New York Journal of Mathematics
New York J. Math.17(2011) 41–49.
A generalization of Jørgensen’s inequality
to infinite dimension
Liulan Li
Abstract. In this paper, we give a generalization of Jørgensen’s in-equality to hyperbolic M¨obius transformations in infinite dimension by using Clifford algebras. We also give an application.
Contents
1. Introduction 41
2. Preliminaries 42
3. The main result and its proof 45
4. An application 47
References 48
1. Introduction
In the theory of discrete groups, the following important and useful in-equality is well known as Jørgensen’s inin-equality, see [5].
Theorem J. Suppose that f, g ∈ M(R2) generate a discrete and
nonele-mentary group hf, gi. Then
|tr2(f)−4|+|tr([f, g])−2| ≥1.
In [4], Hersonsky gave a partial generalization of Theorem J to M¨obius transformations in Rn by using Clifford algebra, which is stated in the fol-lowing form.
Theorem H. Let f, g ∈M(Rn) such that f and [f, g] are hyperbolic, and
suppose that hf, gi is a discrete and nonelementary group. Then
|tr2
(f)−4|+|tr([f, g])−2| ≥1.
Received September 16, 2010.
2000Mathematics Subject Classification. Primary: 30F40; Secondary: 20H10.
Key words and phrases. Jørgensen’s inequality, M¨obius transformation in infinite di-mension, Clifford algebra.
The research was supported by the Science and Technology Development Program of Hengyang (No. 2010KJ22) and NSF of Hunan (No. 10JJ4005).
ISSN 1076-9803/2011
In [12], Waterman generalized Jørgensen’s inequality to high dimensional groups and obtained
Theorem WA. Letf,g∈M(Rn). Ifhf, gi is discrete and nonelementary, then
kf −Ik · kg−Ik ≥ 1 32.
In [11], Wang also studied the generalization of Jørgensen’s inequality to hyperbolic M¨obius transformations in high dimension, giving the following generalization of Theorem H.
Theorem W. Let f, g ∈ M(Rn) such that f is hyperbolic and [f, g] is
vectorial, and suppose that hf, gi is a discrete and nonelementary group. Then
|tr2(f)−4|+|tr([f, g])−2| ≥1.
We refer to [6,9,10,11,12,13] for related investigations in this direction. The main aim of this paper is to establish Jørgensen’s inequality in the infinite dimensional case. Our main result is Theorem3.1, which is a gener-alization of Theorems H and W and a partial genergener-alization of Theorem J to infinite dimension. We will state and prove it in Section 3. In Section 4
we will give an application of Theorem 3.1.
2. Preliminaries
The Clifford algebra ℓ is the associative algebra over the real field R, generated by a countable family{ik}∞k=1 subject to the following relations:
ihik=−ikih (h=6 k), i2k=−1, ∀h, k≥1
and no others. Every element ofℓ can be expressed of the following type
a=XaII,
where I =iv1iv2. . . ivp,1≤v1 < v2 <· · ·< vp,p ≤n, n is a fixed natural
number depending ona,aI ∈Rare the coefficients and
P
Ia
2
I<∞. IfI =∅,
thenaI is called the real part ofaand denoted by Re(a); the remaining part
is called the imaginary part ofaand denoted by Im(a). Inℓ, the Euclidean norm is expressed by
|a|=
s X
I
a2
I=
p
|Re(a)|2+|Im(a)|2.
The algebraℓhas three important involutions: (1) “′”: replacing eachi
k (k≥1) of aby −ik, we get a new number a′.
a7→a′ is an isomorphism ofℓ:
(ab)′ =a′b′, (a+b)′ =a′+b′,
(2) “*”: replacing eachiv1iv2. . . ivpofabyivpivp−1. . . iv1. We know that
a7→a∗ is an anti-isomorphism ofℓ:
(ab)∗ =b∗a∗, (a+b)∗ =b∗+a∗.
(3) “¯”: ¯a = (a∗)′ = (a′)∗. It is obvious that a 7→ ¯a is also an anti-isomorphism ofℓ.
We refer to elements of the following type as vectors:
x=x0+x1i1+· · ·+xnin+· · · ∈ℓ.
The set of all such vectors is denoted by ℓ2 and we let ℓ2 = ℓ2S{∞}. For any x ∈ ℓ2, we have x∗ = x and ¯x = x′. For x, y ∈ ℓ2, the inner product (x·y) of xand y is given by
(x·y) =x0y0+x1y1+· · ·+xnyn+. . . ,
wherex=x0+x1i1+· · ·+xnin+. . . , y=y0+y1i1+· · ·+ynin+. . . .
Obviously, any nonzero vector x is invertible in ℓ with x−1 = x¯
|x|2. The
inverse of a vector is invertible too. Since any product of nonzero vectors is invertible, we conclude that any product of nonzero vectors is invertible in ℓ. The set of products of finitely many nonzero vectors is a multiplicative group, called Clifford group and denoted by Γ.
