# Algebraic Characterization of Isometries of the Hyperbolic 4-Space

Authors: Krishnendu Gongopadhyay
Journal: ISRN Geometry
 (2012)

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## Abstract

We classify isometries of the real hyperbolic 4-space by their conjugacy
classes of centralizers. We use the representation of the isometries by2×2 quaternionic matrices to obtain this characterization. Another characterization in terms
of conjugacy invariants is also given.


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## Body

## 1. Introduction

LetHn 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,ℝ). 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,ℂ) 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 ofH4 and H5 can be considered as 2×2 matrices over the quaternions ℍ, where the respective isometry group acts by the quaternionic Möbius transformations:(1.1)(abcd):z⟼(az+b)(cz+d)-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 elementsx, 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 thez-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(ℂ) 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 isometryT of Hn, one associates an orthogonal transformation To in SO(n). For each pair of complex conjugate eigenvalues {eiθ,e-iθ}, 0<θ≤π, one associates a rotation angleθ 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.(Forn=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 quaternionsℍ 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 ℍ*. For a quaternion x=x0+x1i+x2j+x3k, we define ℜx=x0 and ℑx=x1i+x2j+x3k. The norm of x is defined as |x|=x02+x12+x22+x32. The conjugate of x is defined by x¯=x0-x1i-x2j-x3k.We chooseℂ to be the subspace of ℍ spanned by {1,i}. With respect to this choice of ℂ, we can write ℍ=ℂ⊕ℂj; that is, every element a in ℍ can be uniquely expressed as a=c0+c1j, where c0, c1 are complex numbers. Similarly we can also write ℍ=ℂ⊕jℂ. For a non-zero quaternion λ, the centralizer of λ in ℍ is Z(λ)=ℝ+ℝλ. If λ∈ℂ, then Z(λ)=ℂ.Definition 2.1.
Two quaternionsa and b are similar if there exists a non-zero quaternion v such that a=vbv-1.Proposition 2.2.
(see [15]). Two quaternions are similar if and only if ℜa=ℜb and |a|=|b|.Corollary 2.3.
The similarity class of every quaternionα contains a pair of complex conjugates with absolute-value |α| and real part equal to ℜα.


### 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+q3k↦q*=qo+qi+q2j-q3k. It determines an antiautomorphism of ℍ: (ab)*=b*a*, (a+b)*=a*+b*,(ii)
′:q=qo+qi+q2j+q3k↦q′=qo-qi-q2j+q3k. It determines an automorphism of ℍ: (ab)′=a′b′, (a+b)′=a′+b′,(iii)
the conjugationq↦q¯. This again gives an anti-automorphism of ℍ. Note that a¯=(a′)*=(a*)′.Following Ahlfors [3] and Waterman [4], we identify ℝ3 with the additive subspace of the quaternions spanned by {1,i,j}, that is,(2.1)R3={q∈H∣q=q0+q1i+q2j}.
We consider the upper half-space model of the hyperbolic space which is given by(2.2)H4={q=q0+q1i+q2j+q3k∈H∣q3>0},
equipped with the metric induced from the differential(2.3)ds=|dq|q3.
The boundary of H4 is identified with 𝕊3=ℝ3∪{∞}.LetSl(2,ℍ) be the subgroup of GL(2,ℍ) given by(2.4)Sl(2,H)={(abcd)∈GL(2,H)∣ab*,cd*,c*a,d*b∈R3,ad*-bc*=1}.
The group Sl(2,ℍ) acts on H4 by the linear fractional transformations:(2.5)(abcd):q⟼(aq+b)(cq+d)-1.
Then, PSl(2,ℍ)=Sl(2,ℍ)/{±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 by(2.6)D4={z∈H∣|z|=1},
equipped with the hyperbolic metric ds=2|dq|/(1-|q|2). The isometry group in the ball model is given by U(1,1;ℍ) which acts by the linear fractional transformations, and an isometry in U(1,1;ℍ) is of the form, see [6, Lemma 1.1],(2.7)A=(abcd);|a|=|d|,|b|=|c|,|a|2-|c|2=1,a¯b=c¯d,ac¯=bd¯.The diffeomorphism which identifies the disk modelD4 to the upper half-space model H4 is given by f:z↦(z+k)(kz+1)-1. The matrix(2.8)fk=(1kk1)
acts as the quaternionic linear fractional transformation f on ℍ̂=ℍ∪{∞}. This implies that PSl(2,ℍ) and PU(1,1;ℍ) are conjugate in PGL(2,ℍ).


