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ISRN Geometry
Volume 2012 (2012), Article ID 757489, 12 pages
http://dx.doi.org/10.5402/2012/757489
Research Article

Algebraic Characterization of Isometries of the Hyperbolic 4-Space

Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, SAS Nagar, Sector 81, P.O. Mohali 140306, India

Received 20 November 2011; Accepted 8 December 2011

Academic Editor: S. Hernández

Copyright © 2012 Krishnendu Gongopadhyay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We classify isometries of the real hyperbolic 4-space by their conjugacy classes of centralizers. We use the representation of the isometries by 2×2 quaternionic matrices to obtain this characterization. Another characterization in terms of conjugacy invariants is also given.

1. Introduction

Let Hn denote the 𝑛-dimensional real hyperbolic space. The isometries of Hn are always assumed to be orientation preserving unless stated otherwise. The isometries of H2 can be identified with the group 𝑃𝑆𝐿(2,). This group acts by the real Möbius transformations or the linear fractional transformations 𝑧(𝑎𝑧+𝑏)/(𝑐𝑧+𝑑) on the hyperbolic plane. Similarly, 𝑃𝑆𝐿(2,) acts on H3 by complex Möbius transformations. Classically the isometries of Hn are classified according to their fixed-point dynamics into three mutually exclusive classes: elliptic, parabolic, and hyperbolic. Recall that an isometry 𝑔 is elliptic if it has a fixed point on Hn; it is parabolic, respectively, hyperbolic if 𝑔 has no fixed point on Hn and exactly one, respectively, two fixed points on the boundary 𝜕Hn. It is well known that for H2 these types are characterized algebraically in terms of their traces, compare with [1]. This classification is of fundamental importance in dynamics, arithmetic, and geometry of H2. Analogous characterization of isometries of H3 in terms of their traces is also well known, compare with [1], also see [2, Appendix-A]. In higher dimensions, the isometries of the 𝑛-dimensional hyperbolic space Hn can be identified with 2×2 matrices over the Clifford numbers, see [3, 4]. Using the Clifford algebraic approach, a characterization of the isometries was obtained by Wada [5]. However, in higher dimensions the above trichotomy of isometries can be refined further, see [2], also see [6, 7]. For a complete understanding of the dynamics and geometry of the isometries, it is desirable to characterize these refined classes. Algebraic characterization of the refined classes using Clifford algebra is a daunting task due to the complicated nature of the Clifford numbers.

Between complex and Clifford numbers, there is an intermediate step involving the quaternions. The isometries of H4 and H5 can be considered as 2×2 matrices over the quaternions , where the respective isometry group acts by the quaternionic Möbius transformations:𝑎𝑏𝑐𝑑𝑧(𝑎𝑧+𝑏)(𝑐𝑧+𝑑)1.(1.1) It is natural to ask for algebraic characterizations of the quaternionic Möbius transformations of H4 and H5 which will generalize the classical trace identities of the real and complex Möbius transformations. Under the above action, the group of invertible quaternionic 2×2 matrices can be identified with the isometries of H5. Using this identification, the author has obtained an algebraic characterization of the isometries of H5, see [8, Theorem-1.1]. The problem of classifying the isometries of H5 has been attempted by many other authors as well, see [7, 9, 10]. In [9, 10], the authors obtained an algebraic characterization of the classical trichotomy of the isometries. Complete characterizations which algebraically classified the refined classes of isometries as well are obtained only in [7, 8]. However, the approaches used in [7, 8] are independent of each other, and hence, the respective characterizations are also of different flavors. Algebraic characterizations of isometries of H4 are also known due to the work of several authors; most notably among them is the work ofCao et al.[6]. Independently of [6], Kido [11] also provided a classification and algebraic characterization of isometries of H4. Kido’s approach is nearly similar to that of Cao et al. However, Kido also provided a clear classification of the fixed points of the quaternionic Möbius transformations of H4. Unfortunately, Kido’s preprint of 2005 was never published until very recently.

