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

Abstract

We obtain an analogue of Jørgensen's inequality in quaternionic hyperbolic space. As an application, we prove that if the r-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.

1. Introduction

Jørgensen's inequality [1] gives a necessary condition for a nonelementary two-generator subgroup of to be discrete, which involves the traces of one of the generators and the commutator of both generators, as follows.

Theorem A. Let . If each two-generator subgroup is discrete and nonelementary, then where is the commutator of and and 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 .

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 . Recently, Cao, Tan, and Parker, [15, 16] obtained analogues of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic -space generated by two elements, one of which is elliptic or loxodromic.

Shimizu’s lemma deals with two-generator subgroup with being parabolic element and there are some generalizations to quaternionic hyperbolic space [7–9] for some special kinds of parabolic elements. But for 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 group .

On the other hand, convergence of nonelementary subgroups of real or complex hyperbolic isometry groups is also another important problem. Let be the -dimensional sense-preserving Möbius group or unitary group .

Definition 1. Let be a sequence of subgroups in group , where is generated by and . If, for each , then one says that algebraically converges to .

The problem that under which condition the limit group is also a Kleinian group if, for each , 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 is -generator Kleinian subgroup of , where is , then the limit group is also a Kleinian group.

However, the examples in [18] show that Theorem B could not be extended to -dimensional cases without any modifications. The reason for this phenomenon is that there is a distinction between the fixed point sets of elliptic elements in and ) (). The reasoning mechanism in mainly relies on the fact that each elliptic element has only two fixed points in . Because the fixed point set of an elliptic element of ) () may be empty set or subset of , we cannot use the same reasoning mechanism as in . 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 ) when .

Martin [2] proved the following theorem.

Theorem C. Let be the algebraic limit group of a sequence of -generator Kleinian groups of of uniformly bounded torsion. Then 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 and be the algebraic limit group of a sequence of -generator Kleinian groups of . If satisfies EP-condition (or condition A), then 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 be the algebraic limit group of complex Kleinian groups of . If satisfies IP-condition, then is a complex Kleinian group.

Here, satisfying IP-condition means that, for any sequence , if, for each , and as with being parabolic or the identity, then 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 subset of is said to have uniformly bounded torsion if there exists an integer such that , And we call a nonelementary and discrete subgroup of a quaternionic Kleinian group.

For a sequence of subgroups of , we introduce the following condition.

Definition 2. One says that satisfies I-condition if any sequence (), satisfying the condition that, for each , and as , has uniformly bounded torsion. Here denotes the cardinality of a set .

Our main results are the following theorems.

Theorem 3. Suppose that and generate a discrete and nonelementary group. Then (i)if is parabolic or loxodromic, one has (ii)if is elliptic, one has where is the Hilbert-Schmidt norm of an element.

Theorem 4. Let be the algebraic limit group of quaternionic Kleinian groups of . If satisfies I-condition, then is a quaternionic Kleinian group.

2. Several Lemmas

Let denote the field , or . We adopt the same notations and definitions as in [7, 16, 22, 23] such as ,  , discrete groups, limit sets, and elementary and nonelementary.

We first discuss some properties of elliptic elements. As in [22], for an elliptic element , let and , be its negative and positive class of eigenvalues, respectively. Let denote the set of fixed point(s) of in . Then the fixed point set of in contains only one fixed point if , , and is a totally geodesic submanifold which is equivalent to (resp., ) if (resp., ) coincides with exact of the class . In the latter case, the fixed point set of in is or , and we define . The elliptic elements with only one fixed point in are called regular elliptic elements, while the other elements are called boundary elliptic elements. We call an elliptic element an irrational rotation if with irrational for some .

Since does not act effectively in , one always consider its projective group . It is well known that the -dimensional Möbius group is isomorphic to the identity component of , the projective orthogonal group. Each elliptic element is conjugate to an element with the form When and is even, there are elliptic elements with and the eigenvalues of positive class form conjugated pairs of complex numbers of norm 1. Those elements correspond to the so-called fixed-point-free elements in . However, when is odd, by our above isomorphism, -dimensional Möbius group cannot contain any fixed-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. Let be an elliptic element of order . If , then there is a constant such that

Proof. Let the right complex eigenvalues of be . By Schur’s unitary triangularization theorem of quaternionic version in [24], there is a matrix such that Hence . It follows from that there is such that and (here and are prime). Hence Set . Then is the desired number.

