Table Of ContentJørgensen’s Inequality and Collars in
n
-dimensional Quaternionic Hyperbolic Space
Wensheng Cao John R. Parker
∗
Department of Mathematics, Department of Mathematical Sciences,
0
1 Wuyi University, Jiangmen, Durham University,
0
Guangdong 529020, P.R. China Durham DH1 3LE, England
2
e-mail: [email protected] email: [email protected]
n
a
J
3
2
Abstract
]
T Inthispaper,weobtainanaloguesofJørgensen’sinequalityfornon-elementarygroupsofisome-
G tries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic.
OurresultgivessomeimprovementoverearlierresultsofKim[10]andMarkham[15]. Theseresults
.
h also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya and
t
a Parker [7].
m Asapplications,weusethequaternionicversionofJørgensen’sinequalitiestoconstructembed-
[ ded collars about short, simple, closed geodesics in quaternionic hyperbolic manifolds. We show
that these canonical collars are disjoint from each other. Our results give some improvement over
4
earlier results of Markham and Parker and answer an open question posed in [16].
v
2
6 Mathematics Subject Classifications (2000): 20H10, 30F40,57S30.
5
Keywords: Quaternionic hyperbolic space; Jørgensen’s inequality, Loxodromic element, Collars.
3
.
6
0 1 Introduction
9
0
: Jørgensen’s inequality [8] gives a necessary condition for a non-elementary two generator subgroup
v
i of PSL(2,C) to be discrete. As a quantitative version of Margulis’ lemma, this inequality has been
X
generalised in many ways. Viewing PSL(2,R), which is isomorphic to PU(1,1), as the holomorphic
r
a isometry group of complex hyperbolic 1-space, we can seek to generalise Jørgensen’s inequality to
PU(n,1) for n > 1, the holomorphic isometry group of higher dimensional complex hyperbolic space.
Examples of this are the stable basin theorem of Basmajian and Miner [1] (see also [20]) and the
complex hyperbolic Jørgensen’s inequality of Jiang, Kamiya and Parker [7].
Kellerhals has generalised Jørgensen’s inequality to PSp(1,1). This group is the isometry group
of quaternionic hyperbolic 1-space H1, which is the same as real hyperbolic 4-space H4. For more
H R
details of PSp(1,1), including a classification of the elements, see [3]. It is interesting to seek gener-
alisations of Jørgensen’s inequality to PSp(n,1) for n > 1, that is to higher dimensional quaternionic
hyperbolic isometries. The first steps in this programme were taken by Kim and Parker [11] who
gave a quaternionic hyperbolic version of Basmajian and Miner’s stable basin theorem. Subsequently,
Markham [15] and Kim [10] independently gave versions of Jørgensen’s inequality for PSp(2,1). Cao
∗SupportedbyNSFsofChina(No.10801107,No.10671004 )andNSFofGuangdongProvince(No.8452902001000043)
1
and Tan [4] obtained an analogue of Jørgensen’s inequality for non-elementary groups of isometries of
quaternionic hyperbolic n-space generated by two elements, one of which is elliptic.
In this paper we consider subgroups of PSp(n,1) with a loxodromic generator. Any loxodromic
element g of PSp(n,1) can be conjugated in Sp(n,1) to the form:
1
diag λ1, λ2, ··· , λn−1,λn,λ−n , (1)
(cid:16) (cid:17)
where λi ∈H for i= 1, ..., n and λ−n1 are right eigenvalues of g with |λi| = 1 for i= 1, ..., n−1 and
λ > 1. We want to consider loxodromic maps that are close to the identity. To make this precise,
n
| |
if g Sp(n,1) is a loxodromic map conjugate to (1), we define the following conjugacy invariants:
∈
1
δ(g) = max{|λi −1| : i= 1,··· ,n−1}, Mg = 2δ(g)+|λn−1|+|λ−n −1| (2)
Observe that M > 0 and that the smaller M is the closer g is to the identity. Note that M is a
g g g
natural generalisation of the invariant
1
Mg = 2|λ1−1|+|λ2−1|+|λ−2 −1|.
defined independently by Kim [10] and Markham [15] for Sp(2,1).
