kluwer Aca-demic, Dordrecht, Netherlands, 1991.

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43 (1976), 829–840. [ Me]Bob Meyerhoff, A lower bound for the volume of hyperbolic 3-manifolds, Canad.

J. Math.39 (1987), 1038–1056. [ N]Toshihiro Nakanishi, The lengths of the closed geodesics on a Riemann surface withself-intersection, Tohoku Math.

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[ Th]William Thurston, The geometry and topology of 3-manifolds, lecture notes, PrincetonUniversity, 1977. [ Y1]Akira Yamada, On Marden’s universal constant of Fuchsian groups, Kodai Math.

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[ Y2]Akira Yamada, On Marden’s universal constant of Fuchsian groups II, J. Analyse Math.41 (1982), 234–248.Mathematics department, University of Oklahoma, Norman, Oklahoma 73019E-mail address: abasmajian@nsfuvax.math.uoknor.edu

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