Inequalities for Discrete Möbius Groups in Infinite Dimension

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.

1. Introduction

Let GM(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]) 𝑑𝑥,𝑦=𝑝(𝑥)−𝑝𝑦, (1) where 𝑝(𝑥)=𝑒_𝑛+1+2𝑥−𝑒_𝑛+1𝑥−𝑒_𝑛+1² (2) is the stereographic projection from ℝ^𝑛 to the unit sphere 𝑆^𝑛 in ℝ^𝑛+1. The chordal metric on GM(𝑛) is defined by 𝑑𝑓,𝑔=sup𝑑𝑓(𝑥),𝑔(𝑥)∶𝑥∈ℝ^𝑛. (3) 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 algebra ℓ is the associative algebra over the real field ℝ, generated by a countable family {𝑖_𝑘}^∞_𝑘=1 subject to the following relations: 𝑖_ℎ𝑖_𝑘=−𝑖_𝑘𝑖_ℎ(ℎ≠𝑘),𝑖²_𝑘=−1,∀ℎ,𝑘≥1 (4) and no others. Every element of ℓ can be expressed by the following type: 𝑎=𝑎_𝐼𝐼, (5) where 𝐼=𝑖_𝑣1𝑖_𝑣2⋯𝑖_𝑣𝑝,  1≤𝑣₁<𝑣₂<⋯<𝑣_𝑝,  𝑝≤𝑛,  𝑛 is a fixed natural number depending on 𝑎, 𝑎_𝐼∈ℝ are the coefficients, and ∑_𝐼𝑎²_𝐼<∞. If 𝐼=∅, then 𝑎_𝐼 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 |𝑎|=_𝐼𝑎²_𝐼=|Re(𝑎)|²+|Im(𝑎)|². (6)

The algebra ℓ has three important involutions.(1) “′”: replacing each 𝑖_𝑘  (𝑘≥1) of 𝑎 by −𝑖_𝑘, we get a new element 𝑎^′. 𝑎↦𝑎^′ is an isomorphism of ℓ: (𝑎𝑏)^′=𝑎^′𝑏^′,(𝑎+𝑏)^′=𝑎^′+𝑏^′, (7) for 𝑎,𝑏∈ℓ.(2) “∗”: replacing each 𝑖_𝑣1𝑖_𝑣2⋯𝑖_𝑣𝑝 of 𝑎 by 𝑖_𝑣𝑝𝑖_𝑣𝑝−1⋯𝑖_𝑣1. 𝑎↦𝑎^∗ is an anti-isomorphism of ℓ: (𝑎𝑏)^∗=𝑏^∗𝑎^∗,(𝑎+𝑏)^∗=𝑏^∗+𝑎^∗. (8)(3) “−": 𝑎=(𝑎^∗)^′=(𝑎^′)^∗. It is obvious that 𝑎↦𝑎 is also an anti-isomorphism of ℓ.

For elements of the following type: 𝑥=𝑥₀+𝑥₁𝑖₁+⋯+𝑥_𝑛𝑖_𝑛+⋯∈ℓ, (9) 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 𝑥⋅𝑦=𝑥₀𝑦₀+𝑥₁𝑦₁+⋯+𝑥_𝑛𝑦_𝑛+⋯, (10) where 𝑥=𝑥₀+𝑥₁𝑖₁+⋯+𝑥_𝑛𝑖_𝑛+⋯,  𝑦=𝑦₀+𝑦₁𝑖₁+⋯+𝑦_𝑛𝑖_𝑛+⋯.

It is easy to verify that 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.

