Discreteness and Convergence of Complex Hyperbolic Isometry Groups

Abstract

We investigate the discreteness and convergence of complex isometry groups and some discreteness criteria and algebraic convergence theorems for subgroups of 𝐏𝐔(𝑛,1) are obtained. All of the results are generalizations of the corresponding known ones.

1. Introduction

In 1976, Jørgensen obtained a very useful necessary condition for two-generator Kleinian groups of 𝑀(ℝ²), which is known as Jørgensen’s inequality. As an application, he obtained the following [1, 2].

Theorem J. A nonelementary subgroup 𝐺 of 𝑀(ℝ²) is discrete if and only if each two-generator subgroup in 𝐺 is discrete.

Furthermore, Gilman [3] and Isachenko [4] showed that the discreteness of all two-generator subgroups of 𝐺, where each generator is loxodromic, is enough to secure the discreteness of 𝐺. 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 𝑀(ℝ^𝑛) (𝑛≥3) have been obtained; see [9–13] and so forth. In 2005, Wang et al. [14] proved the following.

Theorem WLC. Let 𝐺⊂𝑀(ℝ^𝑛) be nonelementary and 𝑓∈𝐺 loxodromic. Then 𝐺 is discrete if and only if 𝑊𝑌(𝐺) is discrete and each nonelementary subgroup ⟨𝑓,𝑔𝑓𝑔⁻¹⟩ is discrete, where 𝑔∈𝐺.

Here 𝑊𝑌(𝐺)=ℎ∶ℎ|_𝑀(𝐺)=𝐼,ℎ∈𝐺, (1) and 𝑀(𝐺) is the smallest 𝐺-invariant hyperbolic subspace whose boundary contains the limit set 𝐿(𝐺) of 𝐺 (cf. [15]).

Since the real hyperbolic plane can be viewed as a complex hyperbolic 1-space ℍ¹_ℂ, it is natural to generalize these results mentioned above to the setting of complex hyperbolic space. Recently, Qin and Jiang [16] proved the following.

Theorem QJ1. Let 𝐺 be an 𝑛-dimensional subgroup of 𝐏𝐔(𝑛,1) and 𝑓 a nonelliptic element in 𝐏𝐔(𝑛,1). If for each loxodromic (resp., regular elliptic) element 𝑔∈𝐺 the two-generator group ⟨𝑓,𝑔⟩ is discrete, then 𝐺 is discrete.

Theorem QJ2. Let 𝐺 be an 𝑛-dimensional subgroup of 𝐏𝐔(𝑛,1) and 𝑓 a regular elliptic element in 𝐏𝐔(𝑛,1). If for each loxodromic (resp., regular elliptic) element 𝑔∈𝐺 the two-generator group ⟨𝑓,𝑔⟩ is discrete, then 𝐺 is discrete.

Here 𝐺 is called 𝑛-dimensional if it does not leave a point in 𝜕ℍ^𝑛_ℂ or a proper totally geodesic submanifold of ℍ^𝑛_ℂ invariant. Obviously, if 𝐺 is 𝑛-dimensional, then 𝐺 is nonelementary and 𝑀(𝐺)=ℍ^𝑛_ℂ.

Motivated by Theorem WLC, a natural question will be asked: can we use the discreteness of subgroups ⟨𝑓,𝑔𝑓𝑔⁻¹⟩ to determine the discreteness of 𝐺 in Theorems QJ1 and QJ2? In this paper, we will give this question a positive answer (see Section 3).

Let 𝔾 be the Möbius group 𝑀(ℝ^𝑛) or the complex hyperbolic isometry group 𝐏𝐔(𝑛,1).