We consider groups generated by g and h that are close to each other. To make this precise, we
use the cross ratio of the fixed points of the two loxodromic maps g and hgh 1. We define the cross
−
ratio in Section 2. The statement of our main theorem is:
Theorem 1.1. Let g be a loxodromic element of Sp(n,1) with M < 1 and with fixed points u, v
g
∈
∂Hn. Let h be any other element of Sp(n,1). If
H
1 M
1/2 1/2 g
[h(u),u,v,h(v)] [h(u),v,u,h(v)] < − (3)
M2
g
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
then the group g,h is either elementary or not discrete.
h i
We remark that this theorem is also valid for SU(n,1) and is stronger than both Theorems 4.1
and 4.2 of [7]. This theorem has some useful corollaries which we gather into a single result:
Corollary 1.2. Let g be a loxodromic element of Sp(n,1) with M < 1 and with fixed points u, v
g
∈
∂Hn. Let h be any other element of Sp(n,1). Suppose that one of the following conditions holds:
H
1 M
1/2 g
[h(u),v,u,h(v)] < − , (4)
M
g
(cid:12) (cid:12) 1 M
(cid:12)[h(u),u,v,h(v)](cid:12)1/2 < − g, (5)
M
g
(cid:12) (cid:12)1/2
(cid:12)[u,v,h(u),h(v)](cid:12) < 1 M , (6)
g
−
2(1 M )
(cid:12) (cid:12) g
[h(u),u,v,h(v)] + [h(u),v,u,h(v)] < − . (7)
(cid:12) (cid:12)
M2
g
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Then the group g,h is either elementary or not discrete.
h i
Whenn = 2thestatementofCorollary1.2withtheconditions(4)and(5)wasgivenindependently
by Kim, Theorem 3.1 of [10], and Markham Theorem 1.1 of [15] and for higher dimensions Cao gave
2
these conditions in an earlier preprint [2]. These results are a direct generalisation of Theorem 4.1 of
[7]. They all follow from Theorem 2.4 of Markham and Parker [17] and the observation (see the proof
of Theorem 1.4 below) that for all z V
0
∈
g(z),z M z,u z,v , (8)
g
≤ h i h i
(cid:12) (cid:12)
(cid:12)(cid:10) (cid:11)(cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
which, in terms of the Cygan metri(cid:12)c, may be(cid:12) rewrit(cid:12)ten as(cid:12)e(cid:12)quatio(cid:12)n (10) of [17] with dg = λn 1/2 and
| |
1/2
m = M .
g g
The statement of Corollary 1.2 with condition (7) is stronger than the corresponding results
in dimension n = 2 given by Kim and Markham. Kim’s criterion, Theorem 3.2 of [10], is M
g
≤
2√2 1 1 and
− −
p
2 2M M2+ 4 8M 8M2 4M3 M4
− g − g − g − g − g − g
[h(u),u,v,h(v)] + [h(u),v,u,h(v)] < .
q 2M2
g
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Markham’s criterion, Theorem 1.2 of [15], is M √2 1 and
g
≤ −
1 M + 1 2M M2
− g − g − g
[h(u),u,v,h(v)] + [h(u),v,u,h(v)] < ,
qM2
g
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
which is a direct generalisation of Theorem 4.2 of [7]. It is easy to see that (when they are defined)
1 M + 1 2M M2
2(1 Mg) − g − g − g
− >
M2 qM2
g g
2 2M M2+ 4 8M 8M2 4M3 M4
− g − g − g − g − g − g
> .
q 2M2
g
Therefore Kim and Markham’s results follow from (7).
Meyerhoff [18] used Jørgensen’s inequality to show that if a simple closed geodesic in a hyper-
bolic 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. In [13, 14]
Kellerhals generalised Meyerhoff’s results to real hyperbolic 4-space and 5-space with the aid of some
properties of quaternions. Markham and Parker [16] used the complex and quaternionic hyperbolic
Jørgensen’s inequality obtained in [7, 15], to give analogues of Meyerhoff’s (and Kellerhals’) results
for short, simple, closed geodesics in 2-dimensional complex and quaternionic hyperbolic manifolds.
They showed that these canonical collars are disjoint from each other and from canonical cusps. For
complex hyperbolic space, by using a lemma of Zagier they also gave an estimate based only on the
length, and left the same question for the case of quaternionic space as an open question.