In all the above works, the authors obtained their characterizations using conjugacy invariants of the isometries. Another approach which has been used recently to characterize the isometries algebraically is in terms of the centralizers, up to conjugacy. We call two elements 𝑥, 𝑦 in a group 𝐺 to be in the same 𝑧-class if their centralizers are conjugate in 𝐺. Kulkarni has proposed that the notion of 𝑧-class may be used to make precise the intuitive idea of “dynamical types” in any “geometry” whose automorphism group contains a copy of 𝐺, see [12]. Motivated by Kulkarni’s proposal, the 𝑧-classes have been used in the classification problem for isometries in [8, 13]. The characterization by 𝑧-classes is based purely in terms of the internal structure of the group alone, and this does not involve any conjugacy invariant. This approach is indeed useful in certain contexts, for example, see Remark 3.3 in this paper. The problem of classifying the 𝑧-classes in a group is a problem of independent interest as well, for example, see [14]. Using the linear or hyperboloid model, the 𝑧-classes in the full isometry group of Hn have been classified and counted in [2]. It would be interesting to classify the 𝑧-classes using the Clifford algebraic representation of the isometry group.

In this paper, we classify the 𝑧-classes of isometries of H4 using the representation of the isometries by quaternionic matrices. We describe the centralizers up to conjugacy in Section 4. The dynamical types of isometries are precisely classified by the isomorphism types of the centralizers, see Theorem 4.1. This demonstrates the usefulness of the 𝑧-classes in the classification problem of the isometries. Apart from the 𝑧-classes, we obtained another characterization of the isometries in terms of conjugacy invariants. One key idea used in [8] was to consider the quaternions as a subring of the 2×2 complex matrices 𝑀2() and then embed the quaternionic matrices into complex matrices. We use this approach for the isometry group of H4. This approach is different from that of Cao et al. or Kido. Using a geometric and simple approach, the conjugacy classes of isometries of H4 are obtained in Section 3. After the conjugacy classes are known, the characterization by conjugacy invariants is obtained essentially as an appendix to the author’s earlier work [8, Theorem-1.1].

2. Preliminaries

2.1. Classification of Isometries

Before proceeding further, we briefly recall the finer classification of isometries from [2]. The basic idea of the classification is the following.

To each isometry 𝑇 of Hn, one associates an orthogonal transformation 𝑇𝑜 in 𝑆𝑂(𝑛). For each pair of complex conjugate eigenvalues {𝑒𝑖𝜃,𝑒𝑖𝜃}, 0<𝜃𝜋, one associates a rotation angle 𝜃 to 𝑇. An isometry is called 𝑘-rotatory elliptic, respectively, 𝑘-rotatory parabolic, respectively, 𝑘-rotatory hyperbolic if it is elliptic, respectively, parabolic, respectively, hyperbolic and has 𝑘-rotation angles. A 0-rotatory hyperbolic is called a stretch, and a 0-rotatory parabolic is called a translation.

To obtain their characterization, Cao et al. [6] also offered a finer classification of the dynamical types of the isometries. The classification of Cao et al. matches with the above classification when restricted to dimension four. However, the terminologies used by these authors are not the same. For the future reader’s convenience, we set up a dictionary between the terminologies of [2] at dimension four and that of Cao et al. in the following.

(For  𝑛=4) Comparison with the classification of  Cao et al. What Cao et al. [6] called simple elliptics are the 1-rotatory elliptics in this paper. The simple parabolics, respectively, simple hyperbolics are the translations, respectively, stretches in this paper. What we call a 2-rotatory elliptic is the compound elliptic in [6]. The compound parabolics, respectively, compound hyperbolics in [6] are the 1-rotatory parabolics, respectively, 1-rotatory hyperbolics here.

2.2. The Quaternions

The space of all quaternions is the four-dimensional real division algebra with basis {1,𝑖,𝑗,𝑘} and multiplication rules 𝑖2=𝑗2=𝑘2=1, 𝑖𝑗=𝑗𝑖=𝑘,𝑗𝑘=𝑘𝑗=𝑖,𝑘𝑖=𝑖𝑘=𝑗. The multiplicative group of nonzero quaternions is denoted by . For a quaternion 𝑥=𝑥0+𝑥1𝑖+𝑥2𝑗+𝑥3𝑘, we define 𝑥=𝑥0 and 𝑥=𝑥1𝑖+𝑥2𝑗+𝑥3𝑘. The norm of 𝑥 is defined as |𝑥|=𝑥20+𝑥21+𝑥22+𝑥23. The conjugate of 𝑥 is defined by 𝑥=𝑥0𝑥1𝑖𝑥2𝑗𝑥3𝑘.

