# Inequalities for Discrete Möbius Groups in Infinite Dimension

Authors: Xi Fu
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

We establish some inequalities for discrete Möbius groups in infinite dimension, which are generalizations of the corresponding results of Coa, 1996 and Gehring and Martin, 1991 in finite dimension.


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## Body

## 1. Introduction

LetGM(n) denotes the group of all Möbius transformations in ℝ-n and M(n) the subgroup of GM(n) consisting of all orientation-preserving Möbius transformations. The chordal distance between two points x,y∈ℝ-n is defined by (see [1])
(1)d(x,y)=|p(x)-p(y)|,
where
(2)p(x)=en+1+2(x-en+1)|x-en+1|2
is the stereographic projection from ℝ-n to the unit sphere Sn in ℝ-n+1. The chordal metric on GM(n) is defined by
(3)d(f,g)=sup{d(f(x),g(x)):x∈ℝ-n}.
By using this metric and Clifford algebra, Gehring et al. established a series of inequalities for discrete subgroups of M(n). For example, Gehring and Martin [2] proved that if 〈f,g〉 is a discrete nonelementary subgroup of M(2), then max{d(f,I),d(g,I)}≥2(2-1); when n>2, by using Clifford algebra, Cao generalized it to M(n) and obtained that if 〈f,g〉 is discrete nonelementary subgroup of M(n) and f is hyperbolic, then max{d(f,I),d(g,I)}≥.683 (see [3]); see [4] for the general cases. In 1991, Frunză [5] introduced the concept of Möbius transformations in infinite dimension and discussed the Clifford matrix representations of Möbius transformations in infinite dimension. Recently, by using the representations, Li [6–9] discussed the properties of Möbius transformations in infinite dimension and obtained several analogous Jørgensen’s inequalities in SL(Γ). In this paper, we continue the study in this direction and some new inequalities for discrete subgroups of SL(Γ) are established. In Section 2, we give the preliminaries. The main results are given in Section 3.


## 2. Preliminaries

The Clifford algebraℓ is the associative algebra over the real field ℝ, generated by a countable family {ik}k=1∞ subject to the following relations:
(4)ihik=-ikih(h≠k),ik2=-1,∀h,k≥1
and no others. Every element of ℓ can be expressed by the following type:
(5)a=∑aII,
where I=iv1iv2⋯ivp,  1≤v1<v2<⋯<vp,  p≤n,  n is a fixed natural number depending on a, aI∈ℝ are the coefficients, and ∑I‍aI2<∞. If I=∅, then aI is called the real part of a and denoted by Re(a); the remaining part is called the imaginary part of a and denoted by Im(a). For each a∈ℓ, the Euclidean norm is expressed by
(6)|a|=∑I‍aI2=|Re(a)|2+|Im(a)|2.The algebraℓ has three important involutions.(1)
“′”: replacing each ik  (k≥1) of a by -ik, we get a new element a′. a↦a′ is an isomorphism of ℓ:
(7)(ab)′=a′b′,(a+b)′=a′+b′,fora,b∈ℓ.(2)
“*”: replacing each iv1iv2⋯ivp of a by ivpivp-1⋯iv1. a↦a* is an anti-isomorphism of ℓ:
(8)(ab)*=b*a*,(a+b)*=b*+a*.(3)
“−":a-=(a*)′=(a′)*. It is obvious that a↦a- is also an anti-isomorphism of ℓ.For elements of the following type:(9)x=x0+x1i1+⋯+xnin+⋯∈ℓ,
we call them vectors. The set of all such vectors is denoted by ℓ2 and we let ℓ2¯=ℓ2⋃‍{∞}. For any x∈ℓ2, we have x*=x and x-=x′. For x,y∈ℓ2, the inner product (x·y) of x and y is given by
(10)(x·y)=x0y0+x1y1+⋯+xnyn+⋯,
where x=x0+x1i1+⋯+xnin+⋯,  y=y0+y1i1+⋯+ynin+⋯.It is easy to verify that any nonzero vectorx 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.If a matrixf=(abcd) satisfies the following:(1)
a,b,c,d∈Γ⋃‍{0};(2)
ad*-bc*=1;(3)
ab*, d*b, cd*, c*a∈ℓ2,then we callf a Clifford matrix in infinite dimension; the set of all such matrices is denoted by SL(Γ). The norm of f is
(11)∥f∥=|a|2+|b|2+|c|2+|d|2.Let(12)f-1=(d*-b*-c*a*).Obviously,ff-1=f-1f=I; that is, f-1 is the inverse of f. By a simple computation, we know that SL(Γ) is a multiplicative group of matrices.For anyf=±(abcd)∈SL(Γ), the corresponding mapping
(13)f(x)=(ax+b)(cx+d)-1
is a bijection of ℓ2¯ onto itself, which we call a Möbius transformation in infinite dimension. Correspondingly, the set of all such mappings is also a group, which is still denoted by SL(Γ).Now, we give a classification to elements ofSL(Γ) as follows. A nontrivial element f∈SL(Γ) is called: (1)
loxodromic if it is conjugate inSL(Γ) to (rλ00r-1λ′), where r∈ℝ∖{±1,0},  λ∈Γ, and |λ|=1; if λ=±1, then f is called hyperbolic;(2)
parabolic if it is conjugate inSL(Γ) to (ab0a′), where a,b∈Γ,  |a|=1,  b≠0, and ab=ba′; if a=±1, then f is called strictly parabolic; if there is a positive integer m such that fm is strictly parabolic and for any positive integer m<k, fm is not strictly, then f is called m-strictly parabolic;(3)
elliptic iff is neither loxodromic nor parabolic.For a subgroupG⊂SL(Γ), we say that G is discrete if and only if f1,f2,…∈G and fi→I imply that fi=I for all sufficiently large i. Otherwise, G is nondiscrete.The following is a necessary condition for a two-generator discrete subgroup ofSL(Γ) in which one generator is strictly parabolic.Lemma 1 (see [9]).
Suppose that〈f,g〉 is a discrete subgroup of SL(Γ) and f,g are represented by
(14)f=(1t01),g=(abcd),
where c≠0. Then
(15)|tc|≥1.Letx,y∈ℓ2¯, the chordal distance between them is defined as follows (see [10]):
(16)d(x,y)=2|x-y|(1+|x|2)(1+|y|2),x,y≠∞,d(x,y)=2(1+|x|2),y=∞.
The chordal metric on SL(Γ) is set to be
(17)d(f,g)=sup{d(f(x),g(x)):x∈ℓ2¯}.
We call d(f)=d(f,I) the chordal norm of f. It is easy to see that 0≤d(f)≤2. The readers can refer to [5, 6, 8–10] for more details about Möbius groups in infinite dimension.