If a matrix 𝑓=^𝑎𝑏_𝑐𝑑 satisfies the following:(1)𝑎,𝑏,𝑐,𝑑∈Γ⋃{0};(2)𝑎𝑑^∗−𝑏𝑐^∗=1;(3)𝑎𝑏^∗, 𝑑^∗𝑏, 𝑐𝑑^∗, 𝑐^∗𝑎∈ℓ₂,

then we call 𝑓 a Clifford matrix in infinite dimension; the set of all such matrices is denoted by SL(Γ). The norm of 𝑓 is 𝑓=|𝑎|²+|𝑏|²+|𝑐|²+|𝑑|². (11)

Let 𝑓⁻¹=𝑑^∗−𝑏^∗−𝑐^∗𝑎^∗. (12)

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 any 𝑓=±^𝑎𝑏_𝑐𝑑∈SL(Γ), the corresponding mapping 𝑓(𝑥)=(𝑎𝑥+𝑏)(𝑐𝑥+𝑑)⁻¹ (13) 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 of SL(Γ) as follows. A nontrivial element 𝑓∈SL(Γ) is called: (1)loxodromic if it is conjugate in SL(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 in SL(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 𝑓^𝑚 is strictly parabolic and for any positive integer 𝑚<𝑘, 𝑓^𝑚 is not strictly, then 𝑓 is called 𝑚-strictly parabolic;(3)elliptic if 𝑓 is neither loxodromic nor parabolic.

For a subgroup 𝐺⊂SL(Γ), we say that 𝐺 is discrete if and only if 𝑓₁,𝑓₂,…∈𝐺 and 𝑓_𝑖→𝐼 imply that 𝑓_𝑖=𝐼 for all sufficiently large 𝑖. Otherwise, 𝐺 is nondiscrete.

The following is a necessary condition for a two-generator discrete subgroup of SL(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 𝑓=1𝑡01,𝑔=𝑎𝑏𝑐𝑑, (14) where 𝑐≠0. Then |𝑡𝑐|≥1. (15)

Let 𝑥,𝑦∈ℓ₂, the chordal distance between them is defined as follows (see [10]): 𝑑=2𝑥,𝑦𝑥−𝑦1+|𝑥|²1+𝑦²𝑑=2,𝑥,𝑦≠∞,𝑥,𝑦1+|𝑥|²,𝑦=∞. (16) The chordal metric on SL(Γ) is set to be 𝑑𝑓,𝑔=sup𝑑𝑓(𝑥),𝑔(𝑥)∶𝑥∈ℓ₂. (17) 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 𝑓=𝑟00𝑟⁻¹,𝑔=𝑎𝑏𝑐𝑑, (18) where 𝑟∈ℝ⧵{±1,0}. If 𝑔({0,∞})∩{0,∞}=∅, then 𝑑𝑓²≥4𝑟−𝑟⁻¹²𝑟+𝑟⁻¹²,𝑑𝑔²≥4|𝑏𝑐|(2||),𝑏𝑐+1𝑟−𝑟⁻¹²(|𝑏𝑐|+1)≥1. (19)(20)(21)