We choose to be the subspace of spanned by {1,𝑖}. With respect to this choice of , we can write =𝑗; that is, every element 𝑎 in can be uniquely expressed as 𝑎=𝑐0+𝑐1𝑗, where 𝑐0, 𝑐1 are complex numbers. Similarly we can also write =𝑗. For a non-zero quaternion 𝜆, the centralizer of 𝜆 in is 𝑍(𝜆)=+𝜆. If 𝜆, then 𝑍(𝜆)=.

Definition 2.1. Two quaternions 𝑎 and 𝑏 are similar if there exists a non-zero quaternion 𝑣 such that 𝑎=𝑣𝑏𝑣1.

Proposition 2.2. (see [15]). Two quaternions are similar if and only if 𝑎=𝑏 and |𝑎|=|𝑏|.

Corollary 2.3. The similarity class of every quaternion 𝛼 contains a pair of complex conjugates with absolute-value |𝛼| and real part equal to 𝛼.

2.3. The Isometry Group Using Quaternions
2.3.1. The Upper Half-Space Model

First we associate three involutions to the quaternions:(i)𝑞=𝑞𝑜+𝑞𝑖+𝑞2𝑗+𝑞3𝑘𝑞=𝑞𝑜+𝑞𝑖+𝑞2𝑗𝑞3𝑘. It determines an antiautomorphism of : (𝑎𝑏)=𝑏𝑎, (𝑎+𝑏)=𝑎+𝑏,(ii)𝑞=𝑞𝑜+𝑞𝑖+𝑞2𝑗+𝑞3𝑘𝑞=𝑞𝑜𝑞𝑖𝑞2𝑗+𝑞3𝑘. It determines an automorphism of : (𝑎𝑏)=𝑎𝑏, (𝑎+𝑏)=𝑎+𝑏,(iii)the conjugation 𝑞𝑞. This again gives an anti-automorphism of . Note that 𝑎=(𝑎)=(𝑎).

Following Ahlfors [3] and Waterman [4], we identify 3 with the additive subspace of the quaternions spanned by {1,𝑖,𝑗}, that is,3=𝑞𝑞=𝑞0+𝑞1𝑖+𝑞2𝑗.(2.1) We consider the upper half-space model of the hyperbolic space which is given by𝐇4=𝑞=𝑞0+𝑞1𝑖+𝑞2𝑗+𝑞3𝑘𝑞3>0,(2.2) equipped with the metric induced from the differential||||𝑑𝑠=𝑑𝑞𝑞3.(2.3) The boundary of H4 is identified with 𝕊3=3{}.

Let 𝑆𝑙(2,) be the subgroup of 𝐺𝐿(2,) given by𝑆𝑙(2,)=𝑎𝑏𝑐𝑑𝐺𝐿(2,)𝑎𝑏,𝑐𝑑,𝑐𝑎,𝑑𝑏3,𝑎𝑑𝑏𝑐=1.(2.4) The group 𝑆𝑙(2,) acts on H4 by the linear fractional transformations:𝑎𝑏𝑐𝑑𝑞(𝑎𝑞+𝑏)(𝑐𝑞+𝑑)1.(2.5) Then, 𝑃𝑆𝑙(2,)=𝑆𝑙(2,)/{±𝐼} is the group of orientation-preserving isometries of H4.

2.3.2. The Ball Model

The ball model of the hyperbolic space is given byD4={𝑧|𝑧|=1},(2.6) equipped with the hyperbolic metric 𝑑𝑠=2|𝑑𝑞|/(1|𝑞|2). The isometry group in the ball model is given by 𝑈(1,1;) which acts by the linear fractional transformations, and an isometry in 𝑈(1,1;) is of the form, see [6, Lemma 1.1],||𝑑||,||𝑏||𝐴=𝑎𝑏𝑐𝑑;|𝑎|==|𝑐|,|𝑎|2|𝑐|2=1,𝑎𝑏=𝑐𝑑,𝑎𝑐=𝑏𝑑.(2.7)

The diffeomorphism which identifies the disk model D4 to the upper half-space model H4 is given by 𝑓𝑧(𝑧+𝑘)(𝑘𝑧+1)1. The matrix𝑓𝑘=1𝑘𝑘1(2.8) acts as the quaternionic linear fractional transformation 𝑓 on ={}. This implies that 𝑃𝑆𝑙(2,) and 𝑃𝑈(1,1;) are conjugate in 𝑃𝐺𝐿(2,).