## 3. The Main Results and Proofs

Now we come to state and prove our main results.Theorem 2.
Suppose that〈f,g〉 is a discrete subgroup of SL(Γ) and f,g are represented by
(18)f=(r00r-1),g=(abcd),
where r∈ℝ∖{±1,0}. If g({0,∞})∩{0,∞}=∅, then
(19)d(f)2≥4(r-r-1)2(r+r-1)2,(20)d(g)2≥4|bc|(2|bc|+1),(21)(r-r-1)2(|bc|+1)≥1.Proof .
Letx=r-1. Then
(22)d2(f(x),x)≥4(|f(x)|-|x|)2(1+|f(x)|2)(1+|x|2)=4(r-r-1)2(r+r-1)2.
This proves (19). For (20), we let x=0,  y=∞. Since
(23)d(g)2≥d2(g(x),x)=4|bd-1|1+|bd-1|2,d(g)2≥d2(g(y),y)=41+|ac-1|2,
we have
(24)4-d(g)2d(g)2≤1|bd-1|2,4-d(g)2d(g)2≤|ac-1|2.
So (20) follows. For (21), we consider the Shimizu-Leutbecher sequence
(25)g0=g,gn+1=gnfgn-1,gn=(anbncndn),n=0,1,….
Then
(26)(an+1bn+1cn+1dn+1)=(randn*-r-1bncn*-ranbn*+r-1bnan*rcndn*-r-1dncn*-rcndn*+r-1dnan*).
It follows that
(27)|bn+1cn+1|=|anbncndn|(r-r-1)2≤|bncn|(|bncn|+1)(r-r-1)2.
Suppose that μ=(1+|bc|)(r-r-1)2<1. We obtain, by induction,
(28)|bn+1cn+1|≤μn+1|bc|.
So,
(29)bncn→0,andn*→1.
The above relation and (26) imply that
(30)an→r,dn→r-1.
Now
(31)|bn+1||bn|=|an||r-r-1|→|r(r-r-1)|<rμ.
Thus, by induction,
(32)bnrn→0,cnrn→0.
Since 〈f,g〉 is discrete and
(33)f-ng2nfn=(a2nr-2nb2nr2nc2nd2n),
we have b2n=c2n=0 for sufficiently large n. Hence g({0,∞})∩{0,∞}≠∅. It is a contradiction. Therefore, (r-r-1)2(|bc|+1)≥1.By a simple caculation, we have the following.Corollary 3.
Suppose that〈f,g〉 is a discrete subgroup of SL(2,Γ). If f is hyperbolic with fix(f)={0,∞} and g({0,∞})∩{0,∞}=∅, then
(34)max{d(f),d(g)}≥63.Theorem 4.
Suppose that〈f,g〉 is a discrete subgroup of SL(Γ) and f,g are represented by
(35)f=(ab0a′),g=(uvst).
If f is m-strictly parabolic and g(∞)≠∞, then
(36)(K6+4K4)(4-K2)2≥2m2,
where K=max{d(f),d(g)}.Proof.
Since〈f,g〉 is discrete, by [9], we know that |bs|≥1/m. It follows from
(37)d2(f(0),0)=|b|2(4-d2(f(0),0)),4|s|2≤∥g∥2d2(g(∞),∞),∥g∥2≤24+d2(g)4-d2(g),
(cf. [11]), that
(38)d2(f)d2(g)(4+d2(g))(4-d2(f))(4-d2(g))≥2m2.
So the inequality follows.Corollary 5.
Suppose that〈f,g〉 is discrete in SL(Γ) and f,g are represented by
(39)f=(ab0a′),g=(uvst),
where f is strictly parabolic. If g(∞)≠∞, then
(40)max{d(f),d(g)}≥1.22.Remark 6.
Following [6–8], we know that there are much differences between subgroups of M(n) and SL(Γ) such as the discreteness and the fixed points sets of elements, which play key roles in the proofs of [3, Theorem 3.11] and [2, Theorem 6.14]. This implies that our results cannot be deduced directly by the same methods used in [2, 3].



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