# Discreteness and Convergence of Complex Hyperbolic Isometry Groups

Authors: Xi Fu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 1. Introduction

In 1976, Jørgensen obtained a very useful necessary condition for two-generator Kleinian groups ofM(ℝ-2), which is known as Jørgensen’s inequality. As an application, he obtained the following [1, 2].Theorem J.
A nonelementary subgroupG of M(ℝ-2) 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 intoM(ℝ-n) (n≥3) have been obtained; see [9–13] and so forth. In 2005, Wang et al. [14] proved the following.Theorem WLC.
LetG⊂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(1)WY(G)={h:h|M(G)=I,h∈G},
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-spaceℍℂ1, 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.
LetG 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.
LetG 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.HereG is called n-dimensional if it does not leave a point in ∂ℍℂn or a proper totally geodesic submanifold of ℍℂn invariant. Obviously, if G is n-dimensional, then G is nonelementary and M(G)=ℍℂn.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).Let𝔾 be the Möbius group M(ℝ-n) or the complex hyperbolic isometry group PU(n,1).Definition 1.
Let{Gr,i} be a sequence of subgroups in 𝔾 and each Gr,i be generated by g1,i,g2,i,…,gr,i. If, for each t∈{1,2,…,r}(r<∞),
(2)gt,i⟶gt∈𝔾asi⟶∞,
then we say that {Gr,i} converges algebraically to Gr=〈g1, g2,…,gr〉 and Gr is called the algebraic limit group of {Gr,i}. If for each i, Gr,i is a Kleinian group, then the question when Gr is still a Kleinian group has attracted much attention. Jørgensen and Klein proved that Gr is still a Kleinian group, when n=2. For the higher dimensional case, there are a number of discussions; see [11, 12, 17].When𝔾=PU(n,1), Cao proved [18] the following.Theorem C1.
Let{Gr,i} be a sequence of groups of 𝔾. If each Gr,i is discrete, then the algebraic limit group Gr of {Gr,i} is either a complex Kleinian group, or it is elementary, or W(Gr) is not finite.Theorem C2.
LetGr be the algebraic limit group of complex Kleinian groups {Gr,i} of 𝔾. If {Gr,i} satisfies IP-condition, then Gr is a complex Kleinian group.Here{Gr,i} satisfies IP-condition means that {Gr,i} satisfies the following conditions: for any sequence {fi},fi∈Gr,i, if card[fix(fi)]=∞ for each i, and fi→f as i→∞ with f being the identity or parabolic, then {fi} has uniformly bounded torsion (see [18]).In this paper, we will discuss the discreteness criteria and algebraic convergence theorems for subgroups ofPU(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

Letℂn,1 be the complex vector space of dimension n+1 with the Hermitian form
(3)〈z,w〉=z1w-1+z2w-2+⋯+znw-n-zn+1w-n+1,
where z, w are the column vectors in ℂn+1. Consider the following subspaces of ℂn,1:
(4)V0={z∈ℂn,1-{0}:〈z,z〉=0},V-={z∈ℂn,1:〈z,z〉<0}.
Let P:ℂn+1-{0}→ℂℙn be the canonical projection from ℂn+1-{0} onto the complex hyperbolic space ℂℙn. The complex hyperbolic space is defined to be ℍℂn=PV- and ∂ℍℂn=PV0 is its boundary. The biholomorphic isometry group of ℍℂn 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 ℍℂn,g is parabolic if it has only one fixed point in ∂ℍℂn, and g is loxodromic if it has exactly two different fixed points in ∂ℍℂn.For elliptic elementg∈PU(n,1), let Λ0 and Λi (i=1,2,…,n) be its negative and positive eigenvalues, respectively. Then the fixed point set of g in ℍℂn contains only one point if Λ0≠Λi and is a totally geodesic submanifold, which is equivalent to ℍℂm if Λ0 coincides with exact m of class Λi (m<n). We call gregular elliptic if Λs≠Λt, where s,t∈{0,1,…,n} and s≠t. Obviously, if g is regular elliptic, then g has only one fixed point in ℍℂn. The following proposition follows directly from [19].Proposition 2.
The regular elliptic (resp., loxodromic) elements ofPU(n,1) form an open set.LetG be a subgroup of PU(n,1). The limit set L(G) of G is defined as
(5)L(G)=G(p)¯∩∂ℍℂn,p∈ℍℂn.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
(6)W(G)={g:g∣M(G)=I,g∈G}.For an elementf∈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 elementsf,g∈PU(n,1) generate a complex Kleinian group.(1)
Iff is parabolic or loxodromic, then
(7)max{N(f),N([f,g])}≥2-3,
where [f,g]=fgf-1g-1 is the commutator of f and g.(2)
Iff is elliptic, then
(8)max{N(f),N([f,gq]):q=1,2,3,…,n+1}≥2-3.