Definition 1. Let {𝐺_𝑟,𝑖} be a sequence of subgroups in 𝔾 and each 𝐺_𝑟,𝑖 be generated by 𝑔_1,𝑖,𝑔_2,𝑖,…,𝑔_𝑟,𝑖. If, for each 𝑡∈{1,2,…,𝑟}(𝑟<∞), 𝑔_𝑡,𝑖⟶𝑔_𝑡∈𝔾as𝑖⟶∞, (2) then we say that {𝐺_𝑟,𝑖} converges algebraically to 𝐺_𝑟=⟨𝑔₁, 𝑔₂,…,𝑔_𝑟⟩ and 𝐺_𝑟 is called the algebraic limit group of {𝐺_𝑟,𝑖}. If for each 𝑖, 𝐺_𝑟,𝑖 is a Kleinian group, then the question when 𝐺_𝑟 is still a Kleinian group has attracted much attention. Jørgensen and Klein proved that 𝐺_𝑟 is still a Kleinian group, when 𝑛=2. For the higher dimensional case, there are a number of discussions; see [11, 12, 17].

When 𝔾=𝐏𝐔(𝑛,1), Cao proved [18] the following.

Theorem C1. Let {𝐺_𝑟,𝑖} be a sequence of groups of 𝔾. If each 𝐺_𝑟,𝑖 is discrete, then the algebraic limit group 𝐺_𝑟 of {𝐺_𝑟,𝑖} is either a complex Kleinian group, or it is elementary, or 𝑊(𝐺𝑟) is not finite.

Theorem C2. Let 𝐺_𝑟 be the algebraic limit group of complex Kleinian groups {𝐺_𝑟,𝑖} of 𝔾. If {𝐺_𝑟,𝑖} satisfies IP-condition, then 𝐺_𝑟 is a complex Kleinian group.

Here {𝐺_𝑟,𝑖} satisfies IP-condition means that {𝐺_𝑟,𝑖} satisfies the following conditions: for any sequence {𝑓_𝑖},𝑓_𝑖∈𝐺_𝑟,𝑖, if card[fix(𝑓_𝑖)]=∞ for each 𝑖, and 𝑓_𝑖→𝑓 as 𝑖→∞ with 𝑓 being the identity or parabolic, then {𝑓_𝑖} has uniformly bounded torsion (see [18]).

In this paper, we will discuss the discreteness criteria and algebraic convergence theorems for subgroups of 𝐏𝐔(𝑛,1) 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 𝐏𝐔(𝑛,1); finally Section 4 is dedicated to three algebraic convergence theorems for complex Kleinian groups.

2. Preliminaries

Let ℂ^𝑛,1 be the complex vector space of dimension 𝑛+1 with the Hermitian form ⟨𝑧,𝑤⟩=𝑧₁𝑤₁+𝑧₂𝑤₂+⋯+𝑧_𝑛𝑤_𝑛−𝑧_𝑛+1𝑤_𝑛+1, (3) where 𝑧, 𝑤 are the column vectors in ℂ^𝑛+1. Consider the following subspaces of ℂ^𝑛,1: 𝑉₀=𝑧∈ℂ^𝑛,1−{0}∶⟨⟩,𝑉𝑧,𝑧=0₋=𝑧∈ℂ^𝑛,1∶⟨𝑧,𝑧⟩<0. (4) Let 𝑃∶ℂ^𝑛+1−{0}→ℂℙ^𝑛 be the canonical projection from ℂ^𝑛+1−{0} onto the complex hyperbolic space ℂℙ^𝑛. The complex hyperbolic space is defined to be ℍ^𝑛_ℂ=𝑃𝑉₋ and 𝜕ℍ^𝑛_ℂ=𝑃𝑉₀ is its boundary. The biholomorphic isometry group of ℍ^𝑛_ℂ is given by the projective unitary group 𝐏𝐔(𝑛,1). For a nontrivial element 𝑔 of 𝐏𝐔(𝑛,1), we say that 𝑔 is 𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐 if it has a fixed point in ℍ^𝑛_ℂ,𝑔 is 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 if it has only one fixed point in 𝜕ℍ^𝑛_ℂ, and 𝑔 is 𝑙𝑜𝑥𝑜𝑑𝑟𝑜𝑚𝑖𝑐 if it has exactly two different fixed points in 𝜕ℍ^𝑛_ℂ.

