# Jørgensen’s Inequality and Algebraic Convergence Theorem in Quaternionic Hyperbolic Isometry Groups

Authors: Huani Qin; Yueping Jiang; Wensheng Cao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We obtain an analogue of Jørgensen's inequality in quaternionic hyperbolic space. As an application, we prove that if ther-generator quaternionic Kleinian group satisfies I-condition, then its algebraic limit is also a quaternionic Kleinian group. Our results are generalizations of the counterparts in the n-dimensional real hyperbolic space.


---

## Body

## 1. Introduction

Jørgensen's inequality [1] gives a necessary condition for a nonelementary two-generator subgroup of SL
(
2
,
C
) to be discrete, which involves the traces of one of the generators and the commutator of both generators, as follows.Theorem A.Let
f
,
g
∈
SL
(
2
,
C
)
. If each two-generator subgroup
〈
f
,
g
〉
is discrete and nonelementary, then
(1)
|
t
r
2
(
f
)
-
4
|
+
|
tr
⁡
(
[
f
,
g
]
)
-
2
|
<
1
,
where
[
f
,
g
]
=
f
g
f
-
1
g
-
1
is the commutator of
f
and
g
and
tr
⁡
(
·
)
is the trace function.Jørgensen’s inequality has been generalized in many ways in real hyperbolic space [2, 3], complex hyperbolic space [4–6], and quaternionic hyperbolic space [7–9] and plays an important role in studying discreteness and algebraic convergence for real, complex, or quaternionic hyperbolic isometry group [10–14]. However, due to the noncommutative multiplication of the quaternions, Jørgensen’s inequality in quaternionic hyperbolic isometry groups is relatively more complicated. To carry the results holding in real or complex hyperbolic geometry over to the quaternionic hyperbolic geometry, one sometimes has to reconsider these results involving the use of commutativity or the fact that purely imaginary complex numbers are isomorphic to R.In quaternionic hyperbolic space, the first step to generalize Jørgensen’s inequality was taken by Kim and Parker [7] who gave a quaternionic hyperbolic version of Basmajian and Miner’s stable basin theorem. Subsequently, Markham [9] and Kim [8] independently gave versions of Jørgensen’s inequality for Sp
(
2,1
). Recently, Cao, Tan, and Parker, [15, 16] obtained analogues of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is elliptic or loxodromic.Shimizu’s lemma deals with two-generator subgroup〈
f
,
g
〉 with g being parabolic element and there are some generalizations to quaternionic hyperbolic space [7–9] for some special kinds of parabolic elements. But for g being screw parabolic, we only have analogues in the setting of 2-dimensional complex hyperbolic space [4, 6] and so forth. This gap is the main obstacle to investigate the discreteness and algebraic convergence theorem of groups in quaternionic hyperbolic space.Our first aim is to erect generalizations of Jørgensen’s inequality for two-generator nonelementary subgroup with some special kinds of elements in higher dimensional quaternionic hyperbolic isometry groupSp
(
n
,
1
).On the other hand, convergence of nonelementary subgroups of real or complex hyperbolic isometry groups is also another important problem. LetG be the n-dimensional sense-preserving Möbius group M
(
R
¯
n
) or unitary group U
(
1
,
n
;
C
).Definition 1.