3. The Conjugacy Classes

3.1. The Conjugacy Classes

Lemma 3.1. Let 𝐴 be an element in 𝑆𝑙(2,).(i)If 𝐴 acts as a 1-rotatory elliptic, then 𝐴 is conjugate to 𝐷𝜃=𝑒𝑖𝜃00𝑒𝑖𝜃, 0<𝜃<𝜋. If 𝜃=0 or 𝜋, 𝐴 acts as the identity.(ii)If 𝐴 acts as a 1-rotatory parabolic, then 𝐴 is conjugate to 𝑇𝜃,𝑗=𝑒𝑖𝜃𝑗0𝑒𝑖𝜃, or 𝑇𝜃,𝑗, 0<𝜃<𝜋.(iii)If 𝐴 acts as a translation, then 𝐴 is conjugate to 𝑇1=1101 or 𝑇1.(iv)If 𝐴 acts as a 1-rotatory hyperbolic, then 𝐴 is conjugate to either 𝐷𝑟,𝜃=𝑟𝑒𝑖𝜃00𝑟1𝑒𝑖𝜃 or 𝐷𝑟,𝜃, 0<𝜃<𝜋, 𝑟>0.(v)If 𝐴 acts as a stretch, then 𝐴 is conjugate to 𝐷𝑟=𝑟00𝑟1, or 𝐷𝑟.(vi)Finally, if 𝐴 acts as a 2-rotatory elliptic, then 𝐴 has a unique fixed point on H4 and 𝐴 cannot conjugate to an upper triangular matrix in 𝑆𝑙(2,). However, in the ball model of H4, 𝐴 is conjugate to 𝐷𝜃,𝜙=𝑒𝑖𝜃00𝑒𝑖𝜙,𝜃±𝜙.(3.1)

