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Discreteness and Convergence of Complex Hyperbolic Isometry Groups
We investigate the discreteness and convergence of complex isometry groups and some discreteness criteria and algebraic convergence theorems for subgroups of are obtained. All of the results are generalizations of the corresponding known ones.
Theorem J. A nonelementary subgroup of is discrete if and only if each two-generator subgroup in is discrete.
Furthermore, Gilman  and Isachenko  showed that the discreteness of all two-generator subgroups of , where each generator is loxodromic, is enough to secure the discreteness of . See [5–8] and so forth for some other discussions along this line.
It is interesting to generalize Theorem J into the higher dimensional case. By adding some conditions, several generalizations of Theorem J into () have been obtained; see [9–13] and so forth. In 2005, Wang et al.  proved the following.
Theorem WLC. Let be nonelementary and loxodromic. Then is discrete if and only if is discrete and each nonelementary subgroup is discrete, where .
Here and is the smallest -invariant hyperbolic subspace whose boundary contains the limit set of (cf. ).
Since the real hyperbolic plane can be viewed as a complex hyperbolic 1-space , it is natural to generalize these results mentioned above to the setting of complex hyperbolic space. Recently, Qin and Jiang  proved the following.
Theorem QJ1. Let be an -dimensional subgroup of and a nonelliptic element in . If for each loxodromic (resp., regular elliptic) element the two-generator group is discrete, then is discrete.
Theorem QJ2. Let be an -dimensional subgroup of and a regular elliptic element in . If for each loxodromic (resp., regular elliptic) element the two-generator group is discrete, then is discrete.
Here is called -dimensional if it does not leave a point in or a proper totally geodesic submanifold of invariant. Obviously, if is -dimensional, then is nonelementary and .
Motivated by Theorem WLC, a natural question will be asked: can we use the discreteness of subgroups to determine the discreteness of in Theorems QJ1 and QJ2? In this paper, we will give this question a positive answer (see Section 3).
Let be the Möbius group or the complex hyperbolic isometry group .
Definition 1. Let be a sequence of subgroups in and each be generated by . If, for each , then we say that converges algebraically to , and is called the algebraic limit group of . If for each , is a Kleinian group, then the question when is still a Kleinian group has attracted much attention. Jørgensen and Klein proved that is still a Kleinian group, when . For the higher dimensional case, there are a number of discussions; see [11, 12, 17].
When , Cao proved  the following.
Theorem C1. Let be a sequence of groups of . If each is discrete, then the algebraic limit group of is either a complex Kleinian group, or it is elementary, or is not finite.
Theorem C2. Let be the algebraic limit group of complex Kleinian groups of . If satisfies IP-condition, then is a complex Kleinian group.
Here satisfies IP-condition means that satisfies the following conditions: for any sequence , if for each , and as with being the identity or parabolic, then has uniformly bounded torsion (see ).
In this paper, we will discuss the discreteness criteria and algebraic convergence theorems for subgroups of further. The rest of this paper is organized as follows: in Section 2, we introduce some preliminary results that we need in the sequel; in Section 3, we show three discreteness criteria for subgroups of ; finally Section 4 is dedicated to three algebraic convergence theorems for complex Kleinian groups.
Let be the complex vector space of dimension with the Hermitian form where , are the column vectors in . Consider the following subspaces of : Let be the canonical projection from onto the complex hyperbolic space . The complex hyperbolic space is defined to be and is its boundary. The biholomorphic isometry group of is given by the projective unitary group . For a nontrivial element of , we say that is if it has a fixed point in is if it has only one fixed point in , and is if it has exactly two different fixed points in .
For elliptic element , let and () be its negative and positive eigenvalues, respectively. Then the fixed point set of in contains only one point if and is a totally geodesic submanifold, which is equivalent to if coincides with exact of class (). We call regular elliptic if , where and . Obviously, if is regular elliptic, then has only one fixed point in . The following proposition follows directly from .
Proposition 2. The regular elliptic (resp., loxodromic) elements of form an open set.
Let be a subgroup of . The limit set of is defined as is called nonelementary if contains more than two points; otherwise, it is called elementary. We call a subgroup of complex Kleinian group if it is discrete and nonelementary. For a nonelementary subgroup of , we denote by the smallest totally geodesic submanifold of whose boundary contains the limit set . It is easy to see that is -invariant since is -invariant. As in , the subgroup of is defined as
For an element , we denote , where is the Hilbert-Schmidt norm. Then we have the following.
3. Discreteness Criteria
In this section, we prove the following theorems.
Theorem 4. Let be an -dimensional subgroup of and a nonelliptic element in . If for each loxodromic (resp., regular elliptic) element the two-generator group is discrete, then is discrete.
Theorem 5. Let be an -dimensional subgroup of and a regular elliptic element with finite order in . If for each loxodromic (resp., regular elliptic) element the two-generator group is discrete, then is discrete.
When is elliptic (may not be regular), we have the following.
Theorem 6. Let be an -dimensional subgroup of and an elliptic element with finite order in . If, for each loxodromic (resp., regular elliptic) element the two-generator group is discrete, then is discrete.
In order to prove the above theorems, we need the following lemma which is a classification of elementary subgroups of .