## 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 isometryT of Hn, one associates an orthogonal transformation To in SO(n). For each pair of complex conjugate eigenvalues {eiθ,e-iθ}, 0<θ≤π, one associates a rotation angleθ 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.(Forn=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 quaternionsℍ 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 ℍ*. For a quaternion x=x0+x1i+x2j+x3k, we define ℜx=x0 and ℑx=x1i+x2j+x3k. The norm of x is defined as |x|=x02+x12+x22+x32. The conjugate of x is defined by x¯=x0-x1i-x2j-x3k.We chooseℂ to be the subspace of ℍ spanned by {1,i}. With respect to this choice of ℂ, we can write ℍ=ℂ⊕ℂj; that is, every element a in ℍ can be uniquely expressed as a=c0+c1j, where c0, c1 are complex numbers. Similarly we can also write ℍ=ℂ⊕jℂ. For a non-zero quaternion λ, the centralizer of λ in ℍ is Z(λ)=ℝ+ℝλ. If λ∈ℂ, then Z(λ)=ℂ.Definition 2.1.
Two quaternionsa and b are similar if there exists a non-zero quaternion v such that a=vbv-1.Proposition 2.2.
(see [15]). Two quaternions are similar if and only if ℜa=ℜb and |a|=|b|.Corollary 2.3.
The similarity class of every quaternionα contains a pair of complex conjugates with absolute-value |α| and real part equal to ℜα.


## 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+q3k↦q*=qo+qi+q2j-q3k. It determines an antiautomorphism of ℍ: (ab)*=b*a*, (a+b)*=a*+b*,(ii)
′:q=qo+qi+q2j+q3k↦q′=qo-qi-q2j+q3k. It determines an automorphism of ℍ: (ab)′=a′b′, (a+b)′=a′+b′,(iii)
the conjugationq↦q¯. This again gives an anti-automorphism of ℍ. Note that a¯=(a′)*=(a*)′.Following Ahlfors [3] and Waterman [4], we identify ℝ3 with the additive subspace of the quaternions spanned by {1,i,j}, that is,(2.1)R3={q∈H∣q=q0+q1i+q2j}.
We consider the upper half-space model of the hyperbolic space which is given by(2.2)H4={q=q0+q1i+q2j+q3k∈H∣q3>0},
equipped with the metric induced from the differential(2.3)ds=|dq|q3.
The boundary of H4 is identified with 𝕊3=ℝ3∪{∞}.LetSl(2,ℍ) be the subgroup of GL(2,ℍ) given by(2.4)Sl(2,H)={(abcd)∈GL(2,H)∣ab*,cd*,c*a,d*b∈R3,ad*-bc*=1}.
The group Sl(2,ℍ) acts on H4 by the linear fractional transformations:(2.5)(abcd):q⟼(aq+b)(cq+d)-1.
Then, PSl(2,ℍ)=Sl(2,ℍ)/{±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 by(2.6)D4={z∈H∣|z|=1},
equipped with the hyperbolic metric ds=2|dq|/(1-|q|2). The isometry group in the ball model is given by U(1,1;ℍ) which acts by the linear fractional transformations, and an isometry in U(1,1;ℍ) is of the form, see [6, Lemma 1.1],(2.7)A=(abcd);|a|=|d|,|b|=|c|,|a|2-|c|2=1,a¯b=c¯d,ac¯=bd¯.The diffeomorphism which identifies the disk modelD4 to the upper half-space model H4 is given by f:z↦(z+k)(kz+1)-1. The matrix(2.8)fk=(1kk1)
acts as the quaternionic linear fractional transformation f on ℍ̂=ℍ∪{∞}. This implies that PSl(2,ℍ) and PU(1,1;ℍ) are conjugate in PGL(2,ℍ).