Let G be a discrete group of n-dimensional quaternionic hyperbolic isometries. Let g G be
∈
loxodromic with axis the geodesic γ. The tube T (γ) of radius r about γ is the collection of points
r
a distance less than r from γ. It is clear that g maps T (γ) to itself. The tube T (γ) is precisely
r r
invariant under the subgroup g of G if h T (γ) is disjoint from T (γ) for all h G g . If T (γ)
r r r
h i ∈ −h i
is precisely invariant under G then C (γ ) = T (γ)/ g is an embedded tubular neighbourhood of the
r ′ r
(cid:0) (cid:1) h i
simple closed geodesic γ = γ/ g . We call C (γ ) the collar of width r about γ .
′ r ′ ′
h i
3
As applications of our quaternionic version Jørgensen’s inequalities, we will give analogues of
Markham and Parker’s results for short, simple, closed geodesics in n-dimensional quaternionic hy-
perbolic manifolds.
Given a loxodromic map g with axis γ and satisfying M < √3 1, we define a positive real
g
−
number r by:
2(1 M )
g
cosh(2r) = − . (9)
M2
g
Then we call the tube T (γ) with r given by (9) the canonical tube about γ. If γ = γ/ g then we
r ′
h i
call the collar C (γ ) with r given by (9) the canonical collar about γ .
r ′ ′
Theorem 1.3. Let G be a discrete, non-elementary, torsion-free subgroup of Sp(n,1). Let g be a
loxodromic element of G with axis the geodesic γ. Suppose that M < √3 1. Then the canonical
g
−
tube T (γ) whose width r is given by (9) is precisely invariant under g in G.
r
h i
In particular, the canonical collar C (γ ) of width r about γ = γ/ g is embedded in the manifold
r ′ ′
h i
= Hn/G.
H
M
Furthermore, we have
Theorem1.4. Let denote aquaternionichyperbolic n-manifold. Thenthecanonical collars around
M
distinct short, simple, closed geodesics in are disjoint.
M
By controlling the rotational part of loxodromic element, we obtain the radius of collars solely in
terms of the length of the corresponding simple closed geodesic as the following, which answers the
open problem posed in [16].
Theorem 1.5. Let N 35 be a positive integer. Let G be a discrete, torsion-free, non-elementary
≥
subgroup of Sp(n,1). Let g be a loxodromic element of G with axis γ having the form (1) and let
l = 2log λ be the length of the closed geodesic γ/ g and suppose that
n
| | h i
Nnl Nnl 2π 2π
R =2 cosh +1 cosh cos +2 2 1 cos < √3 1. (10)
N
s 2 2 − N s − N −
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Define the positive number r by
2(1 R )
N
cosh(2r) = − .
R2
N
Then the tube T (γ) is precisely invariant under G.
r
Corollary 1.6. Let G be a discrete, torsion-free, non-elementary subgroup of Sp(2,1). Let g be a
loxodromic element of G with axis γ having the form (1). Suppose that l = 2log λ < 0.00017681.
2
| |
Let r be a positive number defined by
2(1 R)
cosh(2r) = −
R2
where
1849l 1849l 2π 2π
R = 2 cosh +1 cosh cos +2 2 1 cos . (11)
s 2 2 − 43 s − 43
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Then the tube T (γ) is precisely invariant under G.
r
The structure of the remainder of this paper is as follows. In Section 2, we give the necessary
background material for quaternionic hyperbolic space. Section 3 contains the proof of Theorem 1.1
and Corollary 1.2. In Section 4, we use Theorem 1.1 to obtain the proof of Theorems 1.3 and 1.4. In
Section5,wegiveanexampletoillustratetheideabehindTheorem1.5. UsingtheadaptedPigeonhole
Principle (cf. [18]), we obtain the proof of Theorem 1.5 and Corollary 1.6.
4
2 Background
We begin with some background material on quaternionic hyperbolic geometry. Much of this can be
found in [5, 6, 11, 19]. Let Hn,1 be the quaternionic vector space of quaternionic dimension n+1 (so
real dimension 4n+4) with the quaternionic Hermitian form
z, w = w Jz = w z + +w z (w z +w z ),
∗ 1 1 n 1 n 1 n n+1 n+1 n
h i ··· − − −
wherez andw arethe columnvectors in Hn,1 with entries z , ,z andw , ,w respectively,
1 n+1 1 n+1
··· ···
denotes quaternionic Hermitian transpose and J is the Hermitian matrix
∗
·
I 0 0
n 1
−
J = 0 0 1 .
