Collars in Complex and Quaternionic Hyperbolic ... - Springer Link

Geometriae Dedicata 97: 199–213, 2003. # 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Collars in Complex and Quaternionic Hyperbolic Manifolds In Memory of Hanna Sandler SARAH MARKHAM and JOHN R. PARKER Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, England. e-mail: {sarah.markham, j.r.parker}@durham.ac.uk (Received: February 2003) Abstract. We use the complex and quaternionic hyperbolic versions of Jørgensen’s inequality to construct embedded collars about short, simple, closed geodesics in complex and quaternionic hyperbolic manifolds. In general, the width of these collars depend both on the length of the geodesic and on the rotational part of the group element uniformising it. For complex hyperbolic space we are able to use a lemma of Zagier to give an estimate based only on the length. We show that these canonical collars are disjoint from each other and from canonical cusps. We also calculate the volumes of these collars. Mathematics Subject Classification (2000). 5IM10. Key words. collars, complex hyperbolic manifold, Jørgensen’s inequality, quaternionic hyperbolic manifold.

Introduction In [6] Meyerhoff used Jørgensen’s inequality to show that if a simple closed geodesic in a hyperbolic 3-manifold is sufficiently short, then there exists an embedded tubular neighbourhood of this geodesic, called a collar, whose width depends only on the length (or the complex length) of the closed geodesic. Moreover, he showed that these collars were disjoint from one another and from the canonical cusps arising from Shimizu’s lemma. He went on to give various applications to the volumes of hyperbolic 3-manifolds. In [3] Kellerhals generalised many of Meyerhoff ’s results to real hyperbolic 4-space, which she views as quaternionic hyperbolic 1-space. The purpose of this paper is to use the complex hyperbolic Jørgensen’s inequality, proved in [2], to give analogues of Meyerhoff ’s (and Kellerhals’) results for short, simple, closed geodesics in complex hyperbolic manifolds. Markham has given a quaternionic hyperbolic Jørgensen’s inequality [5] and we use this to state quaternionic hyperbolic analogues of the main theorems. The proofs of these will be very similar to the complex hyperbolic proofs and so we will not give them, but will refer to [5]. Let G be a discrete group of complex or quaternionic hyperbolic isometries. Let A 2 G be loxodromic with axis the geodesic g. The tube Tr ðgÞ of radius r about g is

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SARAH MARKHAM AND JOHN R. PARKER

the collection of points a distance less than r from g. It is clear that A maps Tr ðgÞ to itself. We wish to study what happens to Tr ðgÞ under elements of G other than A. The tube Tr ðgÞ is precisely invariant under (the subgroup hAi of) G if BðTr ðgÞÞ is disjoint from Tr ðgÞ for all B 2 G hAi. If Tr ðgÞ is precisely invariant under G then Cr ðg0 Þ ¼ Tr ðgÞ=hAi is an embedded tubular neighbourhood of the simple closed geodesic g0 ¼ g=hAi. We call Cr ðg0 Þ the collar of width r about g0 . We now state our main theorems. THEOREM 2.1. Let G be a discrete, nonelementary, torsion-free subgroup of PUð2; 1Þ. Let A be a loxodromic element of G with axis the geodesic g and multiplier leiy . Define M ¼ jleiy 1j þ jl1 eiy 1j and suppose that M < 1=2. Let r 2 Rþ be defined by coshðrÞ ¼ ð1 MÞ=M. Then the tube Tr ðgÞ is precisely invariant under G. COROLLARY 2.2. Let G, A and r be as in Theorem 2:1: Then in the manifold M ¼ H2C =G the simple closed geodesic g0 ¼ g=hAi has an embedded collar Cr ðg0 Þ of width r. The same arguments, being careful about which order we multiply, give the quaternionic analogue of these two results. THEOREM 2.3. Let G be a discrete, torsion-free, nonelementary subgroup of PSpð2; 1Þ. Let A be a loxodromic element of G with axis g and eigenvalues la, l1 a, ab. Suppose that M ¼ jla 1j þ 2jab 1j þ jl1 a 1j < 1=2. Let r 2 Rþ be defined by coshðrÞ ¼ ð1 MÞ=M. Then the tube Tr ðgÞ, is precisely invariant under G. COROLLARY 2.4. Let G, A and r be as in Theorem 2:3: Then in the manifold M ¼ H2H =G the simple closed geodesic g0 ¼ g=hAi has an embedded collar Cr ðg0 Þ of width r. In the complex case, by using a lemma of Zagier, we are able to express these results entirely in terms of the length l ¼ 2 log l of the simple closed geodesic g0 ¼ g=hAi. THEOREM 2.6. Let G be a discrete, torsion-free, nonelementary subgroup of PUð2; 1Þ. Let A be a loxodromic element of G with axis g and multiplier el=2 eiy where pffiffiffi pffiffiffiffiffi 2 3 17 þ 1 log l< : 2p 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi Let r 2 Rþ be defined by coshðrÞ þ 1 ¼ 1=2sinh 2pl= 3. Then Tr ðgÞ, the tubular neighbourhood of g of radius r, is precisely invariant under G. COROLLARY 2.7. Let M be a complex hyperbolic 2-manifold and let g0 be a simple closed geodesic of length l where