## 2.3.1. The Upper Half-Space Model

First we associate three involutions to the quaternions:(i)
*:q=qo+qi+q2j+q3k↦q*=qo+qi+q2j-q3k. It determines an antiautomorphism of ℍ: (ab)*=b*a*, (a+b)*=a*+b*,(ii)
′:q=qo+qi+q2j+q3k↦q′=qo-qi-q2j+q3k. It determines an automorphism of ℍ: (ab)′=a′b′, (a+b)′=a′+b′,(iii)
the conjugationq↦q¯. This again gives an anti-automorphism of ℍ. Note that a¯=(a′)*=(a*)′.Following Ahlfors [3] and Waterman [4], we identify ℝ3 with the additive subspace of the quaternions spanned by {1,i,j}, that is,(2.1)R3={q∈H∣q=q0+q1i+q2j}.
We consider the upper half-space model of the hyperbolic space which is given by(2.2)H4={q=q0+q1i+q2j+q3k∈H∣q3>0},
equipped with the metric induced from the differential(2.3)ds=|dq|q3.
The boundary of H4 is identified with 𝕊3=ℝ3∪{∞}.LetSl(2,ℍ) be the subgroup of GL(2,ℍ) given by(2.4)Sl(2,H)={(abcd)∈GL(2,H)∣ab*,cd*,c*a,d*b∈R3,ad*-bc*=1}.
The group Sl(2,ℍ) acts on H4 by the linear fractional transformations:(2.5)(abcd):q⟼(aq+b)(cq+d)-1.
Then, PSl(2,ℍ)=Sl(2,ℍ)/{±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 by(2.6)D4={z∈H∣|z|=1},
equipped with the hyperbolic metric ds=2|dq|/(1-|q|2). The isometry group in the ball model is given by U(1,1;ℍ) which acts by the linear fractional transformations, and an isometry in U(1,1;ℍ) is of the form, see [6, Lemma 1.1],(2.7)A=(abcd);|a|=|d|,|b|=|c|,|a|2-|c|2=1,a¯b=c¯d,ac¯=bd¯.The diffeomorphism which identifies the disk modelD4 to the upper half-space model H4 is given by f:z↦(z+k)(kz+1)-1. The matrix(2.8)fk=(1kk1)
acts as the quaternionic linear fractional transformation f on ℍ̂=ℍ∪{∞}. This implies that PSl(2,ℍ) and PU(1,1;ℍ) are conjugate in PGL(2,ℍ).