## 3. Discreteness Criteria

In this section, we prove the following theorems.Theorem 4.
LetG 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.
LetG 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.Whenf is elliptic (may not be regular), we have the following.Theorem 6.
LetG 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 ofPU(n,1).Lemma 7.
LetG be a subgroup of PU(n,1).(1)
IfG contains a loxodromic element, then G is elementary if and only if it fixes a point in ∂ℍℂn or a point-pair {x,y} in ∂ℍℂn.(2)
IfG contains a parabolic element but no loxodromic element, then G is elementary if and only if it fixes a point in ∂ℍℂn.(3)
IfG is purely elliptic, then G fixes a point in ℍ¯ℂn.Proof of Theorem4.
Firstly, we prove the case when eachg 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 {gi} in G such that each gi is loxodromic and gi→I as i→∞. Then, for large enough i, we have
(9)N(gifgi-1f-1)+∑q=1n+1‍N([gifgi-1f-1,fq])<2-3.
Since f is nonelliptic and 〈f,gifgi-1f-1〉=〈f,gifgi-1〉, by Lemma 3, we know that, for all large enough i, 〈f,gifgi-1〉 are elementary. This implies that
(10)fix(f)∩fix(gi)≠∅.
Since G is nonelementary, we can find three loxodromic elements hs(s=1,2,3) in G such that
(11)fix(f)∩fix(hs)=∅,fix(hj)∩fix(hk)=∅,
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,
(12)fix(f)∩fix(hsgihs-1)≠∅,s=1,2,3.
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 eachg is regular elliptic. Suppose that G is nondiscrete. Similarly, by Proposition 2, we can find a sequence {gi} in G such that each gi is regular elliptic and gi→I as i→∞. This implies that, for sufficiently large i, the subgroups 〈f,gifgi-1〉 are elementary. It follows that
(13)fix(f)=fix(gi).
It is a contradiction since f is nonelliptic and gi is regular elliptic.
This completes the proof.Proof of Theorem5.
The proof of Theorem5 follows from a discussion similar to that in the proof of Theorem 4.Proof of Theorem6.
We only prove the case wheng 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 {gi}⊂G such that, for each i, gi is loxodromic and
(14)gi⟶Iasi⟶∞.
Since G is n-dimensional, we can find finitely many loxodromic elements h1,h2,…,ht in G such that the set S={Afix(h1),Afix(h2),…,Afix(ht)} can span the whole complex hyperbolic space ℍℂn, where Afix(h) is the attractive fixed point of h. For each k(k=1,2,…,t), let UAfix(hk) be a small neighbourhood of Afix(hk) in ℍ¯ℂn; then there exists an integer N such that
(15)hkN(fix(f))⊂UAfix(hk).
Since
(16)〈hkNfhk-N,gi〉=hkN〈f,hk-NgihkN〉hk-N,max{N(hk-NgihkN),N([hk-NgihkN,f])}<2-3,
for large enough i, we can see that the subgroups 〈hkNf2hk-N,gi〉 are elementary. By Lemma 7, we know that, for each k, (k=1,2,…,t),
(17)fix(gi)∩UAfix(hk)≠∅.
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{Gr,i} be a sequence of groups of PU(n,1) and Gr be its algebraic limit group. Then we have the following.(1)
If, for eachi, Gr,i is a complex Kleinian group, then Gr is nonelementary and Gr is discrete if and only if each one-generator subgroup of W(Gr) is discrete.(2)
If, for eachi, Gr,i is discrete, then Gr is elementary if and only if for large enough i, all Gr,i are elementary.Proof.
The proof of (1). The nonelementariness ofGr follows from [21, Theorem 1.4]. Now, we come to prove that if Gr is nondiscrete, then there is an element f∈W(Gr) such that the subgroup 〈f〉 is nondiscrete. Suppose that Gr is nondiscrete. Since r<∞ (that is, Gr is finitely generated), by Selberg’s Lemma we know that Gr contains a torsion free subgroup Gr1 with finite index which is nonelementary and nondiscrete either. Then there exists a sequence {fj} in Gr1 such that
(18)fj⟶Iasj⟶∞.