COLLARS IN COMPLEX AND QUATERNIONIC HYPERBOLIC MANIFOLDS

l
0 by coshðrj Þ ¼ ð1 Mj Þ=Mj . Without loss of generality A1 has fixed points o and 1 and suppose A2 has fixed points p and q. We shall use the Cygan metric r0 on @H2C f1g (see [2, 8] for example). Then, as A1 fixes o and 1, we have, for all r0 ðA1 z; A1 wÞ ¼ l1 r0 ðz; wÞ for all 1=2 1=2 z; w 2 @H2C f1g. From Lemma 2.1 of [8] we have r0 ðA1 z; zÞ 4 M1 l1 r0 ðz; oÞ 2 for all z 2 @HC f1g. Thus we have ½A1 ð pÞ; q; p; A1 ðqÞ1=2 ¼ r0 ðA1 ð pÞ; pÞr0 ðA1 ðqÞ; qÞ r0 ð p; qÞr0 ðA1 ð pÞ; A1 ðqÞÞ r0 ð p; oÞr0 ðq; oÞ r0 ð p; qÞ2 r0 ð p; oÞ r0 ð p; oÞ þ1 4 M1 r0 ð p; qÞ r0 ð p; qÞ 4 M1

¼ M1 ðj½o; q; p; 1j1=2 Þðj½o; q; p; 1j1=2 þ 1Þ: Therefore, by the complex hyperbolic Jørgensen’s inequality, we have 1 M2 M1 j½o; q; p; 1j1=2 ðj½o; q; p; 1j1=2 þ 1Þ 5 j½A1 ðpÞ; q; p; A1 ðqÞj1=2 5 M2 which implies 1=2

j½o; q; p; 1j

1=2

ðj½o; q; p; 1j

1 M1 þ1 : M1

1 M2 þ1 : M2

1 M2 þ 1Þ 5 M2

By symmetry 1=2

j½o; q; p; 1j Therefore

1=2

ðj½o; q; p; 1j

1 M1 þ 1Þ 5 M1

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j½o; q; p; 1j1=2 ðj½o; q; p; 1j1=2 þ 1Þ ! 1 M2 1=2 1 M1 1=2 1 M2 1=2 1 M1 1=2 þ1 : 5 M2 M1 M2 M1 Since x1=2 ðx1=2 þ 1Þ increases monotonically for x > 0, we have 1 M1 1 M2 j½o; q; p; 1j 5 M1 M2 ¼ coshðr1 Þ coshðr2 Þ: Moreover, using a similar argument to that used in the proof of Theorem 2.1, we have cosh2 ð rðg1 ; g2 Þ=2Þ 5 j½o; q; p; 1j. Therefore rðg1 ; g2 Þ cosh2 5 j½o; q; p; 1j 2 5 coshðr1 Þ coshðr2 Þ coshðr1 þ r2 Þ þ coshðr1 r2 Þ ¼ 2 coshðr1 þ r2 Þ þ 1 5 2 2 r1 þ r2 ¼ cosh : 2 Thus we have rðg1 ; g2 Þ 5 r1 þ r2 . This gives the result.