Proof. Let 𝐴=𝑎𝑏𝑐𝑑. The induced linear fractional transformation is given by 𝑓𝐴𝑞(𝑎𝑞+𝑏)(𝑐𝑞+𝑑)1.(3.2) Now there are two cases.Case 1. Suppose 𝑓𝐴 has a fixed point on 𝕊3. This is the case precisely when 𝐴 acts as a parabolic, hyperbolic, or 1-rotatory elliptic isometry. Up to conjugacy, we assume the fixed point to be . So up to conjugacy we can assume 𝑐=0, and hence 𝐴=𝑎𝑏0𝑑, 𝑎𝑑=1. Since every quaternion is conjugate to an element in , let 𝑎=𝑣𝑙𝑒𝑖𝜃𝑣1, where 𝑙𝑒𝑖𝜃 is a nonzero complex number. We can further consider 𝑣 to have unit norm, that is, |𝑣|=1. Using the relation 𝑎𝑑=1, we see that 𝑑=𝑣𝑙1𝑒𝑖𝜃(𝑣)1. Let 𝐶=𝑣00𝑣. Since 𝑣(𝑣)=𝑣𝑣=1, hence 𝐶 is an element in 𝑆𝑙(2,), and 𝐶1𝐴𝐶=𝑙𝑒𝑖𝜃𝑣1𝑏𝑣0𝑙1𝑒𝑖𝜃. Since 𝑒𝑖𝜃=𝑗𝑒𝑖𝜃𝑗, hence, for 𝜋<𝜃<0, conjugating 𝐶1𝐴𝐶 by 𝐽=𝑗00𝑗, we can further take 𝜃 to be in the interval [0,𝜋]. Hence, up to conjugacy, every element in 𝑆𝑙(2,) which has a fixed point on 𝕊3 is of the form 𝑇𝑙,𝜃,𝑞=𝑙𝑒𝑖𝜃𝑞0𝑙1𝑒𝑖𝜃, where 𝑙>0, 0𝜃𝜋, 𝑞. When 𝑞=0, 𝑙=1, we denote it by 𝐷𝜃, and when 𝑙1, 𝜃=0, we denote it by 𝐷𝑙,𝜃.
Now consider an element 𝑇𝑙,𝜃,𝑞 as above, where 𝑞=𝑞0+𝑞1𝑗 and 𝑙=1. Then, 𝑇1,𝜃,𝑞 acts on H4 as 𝑓𝑞𝑧𝑒𝑖𝜃𝑧𝑒𝑖𝜃+𝑞𝑒𝑖𝜃.(3.3) Let 𝑞0=𝑞0𝑒𝑖𝜃, 𝑞1=𝑞1𝑒𝑖𝜃, 𝑦=𝑞0(1𝑒2𝑖𝜃)1. Conjugating 𝑓𝑞 by the map 𝜏𝑦𝑧𝑧𝑦=𝑥=𝑥0+𝑥1𝑗,(3.4) we have 𝜏𝑦𝑓𝑞𝜏1𝑦(𝑥)=𝜏𝑦𝑓(𝑥+𝑦)=𝜏𝑦𝑒𝑖2𝜃𝑥0+𝑦+𝑞0+𝑥1+𝑞1𝑗=𝑒𝑖2𝜃𝑥0+𝑐0+𝑥1+𝑞1𝑗=𝑒𝑖𝜃𝑥𝑒𝑖𝜃+𝑞1𝑗𝑒𝑖𝜃since𝑐0=𝑒𝑖2𝜃𝑦+𝑞0𝑦=0.(3.5) Thus, 𝑇1,𝜃,𝑞 is conjugate to 𝑇1,𝜃,𝑞1𝑗, where 𝑞1. If 𝐴 acts as an elliptic, 𝑞1=0. In this case we denote it by 𝐷𝜃. If 𝑞10, then 𝐴 acts as a 1-rotatory parabolic when 𝜃0,𝜋. Conjugating it further by a transformation of the form 𝑧𝑞11𝑧, we can further assume, 𝑞1=1; that is, 𝐴 is conjugate to 𝑇𝜃,𝑗. If 𝜃=0 or 𝜋, 𝐴 acts as a translation. In this case, we denote it by 𝑇𝑞1, and further conjugating it by 𝑅𝑗𝑧𝑗𝑧 we get 𝑅𝑗𝑇𝑞1𝑅𝑗1𝑧𝑧+1. Thus, up to conjugation, a translation can be taken as 𝑧𝑧+1.
Let 𝑙1 in the expression of 𝑇𝑙,𝜃,𝑞. Then, it acts as 𝑔𝑞𝑧𝑙2𝑒i𝜃𝑧𝑒𝑖𝜃+𝑙𝑞𝑒𝑖𝜃.(3.6) Let 𝑞0=𝑙𝑞0𝑒𝑖𝜃,𝑞1=𝑙𝑞1𝑒𝑖𝜃. Let 𝑤0=𝑞0(1𝑙2𝑒2𝑖𝜃)1,𝑤1=𝑞1(1𝑙2)1, 𝑤=𝑤0+𝑤1𝑗. Conjugating 𝑔𝑞 by the map 𝜏𝑤𝑧𝑧𝑤=𝑥=𝑥0+𝑥1𝑗(3.7) we have 𝜏𝑤𝑔𝑞𝜏1𝑤(𝑥)=𝜏𝑤𝑔𝑞(𝑥+𝑤)=𝜏𝑤𝑙2𝑒𝑖2𝜃𝑥0+𝑤0+𝑞0+𝑙2𝑥1+𝑤1+𝑞1𝑗=𝑙2𝑒𝑖2𝜃𝑥0+𝑐0+𝑙2𝑥1+𝑐1𝑗.(3.8) Thus, 𝑇𝑙,𝜃,𝑞 is conjugate to 𝐷𝑙,𝜃. When 𝜃0,𝜋, it acts as a 1-rotatory hyperbolic. Otherwise, it acts as a stretch.
Case 2. Suppose 𝑓𝐴 has no fixed point on 𝕊3. We claim that the fixed point of 𝑓𝐴 is unique. To see this, if possible suppose that 𝑓𝐴 has at least two fixed points 𝑥 and 𝑦 on H4. Since between two points there is a unique geodesic, 𝑓𝐴 must fixes the end points of the geodesic joining 𝑥 and 𝑦; consequently, 𝑓𝐴 pointwise fixes the geodesic. Thus, the associated orthogonal transformation must have an eigenvalue 1. Since 𝑓𝐴 is orientation preserving, in the hyperboloid model, its representation must have an eigenvalue 1 of multiplicity at least 3, and hence the number of rotation angles can be at most one. Hence, 𝑓𝐴 must have a fixed point on 𝕊3. This is a contradiction. Thus, 𝑓𝐴 must be a 2-rotatory elliptic with a unique fixed point.
Let 𝐴 be a 2-rotatory elliptic. Then, 𝐴 cannot be conjugated to an upper triangular matrix in 𝑆𝑙(2,). In this case, for computational purpose, it is easier to use the ball model of the hyperbolic space. Up to conjugation, we assume that, in the ball model, 𝐴 has the unique fixed point 0. Hence 𝑏=0. This implies 𝑐=0. It follows from [6, Proposition 3.2], that we must have (𝑎)(𝑑); see Section  3.1 of [6]. In particular, 𝑎 is not similar to 𝑑. As in the proof of Lemma 3.1, we may assume by further conjugation that 𝑎,𝑑, |𝑎|=|𝑑|=1, and thus we assume 𝑒𝐴=𝑖𝜃00𝑒𝑖𝜙,𝜃±𝜙.(3.9) This completes the proof.