Lemma 7. Let be a subgroup of .(1)If contains a loxodromic element, then is elementary if and only if it fixes a point in or a point-pair in .(2)If contains a parabolic element but no loxodromic element, then is elementary if and only if it fixes a point in .(3)If is purely elliptic, then fixes a point in .
Proof of Theorem 4. Firstly, we prove the case when each is loxodromic. Suppose not. Then is dense in according to Corollary 4.5.1 of . By Proposition 2, there exists a sequence in such that each is loxodromic and as . Then, for large enough , we have
Since is nonelliptic and , by Lemma 3, we know that, for all large enough , are elementary. This implies that
Since is nonelementary, we can find three loxodromic elements in such that
where and . It follows from a discussion similar to the above that we can obtain that, for large enough ,
Since is nonelliptic, that is, contains less than three points; it is a contradiction.
Now, we come to prove the case when each is regular elliptic. Suppose that is nondiscrete. Similarly, by Proposition 2, we can find a sequence in such that each is regular elliptic and as . This implies that, for sufficiently large , the subgroups are elementary. It follows that It is a contradiction since is nonelliptic and is regular elliptic.
This completes the proof.
Proof of Theorem 6. We only prove the case when is loxodromic; similar arguments can be applied to the case when is regular elliptic. Suppose that is nondiscrete. Then there exists a sequence such that, for each , is loxodromic and Since is -dimensional, we can find finitely many loxodromic elements in such that the set can span the whole complex hyperbolic space , where is the attractive fixed point of . For each , let be a small neighbourhood of in ; then there exists an integer such that Since for large enough , we can see that the subgroups are elementary. By Lemma 7, we know that, for each , , Obviously, it is a contradiction.
4. Algebraic Convergence
In this section, we discuss the algebraic convergence of complex hyperbolic Kleinian groups. Firstly, we generalize Theorem C1 into the following form.
Theorem 8. Let be a sequence of groups of and be its algebraic limit group. Then we have the following.(1)If, for each , is a complex Kleinian group, then is nonelementary and is discrete if and only if each one-generator subgroup of is discrete.(2)If, for each , is discrete, then is elementary if and only if for large enough , all are elementary.
Proof. The proof of (1). The nonelementariness of follows from [21, Theorem 1.4]. Now, we come to prove that if is nondiscrete, then there is an element such that the subgroup is nondiscrete. Suppose that is nondiscrete. Since (that is, is finitely generated), by Selberg’s Lemma we know that contains a torsion free subgroup with finite index which is nonelementary and nondiscrete either. Then there exists a sequence in such that
As is nonelementary, we can find finitely many loxodromic elements , in such that the set , spans , the boundary of . Then, for large enough , we have
Let and be the corresponding elements of and in , respectively. Then, for large enough and ,
Lemma 3 implies that, for large enough and , the subgroups are elementary. Since the loxodromic elements of form an open set, we know that, for sufficiently large , are loxodromic as well. It follows that
which shows that, for and all sufficiently large ,
Thus, for all sufficiently large ,
Since is torsion free, we know that there exists an element such that is nondiscrete. Note that , so . Hence, the conclusion of (1) follows.
The proof of (2). We only need to prove that if, for large enough , all are elementary, then is since the converse is trivial by (1). Suppose that is nonelementary. Then we can find two loxodromic elements and in such that Let and be the corresponding elements of and in , respectively. Then, for large enough , we have It follows a discussion similar to that in the proof of (1) that, for large enough , both and are loxodromic. This shows that, for large enough , all are nonelementary. It is a contradiction.
Definition 9. Let be a sequence of complex Kleinian groups of . We say that satisfies E-condition if there is no sequence such that as , where is an elliptic element with infinite order.
In the following, we give an example which shows that, if the sequence does not satisfy IP-condition but -condition, then the limit group is still a complex Kleinian groups.
Example 10. Suppose that is a purely loxodromic nonelementary subgroup of and, for each , Let be the Poincaré extension of in and . Then it is easy to see that the algebraic limit group of is a complex Kleinian group. Note that as ; we know that does not satisfy IP-condition but -condition.
As applications of Theorem 8 and -condition, we have the following.
Theorem 11. Let be the algebraic limit group of complex Kleinian groups of . If satisfies E-condition, then is a complex Kleinian group.
Proof. By Theorem 8(1), we know that is nonelementary. Suppose that is nondiscrete. Then there exist an elliptic element and an integer sequence such that and
For each , let be the corresponding element of in . By [21, Lemma 4.2], we know that . It follows from the hypothesis that satisfies -condition; we have for large enough . This implies that . It is a contradiction.
The proof is completed.
When , Wang  proved the following.
Theorem W. Let . If the generator system of satisfies that none are elliptic and no two have any fixed point in common, and, if all are Kleinian groups, then(1)all the generators are neither elliptic nor identity;(2)if is nonelementary and is discrete, then is discrete.
It easily follows a similar argument as in the proof of Theorem 8 and we can obtain the following.
Theorem 12. Let . If the generator system of satisfies that none are elliptic and no two have any fixed point in common, and, if all are discrete, then(1) is nonelementary;(2) is discrete if and only if is discrete.
The research was partly supported by Tian-Yuan Foundation (no. 11226096).
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