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Let M ¼ H2C =G be a finite volume complex hyperbolic 2-manifold with a cusp. Suppose that G has a parabolic fixed point at 1. In [7] Parker defined a canonical horoball H, defined in terms of the (Cygan) translation lengths of elements of G1 , the stabiliser of 1 in G. We define a canonical cusp to be H=G1 . Let B be any element of G not fixing 1 and written in the form (1). Then the appropriate version of Shimizu’s lemma (see Theorem 2.2 of [7]) says that 1=jgj 4 u0 where u0 is the height of the canonical horoball. In Proposition 2.4 of [7] it is shown that canonical horoballs at distinct parabolic fixed points are disjoint. We now show that canonical horoballs at parabolic fixed points are disjoint from canonical collars about short geodesics. THEOREM 3.2. Let M denote a noncompact complex hyperbolic 2-manifold of finite volume. Then the canonical cusps and the canonical collars around simple short closed geodesics in M are disjoint. Proof. Let M ¼ H2C =G where G is a discrete non-elementary subgroup of PUð2; 1Þ and assume without loss of generality that G contains a parabolic element fixing 1. Let u0 be the height of the canonical horoball based at 1. Let B 2 G be loxodromic with multiplier leiy and define M ¼ jleiy 1jþ 1 iy jl e 1j. Suppose that M 4 1=2. Since 1 is a parabolic fixed point, B cannot fix 1. Thus, if B is written in the form (1) we have 1=jgj 4 u0 (Theorem 2.2 of [7]).


Suppose that 2 0 a b0 B ¼ 4 d0 e 0 g0 h0

32 c0 l f 0 54 0 0 j0

0 eiy 0

32 0 j 0 0 54 h0 l1 g 0

f 0 e0 d 0

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3 c0 b0 5: a 0

Then we have jgj ¼ g0 j 0 l þ jh0 j2 eiy þ j 0 g0 l1 ¼ g0 j 0 ðl eiy Þ þ j 0 g0 ðl1 eiy Þ 4 Mj j0 j jg0 j: Since B does not fix 1 we have g0 ; j 0 6¼ 0. The axis g of B is given by points ps for s 2 Rþ where 2 0 32 3 2 3 a b0 c 0 s a0 s þ c0 ps ¼ 4 d 0 e0 f 0 54 0 5 ¼ 4 d 0 s þ f 0 5: 1 g0 h0 j 0 g0 s þ j 0 Applying P and putting this into horospherical coordinates, we see that ps ¼ ðzs ; vs ; us Þ where s us ¼ 0 2 2 : jg j s þ jh0 j2 s þ j j0 j2 Elementary calculus shows that us is maximised by s ¼ s0 ¼ j j0 j=jg0 j. Thus us 0 ¼

1 : þ jh0 j2

2jj0 jjg0 j

Therefore the maximum height of a point in Tr ðgÞ is u¼

er : 2j j0 jjg0 j þ jh0 j2

Let coshðrÞ ¼ ð1 MÞ=M. We claim that er 1 : < 2 0 0 0 jgj 2j j jjg j þ jh j Since 1=jgj 4 u0 we see that this claim implies that the maximum height of a point on the canonical tube about g is less than u0 . Thus the canonical tube and canonical horoball are disjoint, which proves the result. So it remains to prove the claim. er jgj < 2 coshðrÞMj j 0 j jg0 j ¼ 2ð1 MÞj j 0 j jg0 j < 2j j 0 j jg0 j 4 2j j 0 j jg0 j þ jh0 j2 : This gives the result.

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For quaternionic hyperbolic manifolds, we define the canonical collar about a short, simple, closed geodesic to be the collar of Corollary 2.4. Then an almost identical argument to that given above yields: THEOREM 3.3. Let M be a quaternionic hyperbolic 2-manifold. Then the canonical collars around short, simple, closed geodesics are disjoint. THEOREM 3.4. Let M denote a noncompact quaternionic hyperbolic 2-manifold of finite volume. Then the canonical cusps and the canonical collars around short, simple, closed geodesics in M are disjoint.