Definition 2.1. If a matrixg=
a b c d
satisfies:
(1) a, b, c, d∈ΓS
{0}, (2) △(g) =ad∗−bc∗ = 1, (3) ab∗, d∗b, cd∗, c∗a∈ℓ2,
then we call g a Clifford matrix in infinite dimension; the set of all such matrices is denoted by SL(Γ).
Let
I =
1 0 0 1
, g−1=
d∗ −b∗
−c∗ a∗
.
Obviously, gg−1
= g−1g
= I, that is, g−1
is the inverse of g. By a simple computation, we know that SL(Γ) is a multiplicative group of matrices.
For anyg=±
a b c d
∈SL(Γ), the corresponding mapping
g(x) = (ax+b)(cx+d)−1
is a bijection of ℓ2 onto itself, which we call a M¨obius transformation in infinite dimension. Correspondingly, the set of all such mappings is also a group, which is still denoted by SL(Γ).
• f isloxodromic if it is conjugate in SL(Γ) to
• fisparabolicif it is conjugate in SL(Γ) to
• Otherwise we sayf iselliptic.
Definition 2.2. Forg=
For a nontrivial element g =
For the trace, we have the following result (see [8]).
Lemma 2.3. Letg=
a b c d
∈SL(Γ).ThenRe(tr(g))is invariant under conjugation.
The following two lemmas come from [8].
Lemma 2.4. g =
Definition 2.6. For a subgroup G⊂SL(Γ), we callGelementary if Ghas
a finiteG-orbit, that is, there exists a point x∈ℓ2 such that
G(x) ={g(x)|g∈G}
is finite; otherwise, we callG nonelementary.
We say thatGis discrete ifg, f1, f2,· · · ∈Gandfi →gimply fi =gfor
all sufficiently large i. Otherwise, Gis not discrete.
Lemma 2.7. Let f ∈ SL(Γ) be not elliptic, and let θ : SL(Γ)→ SL(Γ) be
defined by
Suppose that there exists n such that θn(g) =f, then the group hf, gi gen-erated by f and g is elementary.
Proof. Define g0 =g andgn=θn(g). So for somem≥0,
gm+1 =gmf g−m1.
Suppose first that f is parabolic. Since f has exactly one fixed point, we may assume that f(∞) = ∞. As g1, . . . , gn are conjugate to f, they
are each parabolic and so have a unique fixed point. Thus if gr+1 fixes ∞, then so doesgr, where r ≥0. As gn(=f) fixes∞, we deduce that each gj
(j = 0,1, . . . , n) fixes∞. This shows thathf, gi is elementary.
Suppose now thatf is loxodromic and the two fixed points off arexand y. Clearly,g1, . . . , gn each have exactly two fixed points. Now suppose that
gr+1 fixes x and y (as doesgn): then
{x, y}={gr(x), gr(y)}.
Since gr cannot interchangex and y forr≥1, we know that ifgr+1 fixes x and y, then so does gr for r ≥1. It follows that g1, . . . , gn each fix x and
y. This shows that f and g leave the set {x, y} invariant and so hf, gi is
elementary.
3. The main result and its proof
Now we come to state and prove our main result.
Theorem 3.1. Let f, g∈SL(Γ) such that f is hyperbolic and [f, g] is
vec-torial, and suppose that hf, gi is discrete and nonelementary, then
(3.1) |tr2
(f)−4|+|tr([f, g])−2| ≥1.
Proof. By Lemmas 2.4, 2.5 and 2.3, we know that tr(f) ∈ R, and tr(f)
and tr([f, g]) are invariant under conjugation. Without loss of generality, we may assume that
f =
τ 0 0 τ−1
, g=
a b c d
,
where τ > 0 and τ 6= 1. Let κ denote the left side of relation (3.1) and suppose that (3.1) fails. Then
(3.2) κ= (τ −τ−1)2
(1 +|bc|)<1.
We let
g0 =g, gm+1 =gmf gm−1, gm=
am bm
cm dm
Then, we have
−1. It is obvious that f(x) is an increasing function on [0,+∞) such that f(x) ≤x on [0, r]. It follows from (3.2) that |bc|< r. The above facts and relations (3.3) show that
|bm+1cm+1∗| ≤f(|bmcm∗|)≤ · · · ≤fm+1(|bc∗|)≤ |bc∗|,
The above relation and (3.3) imply that
lim
So for sufficiently large m, we have
In a very similar way, we get that
lim
hf, gimust be elementary, which violates the assumption. The contradiction
Remark 3.2. Theorem3.1is a generalization of TheoremB in [4] and the
corresponding result in [11].
4. An application
∈SL(Γ)(U is called unitary),
g=
Lemma 4.2. Let f ∈SL(Γ) be hyperbolic. Then
kf −Ik2 ≥ 1 2|tr
2
(f)−4|.