For elliptic element 𝑔∈𝐏𝐔(𝑛,1), let Λ₀ and Λ_𝑖 (𝑖=1,2,…,𝑛) be its negative and positive eigenvalues, respectively. Then the fixed point set of 𝑔 in ℍ^𝑛_ℂ contains only one point if Λ₀≠Λ_𝑖 and is a totally geodesic submanifold, which is equivalent to ℍ^𝑚_ℂ if Λ₀ coincides with exact 𝑚 of class Λ_𝑖 (𝑚<𝑛). We call 𝑔 regular elliptic if Λ_𝑠≠Λ_𝑡, where 𝑠,𝑡∈{0,1,…,𝑛} and 𝑠≠𝑡. Obviously, if 𝑔 is regular elliptic, then 𝑔 has only one fixed point in ℍ^𝑛_ℂ. The following proposition follows directly from [19].

Proposition 2. The regular elliptic (resp., loxodromic) elements of 𝐏𝐔(𝑛,1) form an open set.

Let 𝐺 be a subgroup of 𝐏𝐔(𝑛,1). The limit set 𝐿(𝐺) of 𝐺 is defined as 𝐿(𝐺)=𝐺(𝑝)∩𝜕ℍ^𝑛_ℂ,𝑝∈ℍ^𝑛_ℂ. (5)𝐺 is called nonelementary if 𝐿(𝐺) contains more than two points; otherwise, it is called elementary. We call a subgroup 𝐺 of 𝐏𝐔(𝑛,1) complex Kleinian group if it is discrete and nonelementary. For a nonelementary subgroup 𝐺 of 𝐏𝐔(𝑛,1), we denote by 𝑀(𝐺) the smallest totally geodesic submanifold of 𝐺 whose boundary contains the limit set 𝐿(𝐺). It is easy to see that 𝑀(𝐺) is 𝐺-invariant since 𝐿(𝐺) is 𝐺-invariant. As in [18], the subgroup 𝑊(𝐺) of 𝐺 is defined as 𝑊(𝐺)=𝑔∶𝑔∣_𝑀(𝐺)=𝐼,𝑔∈𝐺. (6)

For an element 𝑓∈𝐏𝐔(𝑛,1), we denote 𝑁(𝑓)=‖𝑓−𝐼‖, where ‖⋅‖ is the Hilbert-Schmidt norm. Then we have the following.

Lemma 3 (see [18, 20]). Suppose that two elements 𝑓,𝑔∈𝐏𝐔(𝑛,1) generate a complex Kleinian group.(1)If 𝑓 is parabolic or loxodromic, then max𝑁𝑓,𝑁𝑓,𝑔≥2−√3, (7) where [𝑓,𝑔]=𝑓𝑔𝑓⁻¹𝑔⁻¹ is the commutator of 𝑓 and 𝑔.(2)If 𝑓 is elliptic, then max𝑁𝑓,𝑁𝑓,𝑔^𝑞∶𝑞=1,2,3,…,𝑛+1≥2−√3. (8)

3. Discreteness Criteria

In this section, we prove the following theorems.

Theorem 4. Let 𝐺 be an 𝑛-dimensional subgroup of 𝐏𝐔(𝑛,1) and 𝑓 a nonelliptic element in 𝐏𝐔(𝑛,1). If for each loxodromic (resp., regular elliptic) element 𝑔∈𝐺 the two-generator group ⟨𝑓,𝑔𝑓𝑔⁻¹⟩ is discrete, then 𝐺 is discrete.