## 3. The Conjugacy Classes

### 3.1. The Conjugacy Classes

Lemma 3.1.
LetA be an element in Sl(2,ℍ).(i)
IfA acts as a 1-rotatory elliptic, then A is conjugate to Dθ=(eiθ00e-iθ), 0<θ<π. If θ=0 or π, A acts as the identity.(ii)
IfA acts as a 1-rotatory parabolic, then A is conjugate to Tθ,j=(eiθj0e-iθ), or -Tθ,j, 0<θ<π.(iii)
IfA acts as a translation, then A is conjugate to T1=(1101) or -T1.(iv)
IfA acts as a 1-rotatory hyperbolic, then A is conjugate to either Dr,θ=(reiθ00r-1e-iθ) or D-r,θ, 0<θ<π, r>0.(v)
IfA acts as a stretch, then A is conjugate to Dr=(r00r-1), or D-r.(vi)
Finally, ifA 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,ℍ). However, in the ball model of H4, A is conjugate to
(3.1)Dθ,ϕ=(eiθ00eiϕ),θ≠±ϕ.Proof.
LetA=(abcd). The induced linear fractional transformation is given by
(3.2)fA:q⟼(aq+b)(cq+d)-1.
Now there are two cases.Case 1.
SupposefA 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 ℂ, let a=vleiθv-1, 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=v′l-1e-iθ(v′)-1. Let C=(v00v′). Since v(v′)*=vv¯=1, hence C is an element in Sl(2,ℍ), and C-1AC=(leiθv-1bv′0l-1e-iθ). Since eiθ=je-iθj, hence, for -π<θ<0, conjugating C-1AC by J=(j00-j), we can further take θ to be in the interval [0,π]. Hence, up to conjugacy, every element in Sl(2,ℍ) which has a fixed point on 𝕊3 is of the form Tl,θ,q=(leiθq0l-1e-iθ), where l>0, 0≤θ≤π, q∈ℍ. When q=0, l=1, we denote it by Dθ, and when l≠1, θ=0, we denote it by Dl,θ.
Now consider an elementTl,θ,q as above, where q=q0+q1j and l=1. Then, T1,θ,q acts on H4 as
(3.3)fq:z⟼eiθzeiθ+qeiθ.
Let q0′=q0eiθ, q1′=q1e-iθ, y=q0′(1-e2iθ)-1. Conjugating fq by the map
(3.4)τ-y:z⟼z-y=x=x0+x1j,
we have
(3.5)τ-yfqτ-y-1(x)=τ-yf(x+y)=τ-y((ei2θ(x0+y)+q0′)+(x1+q1′)j)=(ei2θx0+c0)+(x1+q1′)j=eiθxeiθ+q1jeiθsincec0={ei2θy+q0′-y}=0.
Thus, T1,θ,q is conjugate to T1,θ,q1j, where q1∈ℂ. If A acts as an elliptic, q1=0. In this case we denote it by Dθ. If q1≠0, then A acts as a 1-rotatory parabolic when θ≠0,π. Conjugating it further by a transformation of the form z↦q1-1z, 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 Rj:z↦-jz we get RjTq1Rj-1:z↦z+1. Thus, up to conjugation, a translation can be taken as z↦z+1.
Letl≠1 in the expression of Tl,θ,q. Then, it acts as
(3.6)gq:z⟼l2eiθzeiθ+lqeiθ.
Let q0′=lq0eiθ,q1′=lq1e-iθ. Let w0=q0′(1-l2e2iθ)-1,w1=q1′(1-l2)-1, w=w0+w1j. Conjugating gq by the map
(3.7)τ-w:z⟼z-w=x=x0+x1j
we have
(3.8)τ-wgqτ-w-1(x)=τ-wgq(x+w)=τ-w((l2ei2θ(x0+w0)+q0′)+(l2(x1+w1)+q1′)j)=(l2ei2θx0+c0)+(l2x1+c1)j.
Thus, Tl,θ,q is conjugate to Dl,θ. When θ≠0,π, it acts as a 1-rotatory hyperbolic. Otherwise, it acts as a stretch.Case 2.
SupposefA 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.
LetA be a 2-rotatory elliptic. Then, A cannot be conjugated to an upper triangular matrix in Sl(2,ℍ). 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 ℜ(a)≠ℜ(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∈ℂ, |a|=|d|=1, and thus we assume
(3.9)A=(eiθ00eiϕ),θ≠±ϕ.
This completes the proof.