As Gr1 is nonelementary, we can find finitely many loxodromic elements g1, g2,…,gk in Gr1 such that the set {fix(g1), fix(g2),…,fix(gk)} spans ∂M(Gr1), the boundary of M(Gr1). Then, for large enough j, we have
(19)N(fj)+∑q=1n+1‍N([fj,gsq])<2-3,s∈{1,2,…,k}.
Let fi,j and gi,s be the corresponding elements of fj and gs in Gr,i, respectively. Then, for large enough i and j,
(20)N(fi,j)+∑q=1n+1‍N([fi,j,gi,sq])<2-3.
Lemma 3 implies that, for large enough i and j, the subgroups 〈fi,j,gi,s〉 are elementary. Since the loxodromic elements of PU(n,1) form an open set, we know that, for sufficiently large i, gi,s are loxodromic as well. It follows that
(21)fix(gi,s)⊂fix(fi,j),
which shows that, for s∈{1,2,…,k} and all sufficiently large j,
(22)fix(gs)⊂fix(fj).
Thus, for all sufficiently large j,
(23)fj∈W(Gr1).
Since Gr1 is torsion free, we know that there exists an element f∈W(Gr1) such that 〈f〉 is nondiscrete. Note that M(Gr)=M(Gr1), so f∈W(Gr). Hence, the conclusion of (1) follows.
The proof of (2). We only need to prove that if, for large enoughi, all Gr,i are elementary, then is Gr since the converse is trivial by (1). Suppose that Gr is nonelementary. Then we can find two loxodromic elements f and g in Gr such that
(24)fix(f)∩fix(g)=∅.
Let fi and gi be the corresponding elements of f and g in Gr,i, respectively. Then, for large enough i, we have
(25)fix(fi)∩fix(gi)=∅.
It follows a discussion similar to that in the proof of (1) that, for large enough i, both fi and gi are loxodromic. This shows that, for large enough i, all Gr,i are nonelementary. It is a contradiction.Definition 9.
Let{Gi} be a sequence of complex Kleinian groups of PU(n,1). We say that {Gi} satisfies E-condition if there is no sequence {fi},fi∈W(Gi) such that fi→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{Gi} does not satisfy IP-condition but E-condition, then the limit group Gr is still a complex Kleinian groups.Example 10.
Suppose thatH is a purely loxodromic nonelementary subgroup of PU(1,1) and, for each j,
(26)fj=[ei(1/2j)00010001].
Let H~ be the Poincaré extension of H in PU(2,1) and Gj=〈H~,fj〉. Then it is easy to see that the algebraic limit group G of {Gj} is a complex Kleinian group. Note that fj→I as j→∞; we know that {Gj} does not satisfy IP-condition but E-condition.As applications of Theorem8 and E-condition, we have the following.Theorem 11.
LetGr be the algebraic limit group of complex Kleinian groups {Gr,i} of  PU(n,1). If  {Gr,i} satisfies E-condition, then Gr is a complex Kleinian group.Proof.
By Theorem8(1), we know that Gr is nonelementary. Suppose that Gr is nondiscrete. Then there exist an elliptic element f∈W(Gr) and an integer sequence {nj} such that ord(f)=∞ and
(27)fnj⟶Iasnj⟶∞.
For each nj, let finj be the corresponding element of fnj in Gr,i. By [21, Lemma 4.2], we know that finj∈W(Gr,i). It follows from the hypothesis that {Gr,i} satisfies E-condition; we have finj=I for large enough i. This implies that fnj=I. It is a contradiction.
The proof is completed.Whenr≤∞, Wang [17] proved the following.Theorem W.
Letr≤∞. If the generator system {gt,i}t=1r of Gr,i satisfies that none are elliptic and no two have any fixed point in common, and, if all Gr,i are Kleinian groups, then(1)
all the generatorsgt=limi→∞gt,i are neither elliptic nor identity;(2)
ifGr=〈g1,g2,…,gr〉 is nonelementary and W(Gr) is discrete, then Gr is discrete.It easily follows a similar argument as in the proof of Theorem8 and we can obtain the following.Theorem 12.
Letr≤∞. If the generator system {gt,i}t=1r of  Gr,i satisfies that none are elliptic and no two have any fixed point in common, and, if all Gr,i are discrete, then(1)
Gr=〈g1,g2,…,gr〉 is nonelementary;(2)
Gr is discrete if and only if  W(Gr) is discrete.



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