Theorem 5. Let 𝐺 be an 𝑛-dimensional subgroup of 𝐏𝐔(𝑛,1) and 𝑓 a regular elliptic element with finite order 𝑘(𝑘≥3) in 𝐏𝐔(𝑛,1). If for each loxodromic (resp., regular elliptic) element 𝑔∈𝐺 the two-generator group ⟨𝑓,𝑔𝑓𝑔⁻¹⟩ is discrete, then 𝐺 is discrete.

When 𝑓 is elliptic (may not be regular), we have the following.

Theorem 6. Let 𝐺 be an 𝑛-dimensional subgroup of 𝐏𝐔(𝑛,1) and 𝑓 an elliptic element with finite order 𝑘(𝑘≥3) in 𝐏𝐔(𝑛,1). If, for each loxodromic (resp., regular elliptic) element 𝑔∈𝐺 the two-generator group ⟨𝑓,𝑔⟩ is discrete, then 𝐺 is discrete.

In order to prove the above theorems, we need the following lemma which is a classification of elementary subgroups of 𝐏𝐔(𝑛,1).

Lemma 7. Let 𝐺 be a subgroup of 𝐏𝐔(𝑛,1).(1)If 𝐺 contains a loxodromic element, then 𝐺 is elementary if and only if it fixes a point in 𝜕ℍ^𝑛_ℂ or a point-pair {𝑥,𝑦} in 𝜕ℍ^𝑛_ℂ.(2)If 𝐺 contains a parabolic element but no loxodromic element, then 𝐺 is elementary if and only if it fixes a point in 𝜕ℍ^𝑛_ℂ.(3)If 𝐺 is purely elliptic, then 𝐺 fixes a point in ℍ^𝑛_ℂ.

Proof of Theorem 4. Firstly, we prove the case when each 𝑔 is loxodromic. Suppose not. Then 𝐺 is dense in 𝐏𝐔(𝑛,1) according to Corollary 4.5.1 of [15]. By Proposition 2, there exists a sequence {𝑔_𝑖} in 𝐺 such that each 𝑔_𝑖 is loxodromic and 𝑔_𝑖→𝐼 as 𝑖→∞. Then, for large enough 𝑖, we have 𝑁𝑔_𝑖𝑓𝑔_𝑖⁻¹𝑓⁻¹+^𝑛+1_𝑞=1𝑁𝑔_𝑖𝑓𝑔_𝑖⁻¹𝑓⁻¹,𝑓^𝑞<2−√3. (9) Since 𝑓 is nonelliptic and ⟨𝑓,𝑔_𝑖𝑓𝑔_𝑖⁻¹𝑓⁻¹⟩=⟨𝑓,𝑔_𝑖𝑓𝑔_𝑖⁻¹⟩, by Lemma 3, we know that, for all large enough 𝑖, ⟨𝑓,𝑔_𝑖𝑓𝑔_𝑖⁻¹⟩ are elementary. This implies that fix𝑓∩fix𝑔_𝑖≠∅. (10) Since 𝐺 is nonelementary, we can find three loxodromic elements ℎ_𝑠(𝑠=1,2,3) in 𝐺 such that fix𝑓∩fixℎ_𝑠=∅,fixℎ_𝑗∩fixℎ_𝑘=∅, (11) where 𝑖,𝑘∈{1,2,3} and 𝑗≠𝑘. It follows from a discussion similar to the above that we can obtain that, for large enough 𝑖, fix𝑓∩fixℎ_𝑠𝑔_𝑖ℎ_𝑠⁻¹≠∅,𝑠=1,2,3. (12) Since 𝑓 is nonelliptic, that is, fix(𝑓) contains less than three points; it is a contradiction.
Now, we come to prove the case when each 𝑔 is regular elliptic. Suppose that 𝐺 is nondiscrete. Similarly, by Proposition 2, we can find a sequence {𝑔_𝑖} in 𝐺 such that each 𝑔_𝑖 is regular elliptic and 𝑔_𝑖→𝐼 as 𝑖→∞. This implies that, for sufficiently large 𝑖, the subgroups ⟨𝑓,𝑔_𝑖𝑓𝑔_𝑖⁻¹⟩ are elementary. It follows that fix𝑓=fix𝑔_𝑖. (13) It is a contradiction since 𝑓 is nonelliptic and 𝑔_𝑖 is regular elliptic.
This completes the proof.