By the above lemma, we know that if the sequence of nontrivial unitary quaternionic transformations converges to the identity, then the orders of converge to infinity. So a family of groups satisfies I-condition if there is no sequence converging to the identity, such that for each .

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 .

The following lemma is a classification of elementary subgroups of .

Lemma 6 (cf. [25]). If contains a parabolic element but no loxodromic element, then is elementary if and only if it fixes a point in ;
if contains a loxodromic element, then is elementary if and only if it fixes a point in or a point-pair ;
   is purely elliptic; that is, each nontrivial element of is elliptic; then is elementary and fixes a point in .

By Lemma 6, we have the following lemmas.

Lemma 7. If is discrete nilpotent group without elliptic element, then is elementary.

Lemma 8 (cf. [2, Lemma 2.8]). Let and be two distinct points in . If interchanges and , then .

The proofs of the following two lemmas follow from similar discussions in [2].

Lemma 9 (cf. [2, Lemma 4.1]). Let be discrete with f being parabolic or loxodromic element. If is elementary, then is also elementary.

Lemma 10 (cf.[2, Lemma 4.2]). Let be discrete with being elliptic element. Let . If is elementary, then is elementary or .

Lemma 11. Suppose that and generate a discrete and nonelementary group. Then where is the Hilbert-Schmidt norm of an element.

Proof of Lemma 11. We can choose to be the subspace of spanned by . With respect to this choice of we can write ; that is, every element can be uniquely expressed as , where .
Similarly, can be expressed as , where , the set of complex matrices. This gives an embedding We call the complex representation of . Obviously, is an isomorphism between and . Let and let . Then is the zassenhaus neighborhood [2] of and we have
Suppose that (12) does not hold. Then By the property of Zassenhaus neighborhood, is nilpotent. Hence is also nilpotent. By Selberg lemma, contains a torsion free subgroup with finite index. Hence is nilpotent. By Lemma 7, is elementary. By [22, Lemma ], , which implies that is elementary. This is a contradiction. The proof is complete.

3. Proofs of Main Results

Proof of Theorem 3. (i) Suppose that (4) does not hold. Then By Lemma 11, is elementary. Since , by Lemma 9, is elementary. This is a contradiction. Similarly, (5) holds.
(ii) Suppose that (6) does not hold. Then where is the dimension of . Let . Then is nilpotent. By the isomorphism of , is also nilpotent. As in the reasoning in Lemma 11, is elementary. By Lemma 10, is elementary. This is a contradiction. Similarly, (7) holds. The proof is complete.

Proof of Theorem 4. We divide our proof into two parts.(1)We first prove that is discrete.
Suppose that is not discrete. Then there is a sequence of such that and we can find a corresponding sequence such that Since satisfies I-condition and is discrete for each , we may assume that, for each , is parabolic, loxodromic, or regular elliptic element.
If is parabolic, loxodromic, for each , there is at least one generator of , say , such that is nonelementary, which is a contradiction to Theorem 3.
If is a regular elliptic element, by Theorem 3, is discrete and elementary for . By Lemma 6, each is purely elliptic or contains a loxodromic element. If the latter case occurs, then 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 shares a fixed point in . This is also a contradiction.
The above proves the discreteness of . (2)We prove that is nonelementary.
We assume that and as ; that is, . The proof of part implies that each is not the identity.
Since is discrete and nonelementary, there exist two loxodromic elements and having no common fixed points. Since and are words of the generators , we can get the limit and by the word convergence of and , respectively. It remains to prove that is nonelementary.
We first show that is parabolic or loxodromic. Since is discrete, cannot be an irrational rotation. Suppose that there is a positive number such that . Then and Hence for sufficiently large , By Theorem 3, , which are subgroups of discrete group , are elementary for sufficiently large . This implies that is elementary, which is a contradiction.
We then show that is nonelementary.
Suppose on the contrary that is elementary. As in [2, Proposition 2.7], we can show that is virtually Abelian. Thus there exist two integers and such that Let . Then As in the proof of part , we can get a contradiction. Thus is nonelementary.
The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by NSFs of China (11071063 and 10801107) and NSF of Guangdong Province (no. S2011010000735).