−
0 1 0
−
We define a unitary quaternionic transformation (or symplectic transformation) g to be an automor-
phism of Hn,1, that is, a linear bijection such that g(z), g(w) = z, w for all z and w in Hn,1. We
h i h i
denote the group of all unitary transformations by Sp(n,1).
Following Section 2 of [5], let
V = z Hn,1 0 : z, z = 0
0
∈ −{ } h i
n o
V = z Hn,1 : z, z < 0 .
− ∈ h i
n o
It is obvious that V and V are invariant under Sp(n,1).
0
We define an equivalen−ce relation on Hn,1 by z w if and only if there exists a non-zero
∼ ∼
quaternion λ so that w = zλ. Let [z] denote the equivalence class of z. Let P : Hn,1 0 HPn
−{ } −→
be the right projection map given by P :z [z]. If z = 0 then P is given by
n+1
7−→ 6
P(z , ..., z ,z )t = (z z 1 , ,z z 1 )t Hn.
1 n n+1 1 n−+1 ··· n n−+1 ∈
We also define
P(0, ..., 0, z , 0)t = . (12)
n
∞
Observe that
zλ,wµ =µw Jzλ =µ z,w λ. (13)
∗
h i h i
We define the Siegel domain model of quaternionic hyperbolic n space to be Hn = P(V ) and its
H
boundary to be ∂Hn = P(V ). It is clear that ∂Hn. Also for all z V we have z −= 0 and
H 0 H n+1
so P is given by the formula above. Likewise fo∞r al∈l z V , either z =∈0 o−r P(z) = . 6
0 n+1
∈ 6 ∞
As in Chapter 19 of [19], the Bergman metric on Hn is given by the distance formula
H
ρ(z,w) z, w w, z
cosh2 = h ih i, where z,w HnH, z P−1(z),w P−1(w).
2 z, z w, w ∈ ∈ ∈
h ih i
This expression is independent of the choice of z and w. Since Sp(n,1) preserves the form , , it
h· ·i
clearly preserves the right hand side of this expression. Therefore g Sp(n,1) acts on Hn ∂Hn as
H H
∈ ∪
follows:
g(z) = PgP 1(z).
−
This formula is well defined provided the action of Sp(n,1) is on the left and the action of projection
P of Sp(n,1) is on the right. It is clear that multiples of g by a non-zero real number act in the same
5
way. Since elements of Sp(n,1) have determinant 1 this real number can only be 1. Therefore we
± ±
define PSp(n,1) = Sp(n,1)/ I . All elements of PSp(n,1) are isometries of Hn. We often find
n+1 H
{± }
it convenient to work with matrices in Sp(n,1) rather than projective mappings in PSp(n,1) and we
will pass between them without comment.
If g Sp(n,1), by definition, g preserves the Hermitian form. Hence
∈
w Jz = z, w = gz, gw = w g Jgz
∗ ∗ ∗
h i h i
for all z and w in Hn,1. Letting z and w vary over a basis for Hn,1, we see that J = g Jg. From this
∗
we find g 1 = J 1g J. That is:
− − ∗
A θ η A α β
∗ ∗ ∗
− −
g 1 = β d b for g = η a b Sp(n,1). (14)
− ∗
− ∈
α c a θ c d
∗
−
Using the identities gg 1 = g 1g = I we obtain:
− − n+1
AA αβ βα = I , (15)
∗ ∗ ∗ n 1
− − −
Aθ +αd+βc = 0, (16)
∗
−
Aη +αb+βa = 0, (17)
∗
−
ηθ +ad+bc = 1, (18)
∗
−
ηη +ab+ba = 0, (19)
∗
−
θθ +cd+dc = 0, (20)
∗
−
A A θ η η θ = I , (21)
∗ ∗ ∗ n 1
− − −
A α θ a η c = 0, (22)
∗ ∗ ∗
− −
A β θ b η d = 0, (23)
∗ ∗ ∗
− −
β α+da+bc = 1, (24)
∗
−
β β+db+bd = 0, (25)
∗
−
α α+ca+ac = 0. (26)
∗
−
Following Chen and Greenberg [5], we say that a non-trivial element g of Sp(n,1) is:
(i) elliptic if it has a fixed point in Hn;
H
(ii) parabolic if it has exactly one fixed point which lies in ∂Hn;
H
(iii) loxodromic if it has exactly two fixed points which lie in ∂Hn.