3.2. Algebraic Characterization

Write =𝑗. Express 𝐴=𝐴1+𝑗𝐴2, where 𝐴1,𝐴2𝑀4(). This gives an embedding 𝐴𝐴 of 𝑆𝑙(2,) into 𝑆𝐿(4,), compare with [8, 15], where𝐴=𝐴1𝐴2𝐴2𝐴1.(3.10)

We will use this embedding to characterize the elements in 𝑆𝑙(2,).

Corollary 3.2. Embed the group 𝑆𝐿(2,) into 𝑆𝐿(4,). Let 𝑓 be an orientation-preserving isometry of H4. Let 𝑓 be induced by 𝐴 in 𝑆𝐿(2,). Let 𝐴 be the corresponding element in 𝑆𝐿(4,). Let the characteristic polynomial of 𝐴 be 𝜒𝐴(𝑥)=𝑥42𝑎3𝑥3+𝑎2𝑥22𝑎1𝑥+1.(3.11) Define 𝑐1=𝑎21,𝑐2=𝑎2,𝑐3=𝑎23.(3.12) Then, one has the following.(i)𝐴 acts as a 2-rotatory elliptic if and only if𝑐1=𝑐3,𝑐2<𝑐1+2.(3.13)(ii)𝐴 acts as a 1-rotatory hyperbolic if and only if𝑐1=𝑐3,𝑐2>𝑐1+2.(3.14)(iii)𝐴 acts as a stretch if and only if𝑐1=𝑐3,𝑐2=𝑐1+2,𝑐1>4.(3.15)(iv)𝐴 acts as a translation if and only if 𝑐1=𝑐3,𝑐2=𝑐1+2,𝑐1=4,(3.16) and 𝐴 is not ±𝐼.(v)𝐴 acts as a 1-rotatory elliptic or a 1-rotatory parabolic if and only if𝑐1=𝑐3,𝑐2=𝑐1+2,𝑐1<4.(3.17)
Moreover, if the characteristic polynomial of 𝐴 is equal to its minimal polynomial, then 𝐴 acts as a 1-rotatory parabolic. Otherwise, it acts as a 1-rotatory elliptic.

Proof. Note that the coefficients 𝑐1, 𝑐2, and 𝑐3 are conjugacy invariants for 𝑆𝑙(2,), in fact, for 𝑃𝑆𝑙(2,). Since 𝑃𝑆𝑙(2,) and 𝑃𝑈(1,1;) are conjugate in 𝑃𝐺𝐿(2,), 𝑐1,𝑐2,and𝑐3 serve as conjugacy invariants in the ball model also. Now the result follows from the above conjugacy classification applying [8, Theorem 1.1].