### 3.2. Algebraic Characterization

Writeℍ=ℂ⊕jℂ. Express A=A1+jA2, where A1,A2∈M4(ℂ). This gives an embedding A↦Aℂ of Sl(2,ℍ) into SL(4,ℂ), compare with [8, 15], where(3.10)AC=(A1-A2¯A2A1¯).We will use this embedding to characterize the elements inSl(2,ℍ).Corollary 3.2.
Embed the groupSL(2,ℍ) into SL(4,ℂ). Let f be an orientation-preserving isometry of H4. Let f be induced by A in SL(2,ℍ). Let Aℂ be the corresponding element in SL(4,ℂ). Let the characteristic polynomial of Aℂ be
(3.11)χAC(x)=x4-2a3x3+a2x2-2a1x+1.
Define
(3.12)c1=a12,c2=a2,c3=a32.
Then, one has the following.(i)
A acts as a 2-rotatory elliptic if and only if(3.13)c1=c3,c2<c1+2.(ii)
A acts as a 1-rotatory hyperbolic if and only if(3.14)c1=c3,c2>c1+2.(iii)
A acts as a stretch if and only if(3.15)c1=c3,c2=c1+2,c1>4.(iv)
A acts as a translation if and only if
(3.16)c1=c3,c2=c1+2,c1=4,
and A is not ±I.(v)
A acts as a 1-rotatory elliptic or a 1-rotatory parabolic if and only if(3.17)c1=c3,c2=c1+2,c1<4.
Moreover, if the characteristic polynomial ofAℂ 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 coefficientsc1, c2, and c3 are conjugacy invariants for Sl(2,ℍ), in fact, for PSl(2,ℍ). Since PSl(2,ℍ) and PU(1,1;ℍ) are conjugate in PGL(2,ℍ), 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 invariantsc1,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.


## 3.1. The Conjugacy Classes

Lemma 3.1.
LetA be an element in Sl(2,ℍ).(i)
IfA acts as a 1-rotatory elliptic, then A is conjugate to Dθ=(eiθ00e-iθ), 0<θ<π. If θ=0 or π, A acts as the identity.(ii)
IfA acts as a 1-rotatory parabolic, then A is conjugate to Tθ,j=(eiθj0e-iθ), or -Tθ,j, 0<θ<π.(iii)
IfA acts as a translation, then A is conjugate to T1=(1101) or -T1.(iv)
IfA acts as a 1-rotatory hyperbolic, then A is conjugate to either Dr,θ=(reiθ00r-1e-iθ) or D-r,θ, 0<θ<π, r>0.(v)
IfA acts as a stretch, then A is conjugate to Dr=(r00r-1), or D-r.(vi)
Finally, ifA 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,ℍ). However, in the ball model of H4, A is conjugate to
(3.1)Dθ,ϕ=(eiθ00eiϕ),θ≠±ϕ.Proof.
LetA=(abcd). The induced linear fractional transformation is given by
(3.2)fA:q⟼(aq+b)(cq+d)-1.
Now there are two cases.Case 1.
SupposefA 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 ℂ, let a=vleiθv-1, 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=v′l-1e-iθ(v′)-1. Let C=(v00v′). Since v(v′)*=vv¯=1, hence C is an element in Sl(2,ℍ), and C-1AC=(leiθv-1bv′0l-1e-iθ). Since eiθ=je-iθj, hence, for -π<θ<0, conjugating C-1AC by J=(j00-j), we can further take θ to be in the interval [0,π]. Hence, up to conjugacy, every element in Sl(2,ℍ) which has a fixed point on 𝕊3 is of the form Tl,θ,q=(leiθq0l-1e-iθ), where l>0, 0≤θ≤π, q∈ℍ. When q=0, l=1, we denote it by Dθ, and when l≠1, θ=0, we denote it by Dl,θ.
Now consider an elementTl,θ,q as above, where q=q0+q1j and l=1. Then, T1,θ,q acts on H4 as
(3.3)fq:z⟼eiθzeiθ+qeiθ.
Let q0′=q0eiθ, q1′=q1e-iθ, y=q0′(1-e2iθ)-1. Conjugating fq by the map
(3.4)τ-y:z⟼z-y=x=x0+x1j,
we have
(3.5)τ-yfqτ-y-1(x)=τ-yf(x+y)=τ-y((ei2θ(x0+y)+q0′)+(x1+q1′)j)=(ei2θx0+c0)+(x1+q1′)j=eiθxeiθ+q1jeiθsincec0={ei2θy+q0′-y}=0.
Thus, T1,θ,q is conjugate to T1,θ,q1j, where q1∈ℂ. If A acts as an elliptic, q1=0. In this case we denote it by Dθ. If q1≠0, then A acts as a 1-rotatory parabolic when θ≠0,π. Conjugating it further by a transformation of the form z↦q1-1z, 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 Rj:z↦-jz we get RjTq1Rj-1:z↦z+1. Thus, up to conjugation, a translation can be taken as z↦z+1.
Letl≠1 in the expression of Tl,θ,q. Then, it acts as
(3.6)gq:z⟼l2eiθzeiθ+lqeiθ.
Let q0′=lq0eiθ,q1′=lq1e-iθ. Let w0=q0′(1-l2e2iθ)-1,w1=q1′(1-l2)-1, w=w0+w1j. Conjugating gq by the map
(3.7)τ-w:z⟼z-w=x=x0+x1j
we have
(3.8)τ-wgqτ-w-1(x)=τ-wgq(x+w)=τ-w((l2ei2θ(x0+w0)+q0′)+(l2(x1+w1)+q1′)j)=(l2ei2θx0+c0)+(l2x1+c1)j.
Thus, Tl,θ,q is conjugate to Dl,θ. When θ≠0,π, it acts as a 1-rotatory hyperbolic. Otherwise, it acts as a stretch.Case 2.
SupposefA 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.
LetA be a 2-rotatory elliptic. Then, A cannot be conjugated to an upper triangular matrix in Sl(2,ℍ). 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 ℜ(a)≠ℜ(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∈ℂ, |a|=|d|=1, and thus we assume
(3.9)A=(eiθ00eiϕ),θ≠±ϕ.
This completes the proof.