H
A subgroup G of Sp(n,1) is called elementary if it has a finite orbit in Hn ∂Hn. If all of its
H H
∪
orbits are infinite then G is non-elementary. In particular, G is non-elementary if it contains two
non-elliptic elements of infinite order with distinct fixed points.
Let o be the origin in Hn and be as defined in (12). Both these points lie on ∂Hn. In what
H
∞
follows we make fixed choices of points in Hn,1 that are preimages of these points. Namely
(0,...,0,0,1)t P 1(o) V , (0,...,0,1,0)t P 1( ) V .
− 0 − 0
∈ ⊂ ∈ ∞ ⊂
Define the stabilisers of the points to be:
G = g Sp(n,1) : g(o) = o , G = g Sp(n,1) : g( ) = , G = G G .
o o, o
{ ∈ } ∞ { ∈ ∞ ∞} ∞ ∩ ∞
6
Note that if g has the form (14) then if g G we have b = 0 and if g G we have c= 0.
o
∈ ∈ ∞
Cross-ratios were generalised to complex hyperbolic space by Kora´nyi and Reimann [12]. We will
generalise this definition of complex cross-ratio to the non commutative quaternion ring.
n
Definition 2.1. The quaternionic cross-ratio of four points z ,z ,w ,w in H is defined as:
1 2 1 2 H
[z1,z2,w1,w2] = w1, z1 w1, z2 −1 w2, z2 w2, z1 −1, (27)
h ih i h ih i
where zi = P−1(zi) and wi P−1(wi) for i = 1, 2.
∈ ∈
Using (13) we see that
[z1λ1,z2λ2,w1µ1,w2µ2] = w1µ1, z1λ1 w1µ1, z2λ2 −1 w2µ2, z2λ2 w2µ2, z1λ2 −1
h ih i h ih i
= λ1hw1, z1iµ1µ−11hw1, z2i−1λ−21λ2hw2, z2iµ2µ−21hw2, z1i−1λ−11
1
= λ1[z1,z2,w1,w2]λ−1 .
The quaternionic cross-ratio [z ,z ,w ,w ] depends on the choice of z P 1(z ). However, its
1 2 1 2 1 − 1
∈
absolute value
w1, z1 w2, z2
[z ,z ,w ,w ] = |h ih i| (28)
1 2 1 2
w1, z2 w2, z1
|h ih i|
(cid:12) (cid:12)
is independent of the preimage o(cid:12)f z and w in(cid:12)Hn,1. The following lemma is easy to prove.
i i
Lemma 2.1. Let o, ∂Hn stand for the images of (0, ,0,1)t and (0, ,0,1,0)t Hn,1 under
H
∞ ∈ ··· ··· ∈
the projection map P, respectively and let h PSp(n,1) be given by (14). Then
∈
[h( ),o, ,h(o)] = bc, (29)
∞ ∞ | |
[h( ), ,o,h(o)] = ad, (30)
(cid:12) (cid:12)
(cid:12) ∞ ∞ (cid:12) | |
bc
(cid:12) (cid:12)
[ ,o,h( ),h(o)] = | |. (31)
(cid:12) (cid:12)
∞ ∞ ad
| |
(cid:12) (cid:12)
The following lemma is crucial f(cid:12)or us to prove The(cid:12)orem 1.1.
Lemma 2.2. Let h be as in (14). Then
β α 2ad1/2 bc1/2, (32)
∗
| | ≤ | | | |
ηθ 2ad1/2 bc1/2, (33)
∗
| | ≤ | | | |
ad1/2 bc1/2 +1, (34)
| | ≤ | |
bc1/2 ad1/2 +1, (35)
| | ≤ | |
1 ad1/2 + bc1/2. (36)
≤ | | | |
Proof. Using (25) and (26), we have
β α2 β β α α = 2 (db)2 (ca) 4ad bc. (37)
∗ ∗ ∗
| | ≤ | || | ℜ ℜ ≤ | || |
This gives (32). Similarly, using (19) and (20), we have
ηθ 2 ηη θθ = 2 (ab)2 (cd) 4ad bc.