Let{
G
r
,
i
}
(
i
=
1,2
,
…
) be a sequence of subgroups in group G, where G
r
,
i is generated by g
1
,
i
,
g
2
,
i
,
…
,
g
r
,
i and r
=
1,2
,
…. If, for each t
(
1
≤
t
≤
r
),
(2)
g
t
,
i
⟶
g
t
∈
G
as
i
⟶
∞
,
then one says that {
G
r
,
i
} algebraically converges to G
r
=
〈
g
1
,
g
2
,
…
,
g
r
〉.The problem that under which condition the limit groupG
r is also a Kleinian group if, for each i, G
r
,
i is a Kleinian group was intensely studied. Using the well-known Jørgensen’s inequality, Jørgensen and Klein [17] proved the following.Theorem B.If each
G
r
,
i
is
r
-generator Kleinian subgroup of
G
, where
G
is
M
(
R
¯
2
)
, then the limit group
G
r
is also a Kleinian group.However, the examples in [18] show that Theorem B could not be extended to n-dimensional cases (
n
≥
3
) without any modifications. The reason for this phenomenon is that there is a distinction between the fixed point sets of elliptic elements in M
(
R
¯
2
) and M
(
R
¯
n) (n
≥
3). The reasoning mechanism in M
(
R
¯
2
) mainly relies on the fact that each elliptic element has only two fixed points in R
2. Because the fixed point set of an elliptic element of M
(
R
¯
n) (n
≥
3) may be empty set or subset of R
¯
n, we cannot use the same reasoning mechanism as in n
=
2. By adding some condition(s) to control the fixed point set of elliptic element and using generalized Jørgensen’s inequality, several authors have obtained their analogues in M
(
R
¯
n) when n
≥
3.Martin [2] proved the following theorem.Theorem C.Let
G
r
be the algebraic limit group of a sequence of
r
-generator Kleinian groups of
M
(
R
¯
n
)
of uniformly bounded torsion. Then
G
r
is a Kleinian group.Martin also asked how one might weaken the hypothesis of uniformly bounded torsion. Fang and Nai [19] first gave condition A to consider such a question. Recently, Wang [20] and Yang [21] used EP-condition and condition A, respectively, to weaken Martin’s uniformly bounded torsion and proved the following.Theorem D.Let
r
<
∞
and
G
r
be the algebraic limit group of a sequence of
r
-generator Kleinian groups
{
G
r
,
i
}
of
M
(
R
¯
n
)
. If
{
G
r
,
i
}
satisfies EP-condition (or condition A), then
G
r
is a Kleinian group.See details for the definitions of uniformly bounded torsion, EP-condition, and condition A in [18, 19, 21].In [10], Cao gave a convergence theorem about algebraic limit group of complex Kleinian groups under IP-condition, as follows.Theorem E.Let
G
r
be the algebraic limit group of complex Kleinian groups
{
G
r
,
i
}
of
U
(
1
,
n
;
C
)
. If
{
G
r
,
i
}
satisfies IP-condition, then
G
r
is a complex Kleinian group.Here,
{
G
r
,
i
}
satisfying IP-condition means that, for any sequence
{
f
i
k
}
(
f
i
k
∈
G
r
,
i
k
)
, if, for each
k
,
card
(
fix
(
f
i
k
)
)
=
∞
and
f
i
k
→
f
as
k
→
∞
with
f
being parabolic or the identity, then
{
f
i
k
}
has uniformly bounded torsion.Our second aim is to investigate analogous condition mentioned above that an algebraic convergence theorem holds in the quaternionic hyperbolic space. We define the concept of uniformly bounded torsion as follows: a subsetH of Sp
(
n
,
1
) is said to haveuniformly bounded torsion if there exists an integer M such that ∀
g
∈
H,
(3)
ord
(
g
)
≤
M
or
ord
(
g
)
=
∞
.
And we call a nonelementary and discrete subgroup G of Sp
(
n
,
1
)
a quaternionic Kleinian group.For a sequence of subgroups{
G
r
,
i
} of Sp
(
n
,
1
), we introduce the following condition.Definition 2.
One says that{
G
r
,
i
} satisfies I-condition if any sequence {
f
i
k
} (f
i
k
∈
G
r
,
i
k), satisfying the condition that, for each k, card
[
fix
(
f
i
k
)
]
=
∞ and f
i
k
→
I as k
→
∞, has uniformly bounded torsion. Here card
(
M
) denotes the cardinality of a set M.Our main results are the following theorems.Theorem 3.