Proof of Theorem 5. The proof of Theorem 5 follows from a discussion similar to that in the proof of Theorem 4.

Proof of Theorem 6. We only prove the case when 𝑔 is loxodromic; similar arguments can be applied to the case when 𝑔 is regular elliptic. Suppose that 𝐺 is nondiscrete. Then there exists a sequence {𝑔_𝑖}⊂𝐺 such that, for each 𝑖, 𝑔_𝑖 is loxodromic and 𝑔_𝑖⟶𝐼as𝑖⟶∞. (14) Since 𝐺 is 𝑛-dimensional, we can find finitely many loxodromic elements ℎ₁,ℎ₂,…,ℎ_𝑡 in 𝐺 such that the set 𝑆={𝐴_fix(ℎ1),𝐴_fix(ℎ2),…,𝐴_{fix(ℎ𝑡)}} can span the whole complex hyperbolic space ℍ^𝑛_ℂ, where 𝐴_fix(ℎ) is the attractive fixed point of ℎ. For each 𝑘(𝑘=1,2,…,𝑡), let 𝑈_{𝐴fix𝑘)(ℎ} be a small neighbourhood of 𝐴_{fix(ℎ𝑘)} in ℍ^𝑛_ℂ; then there exists an integer 𝑁 such that ℎ^𝑁_𝑘fix𝑓⊂𝑈_{𝐴fix𝑘)(ℎ}. (15) Since ℎ^𝑁_𝑘𝑓ℎ_𝑘^−𝑁,𝑔_𝑖=ℎ^𝑁_𝑘𝑓,ℎ_𝑘^−𝑁𝑔_𝑖ℎ^𝑁_𝑘ℎ_𝑘^−𝑁,max𝑁ℎ_𝑘^−𝑁𝑔_𝑖ℎ^𝑁_𝑘,𝑁ℎ_𝑘^−𝑁𝑔_𝑖ℎ^𝑁_𝑘,𝑓<2−√3, (16) for large enough 𝑖, we can see that the subgroups ⟨ℎ^𝑁_𝑘𝑓²ℎ_𝑘^−𝑁,𝑔_𝑖⟩ are elementary. By Lemma 7, we know that, for each 𝑘, (𝑘=1,2,…,𝑡), fix𝑔_𝑖∩𝑈_{𝐴fix𝑘)(ℎ}≠∅. (17) 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 {𝐺_𝑟,𝑖} be a sequence of groups of 𝐏𝐔(𝑛,1) and 𝐺_𝑟 be its algebraic limit group. Then we have the following.(1)If, for each 𝑖, 𝐺_𝑟,𝑖 is a complex Kleinian group, then 𝐺_𝑟 is nonelementary and 𝐺_𝑟 is discrete if and only if each one-generator subgroup of 𝑊(𝐺_𝑟) is discrete.(2)If, for each 𝑖, 𝐺_𝑟,𝑖 is discrete, then 𝐺_𝑟 is elementary if and only if for large enough 𝑖, all 𝐺_𝑟,𝑖 are elementary.