∗ ∗ ∗
| | ≤ | || | ℜ ℜ ≤ | || |
This gives (33).
7
Next, using (24) and (37), we have
4 (db) (ca) β α2
∗
ℜ ℜ ≥ | |
= da+bc 12
| − |
= 1+ ad2 + bc2 2 (da) 2 (bc)+2 (dacb).
| | | | − ℜ − ℜ ℜ
Thus
1+ ad2+ bc2 2 (da)+2 (bc) 2 (dacb)+4 (db) (ca)
| | | | ≤ ℜ ℜ − ℜ ℜ ℜ
= 2 (da)+2 (bc)+2 (bdca)
ℜ ℜ ℜ
2ad +2bc +2ad bc.
≤ | | | | | || |
We can rearrange this expression to obtain
2
1 ad bc 4ad bc.
−| |−| | ≤ | || |
(cid:0) (cid:1)
Taking square roots gives
2ad1/2 bc1/2 1 ad bc 2ad1/2 bc1/2.
− | | | | ≤ −| |−| | ≤ | | | |
Rearranging gives
ad1/2 bc1/2 2 1 ad1/2 + bc1/2 2.
| | −| | ≤ ≤ | | | |
Taking square roots of both s(cid:0)ides, including b(cid:1)oth choic(cid:0)es of sign in the(cid:1)left handinequality, gives (34),
(35) and (36). (cid:3)
3 The proof of Jørgensen’s inequality
Proof of Theorem 1.1. Since (3) is invariant under conjugation, we may assume that g is of the
form (1) and h is of the form (14). Using (29) and (30) our hypothesis (3) can be rewritten as
1 M
ad1/2 bc1/2 < − g. (38)
| | | | M2
g
Let h = h and h = h gh 1. We write
0 k+1 k −k
A α β
k k k
h = η a b .
k k k k
θ c d
k k k
Then
A α β
k+1 k+1 k+1
h = η a b
k+1 k+1 k+1 k+1
θ c d
k+1 k+1 k+1
A α β L 0 0 A θ η
k k k ∗k − k∗ − k∗
= ηk ak bk 0 λn 01 −βk∗ dk bk ,
θk ck dk 0 0 λ−n −α∗k ck ak
8
where L = diag(λ , λ , , λ ). Therefore
1 2 n 1
··· −
1
ak+1 = −ηkLθk∗+akλndk +bkλ−n ck, (39)
1
bk+1 = −ηkLηk∗+akλnbk +bkλ−n ak, (40)
1
ck+1 = −θkLθk∗+ckλndk +dkλ−n ck, (41)
1
dk+1 = −θkLηk∗+ckλnbk +dkλ−n ak. (42)
Claim 1: We claim that if ad1/2 bc1/2 < (1 M )/M2 then b c tends to 0 as k tends to infinity.
| | | | − g g | k k|
By (19) and (40), we have
1
|bk+1| = |ηk(In−1−L)ηk∗ +ak(λn −1)bk +bk(λ−n −1)ak|
1
≤ δ(g)ηkηk∗+(|λn−1|+|λ−n −1|)|bkak|
1
= δ(g)2ℜ(akbk)+(|λn −1|+|λ−n −1|)|bkak|
1
≤ (2δ(g)+|λn−1|+|λ−n −1|)|bkak|
= M b a .
g k k
| |
Similarly, by (20) and (41) we have
|ck+1| = |θ(k)(In−1−L)θk∗+ck(λn−1)dk +dk(λ−n1−1)ck|≤ Mg|ckdk|.
Therefore, for all k 0 we have
≥
b c 1/2 M a d 1/2 b c 1/2. (43)
k+1 k+1 g k k k k
| | ≤ | | | |
Using our hypothesis (38) with k = 0 this immediately gives
1 M
b c 1/2 M a d 1/2 b c 1/2 < − g.
1 1 g 0 0 0 0
| | ≤ | | | | M
g
In particular,
M 1+ b c 1/2 < 1.
g 1 1
| |
From this point on the proof closely follow(cid:0)s the proof(cid:1)of the similar result for complex hyperbolic
space given by Jiang, Kamiya and Parker [7].