Suppose thatf and g
∈
Sp
(
n
,
1
) generate a discrete and nonelementary group. Then (i)
iff is parabolic or loxodromic, one has
(4)
max
⁡
{
∥
f
-
I
n
+
1
∥
,
∥
[
f
,
g
]
-
I
n
+
1
∥
}
≥
(
2
-
3
)
2
,
(5)
max
⁡
{
∥
f
-
I
n
+
1
∥
,
∥
g
-
1
f
g
-
I
n
+
1
∥
}
≥
(
2
-
3
)
2
;
(ii)
iff is elliptic, one has
(6)
max
⁡
{
∥
f
-
I
n
+
1
∥
,
∥
[
f
,
g
i
]
-
I
n
+
1
∥
:
i
=
1,2
,
…
,
n
}
≥
(
2
-
3
)
2
,
(7)
max
⁡
{
∥
g
-
i
f
g
i
-
I
n
+
1
∥
:
i
=
0,1
,
…
,
n
}
≥
(
2
-
3
)
2
,
where ∥
·
∥ is the Hilbert-Schmidt norm of an element.Theorem 4.
LetG
r be the algebraic limit group of quaternionic Kleinian groups {
G
r
,
i
} of Sp
(
n
,
1
). If {
G
r
,
i
} satisfies I-condition, then G
r is a quaternionic Kleinian group.


## 2. Several Lemmas

LetF denote the field R
,
C, or H. We adopt the same notations and definitions as in [7, 16, 22, 23] such as H
H
n,  U
(
1
,
n
+
1
;
F
), discrete groups, limit sets, and elementary and nonelementary.We first discuss some properties of elliptic elements. As in [22], for an elliptic element g, let Λ
0 and Λ
i, i
=
1,2
,
…
,
n be its negative and positive class of eigenvalues, respectively. Let fix
(
f
) denote the set of fixed point(s) of f in H
F
n. Then the fixed point set of g in H
F
n contains only one fixed point if Λ
0
≠
Λ
i, i
=
1,2
,
…
,
n, and is a totally geodesic submanifold which is equivalent to H
F
m (resp., H
C
m ) if Λ
0
⊂
R (resp., Λ
0
⊈
R) coincides with exact m of the class Λ
i
,
i
=
1,2
,
…
,
n. In the latter case, the fixed point set of f in H
F
n
¯ is H
F
m
¯ or H
C
m
¯, and we define dim
⁡
(
fix
(
f
)
)
=
m. The elliptic elements with only one fixed point in H
F
m are called regular elliptic elements, while the other elements are called boundary elliptic elements. We call an elliptic element g an irrational rotation if e
i
θ
∈
Λ
t with irrational θ for some t.SinceU
(
1
,
n
+
1
;
F
) does not act effectively in H
F
n, one always consider its projective group PU
(
1
,
n
;
F
)
=
U
(
1
,
n
;
F
)
/
Z
(
1
,
n
;
F
). It is well known that the n-dimensional Möbius group M
(
n
) is isomorphic tothe identity component of PU
(
1
,
n
+
1
;
R
), the projective orthogonal group. Each elliptic element is conjugate to an element with the form
(8)
diag
⁡
(
λ
0
,
A
)
∈
U
(
1
;
F
)
×
U
(
n
;
F
)
,
where
λ
0
∈
Λ
0
.
When F
=
R and n is even, there are elliptic elements with λ
0
=
1 and the eigenvalues of positive class form n
/
2 conjugated pairs of complex numbers of norm 1. Those elements correspond to the so-calledfixed-point-free elements in M
(
R
¯
n
-
1
). However, when n is odd, by our above isomorphism, (
n
-
1
)-dimensional Möbius group M
(
n
-
1
) cannot contain anyfixed-point-free elements. In contrast to real hyperbolic space, we have regular elliptic elements in any dimensional complex and quaternionic hyperbolic space.Using the quaternionic version in [24] of Schur’s unitary triangularization theorem, we can prove the following lemma.Lemma 5.
Letf
∈
Sp
(
n
,
1
) be an elliptic element of order m. If 2
≤
m
<
M, then there is a constant δ
(
M
) such that
(9)
∥
f
-
I
n
+
1
∥
>
δ
(
M
)
.Proof.