Remark 3.3. Observe that the conjugacy invariants 𝑐1,𝑐2,and𝑐3 alone cannot distinguish between a 1-rotatory parabolic and a 1-rotatory elliptic with the same “rotation angle” 𝜃; we are required to refer to the respective minimal polynomials to distinguish these classes further. However, without getting into the conjugacy invariants, one may also distinguish them algebraically by their centralizers. As we will see, their centralizers, up to conjugacy, are different and this gives another characterization of the isometries in terms of the 𝑧-classes.

4. The Centralizers, up to Conjugacy

Theorem 4.1. There are seven 𝑧-classes of isometries of H4. The representative for each class and the isomorphism type of the centralizers in each class are given as follows:(i)the trivial class: the identity map,(ii)the 2-rotatory elliptics 𝐷𝜃,𝜙𝑍(𝐷𝜃,𝜙)𝕊1×𝕊1,(iii)the 1-rotatory elliptics 𝐷𝜃𝑍(𝐷𝜃)𝑆𝐿(2,),(iv)the 1-rotatory hyperbolics 𝐷𝑟,𝜃𝑍(𝐷𝑟,𝜃),(v)the stretches 𝐷𝑟𝑍(𝐷𝑟),(vi)the translations 𝑇1𝑍(𝑇1)𝕊13,(vii)the 1-rotatory translations 𝑇𝜃𝑍(𝑇𝜃)𝕊1×.Thus, the isometries are classified by the isomorphism classes of the centralizers.