Proof. Sincekf−Ikand tr2
(f) are invariant under conjugation by unitary transformations by Lemmas2.3,2.4,2.5and 4.1, without loss of generality, we may assume that
whereu >1. By a simple computation, the conclusion follows.
Lemma 4.3. Let f, g ∈ SL(Γ) be hyperbolic such that [f, g] is vectorial. Then
kf−Ik2· kg−Ik2 ≥ |tr([f, g])−2|.
Proof. Since the two sides of the above inequality are invariant under
con-jugation by unitary transformations, we may assume that
f =
whereu >1. By computation, we see that
[f, g] =
Theorem 4.4. Let f, g∈SL(Γ) be hyperbolic such that[f, g] and [g, f]are
vectorial. If hf, gi is discrete and nonelementary, then
kf −Ik · kg−Ik ≥√2−1.
Proof. Let x= min{|tr2(f)
−4|, |tr2(g)
−4|}.
We first suppose thatx≤2√2−2. By assumptions and Theorem 3.1,
|tr2
(f)−4|+|tr([f, g])−2| ≥1, |tr2
(g)−4|+|tr([g, f])−2| ≥1.
Therefore, by Lemma 4.3, we have that
kf−Ik2· kg−Ik2≥ |tr([f, g])−2| ≥1− |tr2(f)−4|, and
kg−Ik2· kf −Ik2 ≥ |tr([g, f])−2| ≥1− |tr2
(g)−4|. Thus,
kf−Ik2· kg−Ik2 ≥1−(2√2−2) = (√2−1)2 . Now we suppose thatx≥2√2−2. By Lemma 4.2, we have
kf−Ik2 ≥ 1 2|tr
2
(f)−4|, kg−Ik2 ≥ 1 2|tr
2
(g)−4|.
We hence know that
kf −Ik2· kg−Ik2 ≥ 1 4|tr
2
(f)−4||tr2
(g)−4| ≥(√2−1)2
.
References
[1] Ahlfors, Lars V.On the fixed points of M¨obius transformations inRn.Ann. Acad.
Sci. Fenn. Ser. A I Math.10(1985), 15–27.MR0802464(87c:20086),Zbl 0586.30045.
[2] Beardon, Alan F. The geometry of discrete groups. Graduate Texts in Math-ematics, 91. Springer-Verlag, New York, 1983. xii+337 pp. ISBN: 0-387-90788-2.
MR0698777(85d:22026),Zbl 0528.30001.
[3] Frunz˘a, Monica. M¨obius transformations in infinite dimension. Analyse com-plexe (Bucharest, 1989). Rev. Roumaine Math. Pures. Appl. 36 (1991), 369–376. MR1144568(93a:58016),Zbl 0753.30035.
[4] Hersonsky, Sa’ar. A generalization of the Shimizu–Leutbecher and Jørgensen in-equalities to M¨obius transformations in RN. Proc. Amer. Math. Soc. 121 (1994),
209–215.MR1182701(94m:30085),Zbl 0812.30017.
[5] Jørgensen, Troels.On discrete groups of M¨obius transformations.Amer. J. Math. 98(1976), 739–749.MR0427627(55 #658),Zbl 0336.30007.
[6] Leutbecher, Armin. Uber Spitzen diskontinuierlicher Gruppen von lineargebroch-¨ enen Transformationen. Math. Z. 100 (1967), 183–200. MR0214763 (35 #5612), Zbl 0157.13502.
[7] Li, Liulan.A class of M¨obius transformations in infinite dimension. (Chinese. English summary.)J. Nat. Sci. Jiangxi Norm. Univ.33(2009), 556–559.Zbl pre05812476.
[8] Li, Liulan; Wang, Xiantao. On M¨obius transformations in infinite dimension. (Chinese. English summary.) J. Nat. Sci. Heilongjiang Univ. 22 (2005), 497–500. MR2175830,Zbl 1087.30041.
[9] Ohtake, Hiromi.On the discontinous subgroups with parabolic transformations of the M¨obius groups.J. Math. Kyoto Uni.25(1985), 807–816.Zbl 0599.30071.
[11] Wang, Xiantao. Generalizations of Shimizu–Leutbecher inequality and Jørgensen inequality to high dimensional M¨obius groups.Chinese Sci. Bull. 22 (1996), 2109–
2110.
[12] Waterman, P. L. M¨obius transformations in several dimensions. Adv. Math. 101
(1993), 87–113.MR1239454(95h:30056),Zbl 0793.15019.
[13] Wielenberg, Norbert J. Discrete Moebius groups: Fundamental polyhedra and convergence. Amer. J. Math. 99 (1977), 861–877. MR0477035 (57 #16579), Zbl 0373.57024.
Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan 421008, People’s Republic of China