We claim that for k 1 we have
≥
k 1
b c 1/2 M 1+ b c 1/2 − b c 1/2. (44)
k k g 1 1 1 1
| | ≤ | | | |
(cid:16) (cid:17)
In particular, (cid:0) (cid:1)
b c 1/2 b c 1/2.
k k 1 1
| | ≤ | |
Certainly (44) is true for k = 1. Assume that (44) is true for some k 1. Then, using (43) and (34),
≥
we have
b c 1/2 M a d 1/2 b c 1/2
k+1 k+1 g k k k k
| | ≤ | | | |
M 1+ b c 1/2 b c 1/2
g k k k k
≤ | | | |
Mg(cid:0)1+ b1c1 1/2(cid:1)bkck 1/2
≤ | | | |
k 1
M (cid:0)1+ b c 1/2(cid:1) M 1+ b c 1/2 − b c 1/2
g 1 1 g 1 1 1 1
≤ | | | | | |
(cid:0) (cid:1)(cid:16) k (cid:0) (cid:1)(cid:17)
= M 1+ b c 1/2 b c 1/2.
g 1 1 1 1
| | | |
(cid:16) (cid:17)
(cid:0) (cid:1)
9
Then (44) is true for k+1. The result follows by induction.
Since M 1+ b c 1/2 < 1, an immediate consequence of (44) is that
g 1 1
| |
(cid:0) (cid:1)
lim b c 1/2 = 0. (45)
k k
k | |
→∞
This proves Claim 1.
Claim 2: If there exists some integer k such that
b c = 0, (46)
k k
then h,g is either elementary or not discrete.
h i
If b = 0 then, by (25), we have β = 0 and h (o) = o. Similarly, if c = 0 then, by (26), we have
k k k k
α = 0 and so h ( ) = . If b c = 0 but either b or c is non-zero then h fixes exactly one of o
k k k k k k k
∞ ∞
and . Hence, g, h is not discrete by Theorem 3.1 of Kamiya [9]. This implies that g, h is not
k
∞ h i h i
discrete.
Suppose then that b = c = 0 for some k 1. Then h fixes both o and . In particular,
k k k
≥ ∞
o = h (o) = h gh 1 (o) and = h ( ) = h gh 1 ( ).
k k−1 −k−1 ∞ k ∞ k−1 −k−1 ∞
This means that g fixes h 1 (o) and h 1 ( ). If k 2 then h is loxodromic and so cannot swap o
and . Thus h (o) = o−k−a1nd h ( −k)−1=∞ . By in≥duction, wke−1find that g fixes h 1(o) and h 1( ).
∞ k−1 k−1 ∞ ∞ −0 −0 ∞
In other words h = h preserves the set o, and so g,h is elementary.
0
{ ∞} h i
This proves Claim 2.
Claim 3: If
lim b c = 0 and b c = 0 for all k 1 (47)
k k k k
k | | 6 ≥
→∞
then h,g is not discrete.
h i
Assume that (47) holds. Then from (34) we have
a d 1/2 b c 1/2+1
k k k k
| | ≤ | |
and so a d is bounded as k tends to infinity. Hence, from (32) and (33) we have
k k
| |
β α 2a d 1/2 b c 1/2 and η θ 2a d 1/2 b c 1/2
| k∗ k|≤ | k k| | k k| | k k∗|≤ | k k| | k k|
and so
lim β α = lim η θ = 0.
k | k∗ k| k | k k∗|
→∞ →∞
Likewise,
lim η Lθ = lim θ Lη = 0.
k k∗ k k∗
k k
→∞ →∞
From (24) we have
lim d a = lim 1+β α b c = 1.
k k k k k∗ k − k k
→∞ →∞
(cid:0) (cid:1)
Therefore, from (39) and (42) we have
1
klim |ak+1| = klim −ηkLθk∗+akλndk +bkλ−n ck = |λn|, (48)
→∞ →∞
klim |dk+1| = klim (cid:12)(cid:12)−θkLηk∗+ckλnbk +dkλ−n1ak(cid:12)(cid:12) = |λn|−1. (49)
→∞ →∞
(cid:12) (cid:12)
(cid:12) (cid:12)
10