4. Volumes of Collars In this section we calculate the volume of collars around simple closed geodesics in complex and quaternionic hyperbolic manifolds. We then apply this to canonical collars about short geodesics. We will first consider the complex hyperbolic case. Let g0 be a simple closed geodesic in a complex hyperbolic 2-manifold M ¼ H2C =G of length l. Let Cr ðg0 Þ ¼ Tr ðgÞ=hAi be a collar about g0 ¼ g=hAi of width r. This collar is formed by identifying the ends of a piece of Tr ðgÞ of core length l, which we denote by Tr;l ðgÞ or just Tr;l . PROPOSITION 4.1. Let Tr ¼ Tr ðgÞ H2C be a tube of width r about a geodesic g. Let Tr;l ¼ Tr;l ðgÞ be a section of Tr of core length l. Then VolðTr;l Þ ¼

32pl sinh3 ðr=2Þ coshðr=2Þ: 3

Proof. Without loss of generality, we take g to be the geodesic with endpoints o and 1. A segment of this geodesic of length l consists of those points with horospherical coordinates ð0; 0; tÞ where t varies between t1 and t2 with t2 =t1 ¼ el . We take t1 ¼ 1 and t2 ¼ el . We now find the distance between a general point ðz; v; uÞ and the geodesic g. We do this by finding the distance between ðz; v; uÞ and ð0; 0; tÞ and then minimising with respect to t. cosh

2

2 2 2 2 jzj þ u þ t þv2 jzj þ u þ t þ iv rððz; v; uÞ; ð0; 0; tÞÞ ¼ : ¼ 2 4ut 4ut

Elementary calculus shows that this function is minimised by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 t¼ jzj þ u þv2 : Therefore


cosh2

rððz; v; uÞ; gÞ ¼ 2

jzj2 þ u þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi jzj2 þ u þv2 2u

211

:

We want to find the set of points ðz; v; uÞ in H2C with two properties: first, the point ð0; 0; tÞ is the point of g closest to ðz; v; uÞ and, secondly, at most ðz;2 v; uÞ2is a 2distance 2 r from ð0; 0; tÞ. These two conditions are, first, that jzj þ u þv ¼ t should be constant and, secondly, that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi jzj2 þ u þ jzj2 þ u þv2 4 cosh2 ðr=2Þ: 2u That our point satisfies the first condition, allows us to introduce polar coordinates z ¼ seiy , jzj2 þ u ¼ s2 þ u ¼ t cosðfÞ, v ¼ t sinðfÞ. In these new coordinates, the volume element in H2C becomes dVol ¼

4 4st du dv dz: ¼ dt df ds dy: 3 u ðt cosðfÞ s2 Þ3

The condition that ðz; v; uÞ ¼ ðseiy ; t sinðfÞ; t cosðfÞ s2 Þ has distance at most r from ð0; 0; tÞ becomes: t cosðfÞ þ t 4 cosh2 ðr=2Þ: 2ðt cosðfÞ s2 Þ Solving for s we find sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tðcoshðrÞ cosðfÞ 1Þ : s4 coshðrÞ þ 1 We define the right-hand side of this equation to be s0 . We see that s0 is defined provided f0 4 f 4 f0 where f0 is given by cosðf0 Þ ¼ 1= coshðrÞ (equivalently tanðf0 Þ ¼ sinhðrÞ). Thus Z el Z f0 Z s0 Z 2p 4st VolðTr;l Þ ¼ dt df ds dy ðt cosðfÞ s2 Þ 3 1 f0 0 0 s0 Z e l Z f0 t dt df ¼ 2p 2 2 1 f0 ðt cosðfÞ s Þ 0 2 ! Z e l Z f0 coshðrÞ þ 1 2 1 t t dt df ¼ 2p 2t cos2 ðf=2Þ t cosðfÞ 1 f0 Z f0 Z el 1 ðcoshðrÞ þ 1Þ2 ðcoshðrÞ þ 1Þ2 sin2 ðf=2Þ 1 dt þ df ¼ 2p 2 ðf=2Þ 4 ðf=2Þ 2 ðfÞ t 4 cos cos 4 cos 1 f0 f0 ðcoshðrÞ þ 1Þ2 ðcoshðrÞ þ 1Þ2 3 ¼ 2pl tanðf=2Þ þ tan ðf=2Þ tanðfÞ 2 6 f0