Proof. The proof of (1). The nonelementariness of 𝐺_𝑟 follows from [21, Theorem 1.4]. Now, we come to prove that if 𝐺_𝑟 is nondiscrete, then there is an element 𝑓∈𝑊(𝐺_𝑟) such that the subgroup ⟨𝑓⟩ is nondiscrete. Suppose that 𝐺_𝑟 is nondiscrete. Since 𝑟<∞ (that is, 𝐺_𝑟 is finitely generated), by Selberg’s Lemma we know that 𝐺_𝑟 contains a torsion free subgroup 𝐺_𝑟1 with finite index which is nonelementary and nondiscrete either. Then there exists a sequence {𝑓_𝑗} in 𝐺_𝑟1 such that 𝑓_𝑗⟶𝐼as𝑗⟶∞. (18) As 𝐺_𝑟1 is nonelementary, we can find finitely many loxodromic elements 𝑔₁, 𝑔₂,…,𝑔_𝑘 in 𝐺_𝑟1 such that the set {fix(𝑔₁), fix(𝑔₂),…,fix(𝑔_𝑘)} spans 𝜕𝑀(𝐺_𝑟1), the boundary of 𝑀(𝐺_𝑟1). Then, for large enough 𝑗, we have 𝑁𝑓_𝑗+^𝑛+1_𝑞=1𝑁𝑓_𝑗,𝑔^𝑞_𝑠<2−√3,𝑠∈{1,2,…,𝑘}. (19) Let 𝑓_𝑖,𝑗 and 𝑔_𝑖,𝑠 be the corresponding elements of 𝑓_𝑗 and 𝑔_𝑠 in 𝐺_𝑟,𝑖, respectively. Then, for large enough 𝑖 and 𝑗, 𝑁𝑓_𝑖,𝑗+^𝑛+1_𝑞=1𝑁𝑓_𝑖,𝑗,𝑔^𝑞_𝑖,𝑠<2−√3. (20) Lemma 3 implies that, for large enough 𝑖 and 𝑗, the subgroups ⟨𝑓_𝑖,𝑗,𝑔_𝑖,𝑠⟩ are elementary. Since the loxodromic elements of 𝐏𝐔(𝑛,1) form an open set, we know that, for sufficiently large 𝑖, 𝑔_𝑖,𝑠 are loxodromic as well. It follows that fix𝑔_𝑖,𝑠⊂fix𝑓_𝑖,𝑗, (21) which shows that, for 𝑠∈{1,2,…,𝑘} and all sufficiently large 𝑗, fix𝑔_𝑠⊂fix𝑓_𝑗. (22) Thus, for all sufficiently large 𝑗, 𝑓_𝑗∈𝑊𝐺_𝑟1. (23) Since 𝐺_𝑟1 is torsion free, we know that there exists an element 𝑓∈𝑊(𝐺_𝑟1) such that ⟨𝑓⟩ is nondiscrete. Note that 𝑀(𝐺_𝑟)=𝑀(𝐺_𝑟1), so 𝑓∈𝑊(𝐺_𝑟). Hence, the conclusion of (1) follows.
The proof of (2). We only need to prove that if, for large enough 𝑖, all 𝐺_𝑟,𝑖 are elementary, then is 𝐺_𝑟 since the converse is trivial by (1). Suppose that 𝐺_𝑟 is nonelementary. Then we can find two loxodromic elements 𝑓 and 𝑔 in 𝐺_𝑟 such that fix𝑓∩fix𝑔=∅. (24) Let 𝑓_𝑖 and 𝑔_𝑖 be the corresponding elements of 𝑓 and 𝑔 in 𝐺_𝑟,𝑖, respectively. Then, for large enough 𝑖, we have fix𝑓_𝑖∩fix𝑔_𝑖=∅. (25) It follows a discussion similar to that in the proof of (1) that, for large enough 𝑖, both 𝑓_𝑖 and 𝑔_𝑖 are loxodromic. This shows that, for large enough 𝑖, all 𝐺_𝑟,𝑖 are nonelementary. It is a contradiction.

Definition 9. Let {𝐺_𝑖} be a sequence of complex Kleinian groups of 𝐏𝐔(𝑛,1). We say that {𝐺_𝑖} satisfies E-condition if there is no sequence {𝑓_𝑖},𝑓_𝑖∈𝑊(𝐺_𝑖) such that 𝑓_𝑖→𝑓 as 𝑖→∞, where 𝑓 is an elliptic element with infinite order.
In the following, we give an example which shows that, if the sequence {𝐺_𝑖} does not satisfy IP-condition but 𝐸-condition, then the limit group 𝐺_𝑟 is still a complex Kleinian groups.

