Algebraic Characterization of Isometries of the Hyperbolic 4-Space

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.


Introduction
Let H n denote the n-dimensional real hyperbolic space.The isometries of H n are always assumed to be orientation preserving unless stated otherwise.The isometries of H 2 can be identified with the group PSL 2, R .This group acts by the real M öbius transformations or the linear fractional transformations z → az b / cz d on the hyperbolic plane.Similarly, PSL 2, C acts on H 3 by complex M öbius transformations.Classically the isometries of H n 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 H n ; it is parabolic, respectively, hyperbolic if g has no fixed point on H n and exactly one, respectively, two fixed points on the boundary ∂H n .It is well known that for H 2 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 H 2 .Analogous characterization of isometries of H 3 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 H n 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 H 4 and H 5 can be considered as 2 × 2 matrices over the quaternions H, where the respective isometry group acts by the quaternionic M öbius transformations: It is natural to ask for algebraic characterizations of the quaternionic M öbius transformations of H 4 and H 5 which will generalize the classical trace identities of the real and complex M öbius transformations.Under the above action, the group of invertible quaternionic 2 × 2 matrices can be identified with the isometries of H 5 .Using this identification, the author has obtained an algebraic characterization of the isometries of H 5 , see 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 H n 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 H 4 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 M 2 C and then embed the quaternionic matrices into complex matrices.We use this approach for the isometry group of H 4 .This approach is different from that of Cao et al. or Kido.Using a geometric and simple approach, the conjugacy classes of isometries of H 4 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 .

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 H n , one associates an orthogonal transformation T o in SO n .For each pair of complex conjugate eigenvalues {e iθ , e −iθ }, 0 < θ ≤ π, 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 0rotatory 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.

The Quaternions
The space of all quaternions H is the four-dimensional real division algebra with basis {1, i, j, k} and multiplication rules i The multiplicative group of nonzero quaternions is denoted by H * .For a quaternion x x 0 x 1 i x 2 j x 3 k, we define x x 0 and x x 1 i x 2 j x 3 k.The norm of x is defined as |x| We choose C 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 c 0 c 1 j, where c 0 , c 1 are complex numbers.Similarly we can also write H C ⊕ jC.For a non-zero quaternion λ, the centralizer of

The Upper Half-Space Model
First we associate three involutions to the quaternions: i * : q q o q i q 2 j q 3 k → q * q o q i q 2 j − q 3 k.It determines an antiautomorphism of H: ii : q q o q i q 2 j q 3 k → q q o − q i − q 2 j q 3 k.It determines an automorphism of H: ab a b , a b a b , iii the conjugation q → q.This again gives an anti-automorphism of H.Note that a a * a * .
Following Ahlfors 3 and Waterman 4 , we identify R 3 with the additive subspace of the quaternions spanned by {1, i, j}, that is, R 3 q ∈ H | q q 0 q 1 i q 2 j .

2.1
We consider the upper half-space model of the hyperbolic space which is given by H 4 q q 0 q 1 i q 2 j q 3 k ∈ H | q 3 > 0 , 2.2 equipped with the metric induced from the differential The boundary of H 4 is identified with The group Sl 2, H acts on H 4 by the linear fractional transformations: Then, PSl 2, H Sl 2, H /{±I} is the group of orientation-preserving isometries of H 4 .

The Ball Model
The ball model of the hyperbolic space is given by The diffeomorphism which identifies the disk model D 4 to the upper half-space model H 4 is given by f : z → z k kz 1 −1 .The matrix acts as the quaternionic linear fractional transformation f on H H ∪ {∞}.This implies that PSl 2, H and PU 1, 1; H are conjugate in PGL 2, H .

The Conjugacy Classes
Lemma 3.1.Let A be an element in Sl 2, H .
i If A acts as a 1-rotatory elliptic, then A is conjugate to D θ e iθ 0 0 e −iθ , 0 < θ < π.If θ 0 or π, A acts as the identity.
iii If A acts as a translation, then A is conjugate to vi Finally, if A acts as a 2-rotatory elliptic, then A has a unique fixed point on H 4 Now there are two cases.
Case 1. Suppose f A has a fixed point on S 3 .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 a b 0 d , ad * 1.Since every quaternion is conjugate to an element in C, let a vle iθ v −1 , where le iθ 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 hence C is an element in Sl 2, H , and . Since e iθ je −iθ j, hence, for −π < θ < 0, conjugating C −1 AC by J j 0 0 −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 S 3 is of the form T l,θ,q le iθ q 0 l −1 e −iθ , where l > 0, 0 ≤ θ ≤ π, q ∈ H.When q 0, l 1, we denote it by D θ , and when l / 1, θ 0, we denote it by D l,θ .Now consider an element T l,θ,q as above, where q q 0 q 1 j and l 1.Then, T 1,θ,q acts on H 4 as f q : z −→ e iθ ze iθ qe iθ .