## 3.2. Algebraic Characterization

Writeℍ=ℂ⊕jℂ. Express A=A1+jA2, where A1,A2∈M4(ℂ). This gives an embedding A↦Aℂ of Sl(2,ℍ) into SL(4,ℂ), compare with [8, 15], where(3.10)AC=(A1-A2¯A2A1¯).We will use this embedding to characterize the elements inSl(2,ℍ).Corollary 3.2.
Embed the groupSL(2,ℍ) into SL(4,ℂ). Let f be an orientation-preserving isometry of H4. Let f be induced by A in SL(2,ℍ). Let Aℂ be the corresponding element in SL(4,ℂ). Let the characteristic polynomial of Aℂ be
(3.11)χAC(x)=x4-2a3x3+a2x2-2a1x+1.
Define
(3.12)c1=a12,c2=a2,c3=a32.
Then, one has the following.(i)
A acts as a 2-rotatory elliptic if and only if(3.13)c1=c3,c2<c1+2.(ii)
A acts as a 1-rotatory hyperbolic if and only if(3.14)c1=c3,c2>c1+2.(iii)
A acts as a stretch if and only if(3.15)c1=c3,c2=c1+2,c1>4.(iv)
A acts as a translation if and only if
(3.16)c1=c3,c2=c1+2,c1=4,
and A is not ±I.(v)
A acts as a 1-rotatory elliptic or a 1-rotatory parabolic if and only if(3.17)c1=c3,c2=c1+2,c1<4.
Moreover, if the characteristic polynomial ofAℂ 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 coefficientsc1, c2, and c3 are conjugacy invariants for Sl(2,ℍ), in fact, for PSl(2,ℍ). Since PSl(2,ℍ) and PU(1,1;ℍ) are conjugate in PGL(2,ℍ), 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 invariantsc1,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 sevenz-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 ellipticsDθ,ϕ:Z(Dθ,ϕ)≈𝕊1×𝕊1,(iii)
the 1-rotatory ellipticsDθ:Z(Dθ)≈SL(2,ℂ),(iv)
the 1-rotatory hyperbolicsDr,θ:Z(Dr,θ)≈ℂ*,(v)
the stretchesDr:Z(Dr)≈ℍ*,(vi)
the translationsT1:Z(T1)≈𝕊1⋉ℝ3,(vii)
the 1-rotatory translationsTθ:Z(Tθ)≈𝕊1×ℂ.Thus, the isometries are classified by the isomorphism classes of the centralizers.Proof.
First we note that the center ofSl(2,ℍ) is given by {I,-I} and they form a single z-class. Now suppose A≠±I is given by
(4.1)A=(abcd),ab*,cd*,c*a,d*b∈R3,ad*-bc*=1.
Let a=a1+a2j,b=b1+b2j, and so forth, where for i=1,2, ai,bi,ci,di∈ℂ.(i)
Centralizer of1-Rotatory Elliptics.Note that ADθ=DθA implies
(4.2)eiθa=aeiθ,eiθb=be-iθ,e-iθc=ceiθ,e-iθd=de-iθ.