Proof . Let 𝑥=𝑟⁻¹. Then 𝑑²𝑓(𝑥)≥4𝑓(𝑥−|𝑥,𝑥)|²1+𝑓(𝑥)²1+|𝑥|²=4𝑟−𝑟⁻¹²𝑟+𝑟⁻¹². (22) This proves (19). For (20), we let 𝑥=0,  𝑦=∞. Since 𝑑𝑔²≥𝑑²𝑔(𝑥)=4,𝑥𝑏𝑑⁻¹1+𝑏𝑑⁻¹²,𝑑𝑔²≥𝑑²𝑔𝑦=4,𝑦1+𝑎𝑐⁻¹², (23) we have 4−𝑑𝑔²𝑑𝑔²≤1𝑏𝑑⁻¹²,4−𝑑𝑔²𝑑𝑔²≤𝑎𝑐⁻¹². (24) So (20) follows. For (21), we consider the Shimizu-Leutbecher sequence 𝑔₀=𝑔,𝑔_𝑛+1=𝑔_𝑛𝑓𝑔_𝑛⁻¹,𝑔_𝑛=𝑎_𝑛𝑏_𝑛𝑐_𝑛𝑑_𝑛,𝑛=0,1,…. (25) Then 𝑎_𝑛+1𝑏_𝑛+1𝑐_𝑛+1𝑑_𝑛+1=𝑟𝑎_𝑛𝑑^∗_𝑛−𝑟⁻¹𝑏_𝑛𝑐^∗_𝑛−𝑟𝑎_𝑛𝑏^∗_𝑛+𝑟⁻¹𝑏_𝑛𝑎^∗_𝑛𝑟𝑐_𝑛𝑑^∗_𝑛−𝑟⁻¹𝑑_𝑛𝑐^∗_𝑛−𝑟𝑐_𝑛𝑑^∗_𝑛+𝑟⁻¹𝑑_𝑛𝑎^∗_𝑛. (26) It follows that 𝑏_𝑛+1𝑐_𝑛+1=𝑎_𝑛𝑏_𝑛𝑐_𝑛𝑑_𝑛𝑟−𝑟⁻¹²≤𝑏_𝑛𝑐_𝑛𝑏_𝑛𝑐_𝑛+1𝑟−𝑟⁻¹². (27) Suppose that 𝜇=(1+|𝑏𝑐|)(𝑟−𝑟⁻¹)²<1. We obtain, by induction, 𝑏_𝑛+1𝑐_𝑛+1≤𝜇^𝑛+1|𝑏𝑐|. (28) So, 𝑏_𝑛𝑐_𝑛→0,𝑎_𝑛𝑑^∗_𝑛→1. (29) The above relation and (26) imply that 𝑎_𝑛→𝑟,𝑑_𝑛→𝑟⁻¹. (30) Now 𝑏_𝑛+1𝑏_𝑛=𝑎_𝑛𝑟−𝑟⁻¹→𝑟𝑟−𝑟⁻¹<𝑟√𝜇. (31) Thus, by induction, 𝑏_𝑛𝑟^𝑛→0,𝑐_𝑛𝑟^𝑛→0. (32) Since ⟨𝑓,𝑔⟩ is discrete and 𝑓^−𝑛𝑔_2𝑛𝑓^𝑛=𝑎_2𝑛𝑟^−2𝑛𝑏_2𝑛𝑟^2𝑛𝑐_2𝑛𝑑_2𝑛, (33) we have 𝑏_2𝑛=𝑐_2𝑛=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 𝑑𝑓𝑔≥√max,𝑑63. (34)

Theorem 4. Suppose that ⟨𝑓,𝑔⟩ is a discrete subgroup of SL(Γ) and 𝑓,𝑔 are represented by 𝑓=𝑎𝑏0𝑎^′,𝑔=𝑢𝑣𝑠𝑡. (35) If 𝑓 is m-strictly parabolic and 𝑔(∞)≠∞, then 𝐾⁶+4𝐾⁴4−𝐾²²≥2𝑚², (36) where 𝐾=max{𝑑(𝑓),𝑑(𝑔)}.

Proof. Since ⟨𝑓,𝑔⟩ is discrete, by [9], we know that |𝑏𝑠|≥1/𝑚. It follows from 𝑑²𝑓(0),0=|𝑏|²4−𝑑²𝑓(0),4|𝑠|,0₂≤𝑔²𝑑²𝑔(∞),𝑔,∞₂≤24+𝑑²𝑔4−𝑑²𝑔, (37) (cf. [11]), that 𝑑²𝑓𝑑²𝑔4+𝑑²𝑔4−𝑑²𝑓4−𝑑²𝑔≥2𝑚². (38) So the inequality follows.

Corollary 5. Suppose that ⟨𝑓,𝑔⟩ is discrete in SL(Γ) and 𝑓,𝑔 are represented by 𝑓=𝑎𝑏0𝑎^′,𝑔=𝑢𝑣𝑠𝑡, (39) where 𝑓 is strictly parabolic. If 𝑔(∞)≠∞, then max𝑑𝑓,𝑑𝑔≥1.22. (40)

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].

Acknowledgment

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