212


f0 ðcoshðrÞ þ 1Þ2 sinðfÞ 1 cosðfÞ 3þ tanðfÞ cosðfÞ þ 1 1 þ cosðfÞ 6 f0 2 ðcoshðrÞ þ 1Þ sinhðrÞ coshðrÞ 1 3þ ¼ 2pl 2 sinhðrÞ coshðrÞ þ 1 coshðrÞ þ 1 3 2pl sinhðrÞð3 coshðrÞ þ 3 þ coshðrÞ 1 6Þ ¼ 3 8pl sinhðrÞðcoshðrÞ 1Þ ¼ 3 32pl ¼ sinh3 ðr=2Þ coshðr=2Þ: 3 ¼ 2pl

This gives the result.

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COROLLARY 4.2. Let g0 be a simple closed geodesic in M ¼ H2C =G corresponding to the loxodromic element A of G. Let A have multiplier leiy and define M ¼ jleiy 1j þ jl1 eiy 1j. Suppose that M < 1=2. Then the canonical collar Cr ðg0 Þ about g0 has volume VolðCr ðg0 ÞÞ ¼

16p logðlÞð1 2MÞ3=2 : 3M2

Proof. We substitute l ¼ el=2 and 2 cosh2 ðr=2Þ ¼ 1=M in the formula of Proposition 4.1. & Similarly, with the collar from Corollary 2.7 we have. COROLLARY 4.3. Let g0 be a simple closed geodesic in M ¼ H2C =G of sufficiently short length l. Then there is an embedded collar C about g0 of volume qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 3=2 2pl 1 4 sinh 2pl= 3 pffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi pffiffiffi 2 2pl þ : VolðCÞ ¼ p ffiffi 3 3 sinh2 2pl= 3 Similarly, we can find the volume in quaternionic hyperbolic 2-space H2H of a tube of width r and core length l about a geodesic segment. The proof follows that for complex hyperbolic space. PROPOSITION 4.4. Let Tr ¼ Tr ðgÞ H2H be a tube of radius r about a geodesic g. Let Tr;l ¼ Tr;l ðgÞ be a section of Tr of core length l. Then VolðTr;l Þ ¼ This leads to

211 p3 l sinh7 ðr=2Þ cosh3 ðr=2Þ: 105


213

COROLLARY 4.5. Let g0 be a simple closed geodesic in M ¼ H2H =G corresponding to the loxodromic element A of G. Suppose that A has eigenvalues la, l1 a and ab. Suppose that M ¼ jla 1j þ 2jab 1j þ jl1 a 1j < 1=2. Then the canonical collar Cr ðg0 Þ about g0 has volume VolðCr ðg0 ÞÞ ¼

27 p3 logðlÞð1 2MÞ7=2 : 105M5

Acknowledgements The results in this paper form part of the first author’s PhD thesis [5]. She would like to acknowledge the support of EPSRC.

References 1. Goldman, W. M.: Complex Hyperbolic Geometry, Oxford Univ. Press, 1999. 2. Jiang, Y., Kamiya, S. and Parker, J. R.: Jørgensen’s inequality for complex hyperbolic space, Geom. Dedicata 97 (2003), 55–80 (this volume) 3. Kellerhals, R.: Collars in PSL(2, H), Ann. Acad. Sci. Fenn. Math. 26 (2001), 51–72. 4. Kim, I. and Parker, J. R.: Geometry of quaternionic hyperbolic manifolds, Math. Proc. Cambridge Philos. Soc. (to appear). 5. Markham, S.: Hypercomplex hyperbolic geometry, PhD Thesis, Univ. Durham, 2003. 6. Meyerhoff, R.: A lower bound for the volume of hyperbolic 3-manifolds, Canad. J. Maths. 39 (1987), 1038–1056. 7. Parker, J. R.: On volumes of cusped, complex hyperbolic manifolds and orbifolds, Duke Math. J. 94 (1998), 433–464. 8. Parker, J. R.: On the stable basin theorem, Canadian Math Bull (to appear). 9. Sandler, H.: Distance formulas in complex hyperbolic space, Forum Math. 8 (1996), 93–106.