Example 10. Suppose that 𝐻 is a purely loxodromic nonelementary subgroup of 𝐏𝐔(1,1) and, for each 𝑗, 𝑓_𝑗=𝑒^{𝑖(1/2𝑗)}00010001. (26) Let 𝐻 be the Poincaré extension of 𝐻 in 𝐏𝐔(2,1) and 𝐺_𝑗=⟨𝐻,𝑓_𝑗⟩. Then it is easy to see that the algebraic limit group 𝐺 of {𝐺_𝑗} is a complex Kleinian group. Note that 𝑓_𝑗→𝐼 as 𝑗→∞; we know that {𝐺_𝑗} does not satisfy IP-condition but 𝐸-condition.

As applications of Theorem 8 and 𝐸-condition, we have the following.

Theorem 11. Let 𝐺_𝑟 be the algebraic limit group of complex Kleinian groups {𝐺_𝑟,𝑖} of  𝐏𝐔(𝑛,1). If  {𝐺_𝑟,𝑖} satisfies E-condition, then 𝐺_𝑟 is a complex Kleinian group.

Proof. By Theorem 8(1), we know that 𝐺_𝑟 is nonelementary. Suppose that 𝐺_𝑟 is nondiscrete. Then there exist an elliptic element 𝑓∈𝑊(𝐺_𝑟) and an integer sequence {𝑛_𝑗} such that ord(𝑓)=∞ and 𝑓^𝑛𝑗⟶𝐼as𝑛_𝑗⟶∞. (27) For each 𝑛_𝑗, let 𝑓^𝑛𝑗_𝑖 be the corresponding element of 𝑓^𝑛𝑗 in 𝐺_𝑟,𝑖. By [21, Lemma 4.2], we know that 𝑓^𝑛𝑗_𝑖∈𝑊(𝐺_𝑟,𝑖). It follows from the hypothesis that {𝐺_𝑟,𝑖} satisfies 𝐸-condition; we have 𝑓^𝑛𝑗_𝑖=𝐼 for large enough 𝑖. This implies that 𝑓^𝑛𝑗=𝐼. It is a contradiction.
The proof is completed.

When 𝑟≤∞, Wang [17] proved the following.

Theorem W. Let 𝑟≤∞. If the generator system {𝑔_𝑡,𝑖}^𝑟_𝑡=1 of 𝐺_𝑟,𝑖 satisfies that none are elliptic and no two have any fixed point in common, and, if all 𝐺_𝑟,𝑖 are Kleinian groups, then(1)all the generators 𝑔_𝑡=lim_𝑖→∞𝑔_𝑡,𝑖 are neither elliptic nor identity;(2)if 𝐺_𝑟=⟨𝑔₁,𝑔₂,…,𝑔_𝑟⟩ is nonelementary and 𝑊(𝐺_𝑟) is discrete, then 𝐺_𝑟 is discrete.

It easily follows a similar argument as in the proof of Theorem 8 and we can obtain the following.

Theorem 12. Let 𝑟≤∞. If the generator system {𝑔_𝑡,𝑖}^𝑟_𝑡=1 of  𝐺_𝑟,𝑖 satisfies that none are elliptic and no two have any fixed point in common, and, if all 𝐺_𝑟,𝑖 are discrete, then(1)𝐺_𝑟=⟨𝑔₁,𝑔₂,…,𝑔_𝑟⟩ is nonelementary;(2)𝐺_𝑟 is discrete if and only if  𝑊(𝐺_𝑟) is discrete.

Acknowledgment

The research was partly supported by Tian-Yuan Foundation (no. 11226096).