3.5
Thus, T 1,θ,q is conjugate to T 1,θ,q 1 j , where q 1 ∈ C. If A acts as an elliptic, q 1 0. In this case we denote it by D θ .If q 1 / 0, then A acts as a 1-rotatory parabolic when θ / 0, π.Conjugating it further by a transformation of the form z → q −1 1 z, we can further assume, q 1 1; that is, A is conjugate to T θ,j .If θ 0 or π, A acts as a translation.In this case, we denote it by T q 1 , and further conjugating it by R j : z → −jz we get R j T q 1 R −1 j : z → z 1.Thus, up to conjugation, a translation can be taken as z → z 1.
Let l / 1 in the expression of T l,θ,q .Then, it acts as g q : z −→ l 2 e iθ ze iθ lqe iθ .
Case 2. Suppose f A has no fixed point on S 3 .We claim that the fixed point of f A is unique.To see this, if possible suppose that f A has at least two fixed points x and y on H 4 .Since between two points there is a unique geodesic, f A must fixes the end points of the geodesic joining x and y; consequently, f A pointwise fixes the geodesic.Thus, the associated orthogonal transformation must have an eigenvalue 1.Since f A 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, f A must have a fixed point on S 3 .This is a contradiction.Thus, f A 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, 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 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 ∈ C, |a| |d| 1, and thus we assume A e iθ 0 0 e iφ , θ / ± φ.

3.9
This completes the proof.

3.12
Then, one has the following.
i A acts as a 2-rotatory elliptic if and only if

3.13
ii A acts as a 1-rotatory hyperbolic if and only if

3.14
iii A acts as a stretch if and only if

3.15
iv A acts as a translation if and only if and A is not ±I.
v A acts as a 1-rotatory elliptic or a 1-rotatory parabolic if and only if

3.17
Moreover, if the characteristic polynomial of A C 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 c 1 , c 2 , and c 3 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 , c 1 , c 2 , and c 3 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 c 1 , c 2 , and c 3 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.ii Centralizer of Translations.Let T denote the group generated by all translations:

The Centralizers, up to Conjugacy
Now let a a 0 a 1 i a 2 j a 3 k.Then, aa * 1 implies a * 1/|a| 2 a which in turn implies that |a| 1 and a 1 0 a 2 .Thus, Further more, we can write

4.17
This completes the proof.

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 D θ,φ : Thus, the isometries are classified by the isomorphism classes of the centralizers.Proof.First we note that the center of Sl 2, H is given by {I, −I} and they form a single z-class.Now suppose A / ± I is given by A a b c d , ab * , cd * , c * a, d * b ∈ R 3 , ad * − bc * 1. 4.1 Let a a 1 a 2 j, b b 1 b 2 j, and so forth, where for i 1, 2, a i , b i , c i , d i ∈ C. i Centralizer of 1-Rotatory Elliptics.Note that AD θ D θ A implies e iθ a ae iθ , e iθ b be −iθ , e −iθ c ce iθ , e −iθ d de −iθ .4.2 This implies a, d ∈ Z e iθ C. Further e iθ b 1 b 2 j b 1 b 2 j e −iθ ⇒ e iθ b 1 b 1 e −iθ .4.3 Since θ / 0, π, this is possible only if b 1 0. Similarly, c 1 0. Hence, Z D θ a bj c dj | a, b, c, d ∈ C, ad − bc −j ≈ SL 2, C .4.4 ISRN Geometry
2 |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 , where A 1 , A 2 ∈ M 4 C .This gives an embedding A → A C of Sl 2, H into SL 4, C , compare with 8, 15 , where Embed the group SL 2, H into SL 4, C .Let f be an orientation-preserving isometry of H 4 .Let f be induced by A in SL 2, H . Let A C be the corresponding element in SL 4, C .Let the characteristic polynomial of A C be We will use this embedding to characterize the elements in Sl 2, H .