Proof. First we note that the center of 𝑆𝑙(2,) is given by {𝐼,𝐼} and they form a single 𝑧-class. Now suppose 𝐴±𝐼 is given by 𝐴=𝑎𝑏𝑐𝑑,𝑎𝑏,𝑐𝑑,𝑐𝑎,𝑑𝑏3,𝑎𝑑𝑏𝑐=1.(4.1) Let 𝑎=𝑎1+𝑎2𝑗,𝑏=𝑏1+𝑏2𝑗, and so forth, where for 𝑖=1,2, 𝑎𝑖,𝑏𝑖,𝑐𝑖,𝑑𝑖.(i)Centralizer of 1-Rotatory Elliptics. Note that 𝐴𝐷𝜃=𝐷𝜃𝐴 implies 𝑒𝑖𝜃𝑎=𝑎𝑒𝑖𝜃,𝑒𝑖𝜃𝑏=𝑏𝑒𝑖𝜃,𝑒𝑖𝜃𝑐=𝑐𝑒𝑖𝜃,𝑒𝑖𝜃𝑑=𝑑𝑒𝑖𝜃.(4.2) This implies 𝑎,𝑑𝑍(𝑒𝑖𝜃)=. Further 𝑒𝑖𝜃𝑏1+𝑏2𝑗=𝑏1+𝑏2𝑗𝑒𝑖𝜃𝑒𝑖𝜃𝑏1=𝑏1𝑒𝑖𝜃.(4.3) Since 𝜃0,𝜋, this is possible only if 𝑏1=0. Similarly, 𝑐1=0. Hence,𝑍𝐷𝜃=𝑎𝑏𝑗𝑐𝑑𝑗𝑎,𝑏,𝑐,𝑑,𝑎𝑑𝑏c=𝑗𝑆𝐿(2,).(4.4)(ii)Centralizer of Translations. Let 𝒯 denote the group generated by all translations: 𝑇𝑥=1𝑥01,𝑥3{0},(4.5) then 𝒯3. Up to conjugacy, we consider 𝑇1. It follows from 𝐴𝑇1=𝑇1𝐴 that 𝑎=𝑑,𝑐=0. Hence, 𝐴 is of the form 𝐴=𝑎𝑏0𝑎,𝑎,𝑏,𝑎𝑏3,𝑎𝑎=1.(4.6) Now let 𝑎=𝑎0+𝑎1𝑖+𝑎2𝑗+𝑎3𝑘. Then, 𝑎𝑎=1 implies 𝑎=(1/|𝑎|2)𝑎 which in turn implies that |𝑎|=1 and 𝑎1=0=𝑎2. Thus, 𝑍𝑇1=𝑎𝑏0𝑎𝑎=𝑎0+𝑎1𝑘,𝑎0,𝑎1,|𝑎|=1,𝑏,𝑎𝑏3.(4.7) Further more, we can write =1𝑎𝑏0𝑎𝑎00𝑎𝑎𝑏01,(4.8) this implies𝑍𝑇1=𝕊1𝒯𝕊13.(4.9)(iii)Centralizer of 1-Rotatory Parabolics. Next consider the 1-rotatory parabolic 𝑇𝜃,𝑗. Conjugating it further, we consider the 1-rotatory parabolic 𝑇𝜃,𝑒𝑖𝜃𝑗=𝑒𝑖𝜃𝑒𝑖𝜃𝑗0𝑒𝑖𝜃.(4.10) Note that 𝑇𝜃,𝑒𝑖𝜃𝑗 has the Jordan decomposition 𝑇𝜃,𝑒𝑖𝜃𝑗=𝐷𝜃𝑇𝑗=𝑇𝑗𝐷𝜃, where 𝑇𝑗=1𝑗01. Hence, 𝑍(𝑇𝜃,𝑒𝑖𝜃𝑗)=𝑍(𝐷𝜃)𝑍(𝑇𝑗). We see from the equation 𝐴𝑇𝑗=𝑇𝑗𝐴 that 𝑐=0, 𝑎𝑗=𝑗𝑑. From (i), we have 𝑎,𝑑, 𝑏1=0. Now, combining the relations 𝑎𝑑=1 and 𝑎𝑗=𝑗𝑑 implies 𝑑=𝑎,|𝑎|=1. Hence, we have 𝑍𝑇𝜃,𝑒𝑖𝜃𝑗=0𝑎𝑏𝑗𝑎𝑎,𝑏,|𝑎|=1.(4.11) Let 𝑃=0𝑎𝑏𝑗𝑎. Then, 𝑃=𝐷𝜃𝑇𝑐𝑗 for some 0𝜃2𝜋, where 𝑎=𝑒𝑖𝜃 and 𝑐, in fact 𝑐=𝑎1𝑏𝑗. Since matrices of the form 𝐷𝜃 commute with matrices of the form 𝑇𝑐𝑗,𝑐, hence we have𝑍𝑇𝜃,𝑒𝑖𝜃𝑗𝕊1×.(4.12)(iv)Centralizer of Hyperbolics. Up to conjugacy, we consider 𝐷𝑟,𝜃=𝑟𝑒𝑖𝜃00𝑟1𝑒𝑖𝜃, 0𝜃𝜋. Then, 𝐴𝐷𝑟,𝜃=𝐷𝑟,𝜃𝐴𝑟𝑒𝑖𝜃𝑎=𝑟𝑎𝑒𝑖𝜃,𝑟𝑒𝑖𝜃𝑏=𝑟1𝑏𝑒𝑖𝜃,𝑟1𝑒𝑖𝜃𝑐=𝑟𝑐𝑒𝑖𝜃,𝑟1𝑒𝑖𝜃𝑑=𝑑𝑟1𝑒𝑖𝜃.(4.13) Since 𝑟1, this implies 𝑏=0=𝑐, 𝑎,𝑑𝑍(𝑒𝑖𝜃). The equation 𝑎𝑑=1 implies 𝑑=(𝑎)1. Note that if 𝜃0, 𝑍(𝑒𝑖𝜃)= and if 𝜃=0, then 𝑍(1)=. Thus, if 𝐴 is 1-rotatory hyperbolic, that is, 𝜃0, then 𝑍𝐷𝑟,𝜃=𝑎00𝑎1𝑎.(4.14)If 𝐴 is stretch, then𝑍𝐷𝑟=0𝑎𝑎01𝑎.(4.15)(v)Centralizer of 2-Rotatory Elliptics. We use the ball model. Up to conjugation, we consider 𝐷𝜃,𝜙=𝑒𝑖𝜃00𝑒𝑖𝜙,𝜃±𝜙.(4.16) Let 𝑇=𝑎𝑏𝑐𝑑 commute with 𝐴. The equation 𝐷𝜃,𝜙𝑇=𝑇𝐷𝜃,𝜙 implies (since 𝜃±𝜙), 𝑎,𝑑, 𝑏=0=𝑐. Hence,𝑍𝐷𝜃,𝜙=||𝑑||𝑎00𝑑𝑎,𝑑,|𝑎|==1𝕊1×𝕊1.(4.17) This completes the proof.

Acknowledgment

The author gratefully acknowledges the support of SERC-DST FAST Grant SR/FTP/MS-004/2010.

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