This implies a,d∈Z(eiθ)=ℂ. Further
(4.3)eiθ(b1+b2j)=(b1+b2j)e-iθ⟹eiθb1=b1e-iθ.
Since θ≠0,π, this is possible only if b1=0. Similarly, c1=0. Hence,(4.4)Z(Dθ)={(abjcdj)∣a,b,c,d∈C,ad-bc̅=-j}≈SL(2,C).(ii)
Centralizer of Translations.Let 𝒯 denote the group generated by all translations:
(4.5)Tx=(1x01),x∈R3-{0},
then 𝒯≈ℝ3. Up to conjugacy, we consider T1. It follows from AT1=T1A that a=d,c=0. Hence, A is of the form
(4.6)A=(ab0a),a,b∈H,ab*∈R3,aa*=1.
Now let a=a0+a1i+a2j+a3k. Then, aa*=1 implies a*=(1/|a|2)a̅ which in turn implies that |a|=1 and a1=0=a2. Thus,
(4.7)Z(T1)={(ab0a)∣a=a0+a1k,a0,a1∈R,|a|=1,b∈H,a̅b∈R3}.
Further more, we can write
(4.8)(ab0a)=(a00a)(1a̅b01),
this implies(4.9)Z(T1)=S1⋉T≈S1⋉R3.(iii)
Centralizer of1-Rotatory Parabolics.Next consider the 1-rotatory parabolic Tθ,j. Conjugating it further, we consider the 1-rotatory parabolic
(4.10)Tθ,eiθj=(eiθeiθj0e-iθ).
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∈ℂ, b1=0. Now, combining the relations ad=1 and aj=jd implies d=a¯,|a|=1. Hence, we have
(4.11)Z(Tθ,eiθj)={(abj0a¯)∣a,b∈C,|a|=1}.
Let P=(abj0a¯). Then, P=DθTcj for some 0≤θ≤2π, where a=eiθ and c∈ℂ, in fact c=a-1bj. Since matrices of the form Dθ commute with matrices of the form Tcj,c∈ℂ, hence we have(4.12)Z(Tθ,eiθj)≈S1×C.(iv)
Centralizer of Hyperbolics.Up to conjugacy, we consider Dr,θ=(reiθ00r-1e-iθ), 0≤θ≤π. Then,
(4.13)ADr,θ=Dr,θA⟹reiθa=raeiθ,reiθb=r-1be-iθ,r-1e-iθc=rceiθ,r-1e-iθd=dr-1e-iθ.
Since r≠1, this implies b=0=c, a,d∈Z(eiθ). The equation ad*=1 implies d=(a*)-1. Note that if θ≠0, Z(eiθ)=ℂ and if θ=0, then Z(1)=ℍ. Thus, ifAis1-rotatory hyperbolic, that is, θ≠0, then
(4.14)Z(Dr,θ)={(a00a-1)∣a∈C*}≈C*.IfAis stretch, then(4.15)Z(Dr)={(a00(a*)-1)∣a∈H*}≈H*.(v)
Centralizer of2-Rotatory Elliptics.We use the ball model. Up to conjugation, we consider
(4.16)Dθ,ϕ=(eiθ00eiϕ),θ≠±ϕ.
Let T=(abcd) commute with A. The equation Dθ,ϕT=TDθ,ϕ implies (since θ≠±ϕ), a,d∈ℂ, b=0=c. Hence,(4.17)Z(Dθ,ϕ)={(a00d)∣a,d∈C,|a|=|d|=1}≈S1×S1.
This completes the proof.



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*Source: 757489-2012-02-29.xml*