Let the right complex eigenvalues off be λ
j
=
e
i
θ
j
(
j
=
1
,
…
,
n
+
1
). By Schur’s unitary triangularization theorem of quaternionic version in [24], there is a matrix U
∈
U
(
n
+
1
;
H
) such that
(10)
U
f
U
¯
T
=
(
λ
1
*
*
*
0
λ
2
*
*
⋮
⋱
⋱
*
0
⋯
0
λ
n
+
1
)
.
Hence ∥
f
-
I
n
+
1
∥
2
≥
∑
j
=
1
n
+
1
‍
|
λ
j
-
1
|
2
=
2
(
n
+
1
)
-
2
∑
j
=
1
n
+
1
‍
cos
⁡
θ
j. It follows from f
m
=
I
n
+
1 that there is j
(
j
=
1,2
,
…
,
n
+
1
) such that |
cos
⁡
θ
j
|
≠
1 and θ
j
=
2
p
π
/
m (here p and m are prime). Hence
(11)
1
-
cos
⁡
θ
j
≥
1
-
|
cos
⁡
θ
j
|
>
1
-
|
cos
⁡
π
m
|
.
Set δ
(
M
)
=
1
-
|
cos
⁡
(
π
/
m
)
|. Then δ
(
M
) is the desired number.By the above lemma, we know that if the sequence{
g
i
} of nontrivial unitary quaternionic transformations converges to the identity, then the orders of g
i converge to infinity. So a family of groups {
G
r
,
i
} satisfies I-condition if there is no sequence {
f
i
k
}
(
f
i
k
∈
G
r
,
i
k
) converging to the identity, such that card
(
fix
(
f
i
k
)
)
=
∞ for each k.When working in the matrix algebra, one has two choices, whether to use the spectral norm or the Hilbert-Schmidt norm. Following the ideas of Martin [2], we choose the Hilbert-Schmidt norm to construct our version of Jørgensen’s inequality (Theorem 3) in Sp
(
n
,
1
).The following lemma is a classification of elementary subgroups ofSp
(
n
,
1
).Lemma 6 (cf. [25]).
(
1
) If G contains a parabolic element but no loxodromic element, then G is elementary if and only if it fixes a point in ∂
H
H
n;
(
2
) if G contains a loxodromic element, then G is elementary if and only if it fixes a point in ∂
H
H
n or a point-pair {
x
,
y
}
⊂
∂
H
H
n;
(
3
)  G is purely elliptic; that is, each nontrivial element of G is elliptic; then G is elementary and fixes a point in H
H
n
¯.By Lemma6, we have the following lemmas.Lemma 7.
IfG
⊂
Sp
(
n
,
1
) is discrete nilpotent group without elliptic element, then G is elementary.Lemma 8 (cf. [2, Lemma 2.8]).
Letx and y be two distinct points in H
H
n
¯. If f
∈
Sp
(
n
,
1
) interchanges x and y, then ∥
f
-
I
n
+
1
∥
≥
2.The proofs of the following two lemmas follow from similar discussions in [2].Lemma 9 (cf. [2, Lemma 4.1]).
Let〈
f
,
g
〉 be discrete with f being parabolic or loxodromic element. If 〈
f
,
g
-
1
f
g
〉 is elementary, then 〈
f
,
g
〉 is also elementary.Lemma 10 (cf.[2, Lemma 4.2]).
Let〈
f
,
g
〉 be discrete with f being elliptic element. Let m
=
dim
⁡
(
fix
(
f
)
). If G
=
〈
g
-
i
f
g
i
:
i
=
0,1
,
…
,
m
+
1
〉 is elementary, then 〈
f
,
g
〉 is elementary or ∥
f
-
I
n
+
1
∥
≥
2.Lemma 11.
Suppose thatf and g
∈
Sp
(
n
,
1
) generate a discrete and nonelementary group. Then
(12)
max
⁡
{
∥
f
-
I
n
+
1
∥
,
∥
g
-
I
n
+
1
∥
}
≥
(
2
-
3
)
2
,
where ∥
·
∥ is the Hilbert-Schmidt norm of an element.Proof of Lemma11.
We can chooseC to be the subspace of H spanned by {
1
,
i
}. With respect to this choice of C we can write H
=
C
⊕
C
j; that is, every element a
∈
H can be uniquely expressed as a
=
a
1
+
a
2
i
+
a
3
j
+
a
4
k
=
(
a
1
+
a
2
i
)
+
(
a
3
+
a
4
i
)
j, where i
2
=
j
2
=
k
2
=
i
j
k
=
-
1.
Similarly,f
∈
Sp
(
n
,
1
) can be expressed as f
=
f
1
+
f
2
j, where f
1
,
f
2
∈
M
n
+
1
(
C
), the set of (
n
+
1
)
×
(
n
+
1
) complex matrices. This gives an embedding
(13)
ψ
:
Sp
(
n
,
1
)
⟶
GL
(
2
n
+
2
,
C
)
,
f
⟼
ψ
(
f
)
=
(
f
1
f
2
-
f
2
¯
f
1
¯
)
.
We call ψ
(
f
) the complex representation of f. Obviously, ψ is an isomorphism between Sp
(
n
,
1
) and ψ
(
Sp
(
n
,
1
)
). Let E
=
diag
⁡
(
I
n
+
1
,
I
n
+
1
) and let Ω
=
{
f
∈
GL
(
2
n
+
2
,
C
)
:
∥
f
-
E
∥
<
2
-
3
}. Then Ω is the zassenhaus neighborhood [2] of GL
(
2
n
+
2
,
C
) and we have
(14)
2
∥
f
-
I
n
+
1
∥
=
∥
ψ
(
f
)
-
E
∥
.
Suppose that (12) does not hold. Then
(15)
max
⁡
{
∥
ψ
(
f
)
-
E
∥
,
∥
ψ
(
g
)
-
E
∥
}
<
2
-
3
.
By the property of Zassenhaus neighborhood, 〈
ψ
(
f
)
,
ψ
(
g
)
〉 is nilpotent. Hence 〈
f
,
g
〉 is also nilpotent. By Selberg lemma, 〈
f
,
g
〉 contains a torsion free subgroup G with finite index. Hence G is nilpotent. By Lemma 7, G is elementary. By [22, Lemma 4.3
.
2], L
(
G
)
=
L
(
〈
f
,
g
〉
), which implies that 〈
f
,
g
〉 is elementary. This is a contradiction. The proof is complete.


## 3. Proofs of Main Results

Proof of Theorem3.
(i) Suppose that (4) does not hold. Then
(16)
max
⁡
{
∥
f
-
I
n
+
1
∥
,
∥
[
f
,
g
]
-
I
n
+
1
∥
}
<
(
2
-
3
)
2
.
By Lemma 11, 〈
f
,
[
f
,
g
]
〉 is elementary. Since 〈
f
,
[
f
,
g
]
〉
=
〈
f
,
g
f
g
-
1
〉, by Lemma 9, 〈
f
,
g
〉 is elementary. This is a contradiction. Similarly, (5) holds.
(ii) Suppose that (6) does not hold. Then
(17)
max
⁡
{
∥
ψ
(
f
)
-
E
∥
,
∥
[
ψ
(
f
)
,
ψ
(
g
i
)
]
-
E
∥
:
i
=
1,2
,
…
,
m
+
1
}
<
2
-
3
,
where m is the dimension of fix
(
f
). Let G
=
〈
ψ
(
f
)
,
[
ψ
(
f
)
,
ψ
(
g
i
)
]
:
i
=
1,2
,
…
,
m
+
1
〉. Then G is nilpotent. By the isomorphism of ψ, 〈
f
,
[
f
,
g
i
]
:
i
=
1,2
,
…
,
m
+
1
〉 is also nilpotent. As in the reasoning in Lemma 11, 〈
f
,
[
f
,
g
i
]
:
i
=
1,2
,
…
,
m
+
1
〉 is elementary. By Lemma 10, 〈
f
,
g
〉 is elementary. This is a contradiction. Similarly, (7) holds. The proof is complete.Proof of Theorem4.
We divide our proof into two parts.(1)
We first prove thatG
r is discrete.
Suppose thatG
r is not discrete. Then there is a sequence {
g
j
} of G
r such that
(18)
g
j
⟶
I
n
+
1
as
j
⟶
∞
,
and we can find a corresponding sequence {
g
j
k
,
i
k
} such that
(19)
g
j
k
,
i
k
∈
G
r
,
i
k
,
g
j
k
,
i
k
⟶
I
n
+
1
as
k
⟶
∞
.
Since {
G
r
,
i
} satisfies I-condition and G
r
,
i is discrete for each i, we may assume that, for each k, g
j
k
,
i
k is parabolic, loxodromic, or regular elliptic element.
Ifg
j
k
,
i
k is parabolic, loxodromic, for each k, there is at least one generator of G
r
,
i
k, say g
1
,
i
k, such that 〈
g
1
,
i
k
,
g
j
k
,
i
k
〉 is nonelementary, which is a contradiction to Theorem 3.
Ifg
j
k
,
i
k is a regular elliptic element, by Theorem 3, 〈
g
t
,
i
k
,
g
j
k
,
i
k
〉 is discrete and elementary for t
=
1
,
…
r. By Lemma 6, each 〈
g
t
,
i
k
,
g
j
k
,
i
k
〉 is purely elliptic or contains a loxodromic element. If the latter case occurs, then {
g
j
k
,
i
k
2
} are sequence of boundary elliptic elements which converges to the identity. This is a contradiction to our assumption of I-condition, while, for the first case, each g
t
,
i
k shares a fixed point in H
H
n. This is also a contradiction.
The above proves the discreteness ofG
r. (2)
We prove thatG
r is nonelementary.
We assume thatG
r
,
i
=
〈
g
1
,
i
,
g
2
,
i
,
…
,
g
r
,
i
〉 and g
t
,
i
→
g
t as i
→
∞; that is, G
r
=
〈
g
1
,
g
2
,
…
,
g
r
〉. The proof of part (
1
) implies that each g
t
,
t
=
1
,
…
,
r is not the identity.
SinceG
r
,
i is discrete and nonelementary, there exist two loxodromic elements f
i and h
i having no common fixed points. Since f
i and h
i are words of the generators {
g
t
,
i
}, we can get the limit f and g by the word convergence of f
i and h
i, respectively. It remains to prove that 〈
f
,
h
〉
⊂
G
r is nonelementary.
We first show thatf is parabolic or loxodromic. Since 〈
f
,
g
〉 is discrete, f cannot be an irrational rotation. Suppose that there is a positive number M such that f
M
=
I
n
+
1. Then f
i
M
≠
I
n
+
1 and
(20)
f
i
M
⟶
I
n
+
1
as
i
⟶
∞
.
Hence for sufficiently large i,
(21)
max
⁡
{
∥
f
i
M
-
I
n
+
1
∥
,
∥
[
f
i
M
,
h
i
t
]
-
I
n
+
1
∥
∣
t
=
1,2
,
…
,
n
+
1
}
<
(
2
-
3
)
2
.
By Theorem 3, 〈
f
i
M
,
h
i
〉, which are subgroups of discrete group 〈
f
i
,
h
i
〉, are elementary for sufficiently large i. This implies that 〈
f
i
,
h
i
〉 is elementary, which is a contradiction.
We then show that〈
f
,
h
〉 is nonelementary.
Suppose on the contrary that〈
f
,
h
〉 is elementary. As in [2, Proposition 2.7], we can show that 〈
f
,
h
〉 is virtually Abelian. Thus there exist two integers t and s such that
(22)
[
f
t
,
h
f
s
h
-
1
]
=
I
n
+
1
.
Let q
i
=
[
f
i
t
,
h
i
f
i
s
h
i
-
1
]. Then
(23)
q
i
∈
〈
f
i
,
h
i
〉
,
q
i
≠
I
n
+
1
,
q
i
⟶
I
n
+
1
as
i
⟶
∞
.
As in the proof of part (
1
), we can get a contradiction. Thus 〈
f
,
h
〉 is nonelementary.
The proof is complete.



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*Source: 684594-2014-05-18.xml*