$z$-Classes of Isometries of The Hyperbolic Space

Let $G$ be a group. Two elements $x, y$ are said to be {\it $z$-equivalent} if their centralizers are conjugate in $G$. The class equation of $G$ is the partition of $G$ into conjugacy classes. Further decomposition of conjugacy classes into $z$-classes provides an important information about the internal structure of the group. Let $I(\H^n)$ denote the group of isometries of the hyperbolic $n$-space. We show that the number of $z$-classes in $I(\H^n)$ is finite. We actually compute their number, cf. theorem 1.3. We interpret the finiteness of $z$-classes as accounting for the finiteness of"dynamical types"in $I(\H^n)$. Along the way we also parametrize conjugacy classes. We mainly use the linear model for the hyperbolic space for this purpose. This description of parametrizing conjugacy classes appears to be new.


Introduction
Let H n denote the n-dimensional hyperbolic space, and I(H n ) its full isometry group.
In the disk-model of the hyperbolic space, an isometry is said to be elliptic if it has a fixed point in the space. A non-elliptic isometry is called hyperbolic, resp. parabolic if it has 2, resp. 1 fixed points on the ideal conformal boundary of the hyperbolic space. We shall build on this classification, and obtain a finer classification up to conjugacy, and beyond. A remarkable feature of H n is that the group I(H n ) is infinite. But the "dynamical types" that our minds can perceive are just finite in number. Can we account for this fact in terms of the internal structure of the group alone?
Let I(H n ) act on itself by conjugacy. For x in I(H n ), let Z(x) denote the centralizer of x in I(H n ). For x, y in I(H n ) we shall say x ∼ y if Z(x) and Z(y) are conjugate in I(H n ). We call the equivalence class defined by this relation, the z-class of x. It turns out that the number of z-classes is finite, and this fact is interpreted as accounting for the finiteness of dynamical types. The z-classes are pairwise disjoint, and are manifolds, cf. theorem 2.1 [8], and so give a stratification of I(H n ) into finitely many strata. Each stratum has two canonical fibrations, and they explain the "spatial" and "numerical" invariants which characterize the transformation. The I(H n ) provides a significant example of the philosophy that was suggested in [8].
In this paper we work mostly in the linear model. Let V be a real vector space of dimension n+1 equipped with a quadratic form of signature (1, n), and O(Q) its full group of isometries. This group has four components. The hyperboloid {v ∈ V | Q(v) = 1} has two components. One of the components is the model for H n . Suggestively, let I(Q) denote the group which preserves the component, and we identify it with I(H n ). Now O(Q) is a linear algebraic group. A significant property of a linear algebraic group over characteristic zero is that each T in a linear algebraic group has the Jordan decomposition T = T s T u where T s is semisimple, (that is, every T s -invariant subspace has a T s -invariant complement), and T u is unipotent, (that is, all eigenvalues of T u are 1) cf. [7]. Then T s and T u are also in O(Q). T s and T u commute. They are polynomials in T . The decomposition is unique. A systematic use of Jordan decomposition leads to a neat and more refined classification of elements of I(Q) up to conjugacy, and as we shall see, also z-equivalence.
Let T be in O(Q). Let V c = V ⊗ R C be its complexification. We identify T with T ⊗ R id and also consider it as an operator on V c . We say that an eigenvalue λ of T is pure if the corresponding eigenspace {v ∈ V|(T − λ)v = 0}, coincides with the generalised eigenspace {v ∈ V|(T − λ) n+1 v = 0}. Otherwise λ is mixed.
The roots of a polynomial of the form x 2 − 2ax + 1, with |a| < 1 are of the form e ±iθ . With the restriction 0 < θ < π, the θ is uniquely determined. When x 2 − 2ax + 1 is a factor of the characteristic polynomial of a transformation, the θ will be called the rotation angle of the transformation. ii) The eigenvectors corresponding to eigenvalue −1 are necessarily space-like. T is orientation-preserving iff the multiplicity of the eigenvalue −1 is even. Equivalently, T is orientation-preserving iff det T = 1.
iii) Suppose T has a time-like eigenvector. Then T is semisimple, and its characteristic polynomial χ T (x) has the form Here a j 's are distinct, and l ≥ 1. Let k = s j=1 r j + [ m 2 ]. We agree to call T a k-rotatory elliptic if m is even, and a k-rotatory elliptic inversion if m is odd. iv) T can have only one pair of real eigenvalues (r, r −1 ), r = ±1. Such r is necessarily positive. The eigenspaces corresponding to r, r −1 are 1-dimensional, and are light-like. Suppose this is the case. Then T is semisimple, and its characteristic polynomial χ T (x) has the form . We agree to call T a k-rotatory hyperbolic if m is even, and a k-rotatory hyperbolic inversion if m is odd. v) Suppose T is not semisimple. Then T has a 1-dimensional light-like eigenspace with eigenvalue 1. Let T = T s T u be its Jordan decomposition, T u = Id. The characteristic polynomial χ T (x) has the form . We agree to call T a k-rotatory parabolic if m is even, and a k-rotatory parabolic inversion if m is odd. T s is elliptic, and the minimal polynomial of T u is (x − 1) 3 .

vi) Every T ∈ I(Q) satisfies either iii), or iv), or v).
A significant consequence of theorem 1.1 is the classification of conjugacy classes. We call T elliptic if it is k-rotatory elliptic or k-rotatory elliptic inversion for some k. Similarly for hyperbolic or parabolic. It will be convenient to call χ oT (x), the reduced characteristic polynomial of T . Its degree is always even. Let k ′ = 1 2 deg χ oT (x). The number of conjugacy classes is infinite. This infinity arises roughly from the eigenvalues r and the rotation-angles θ j with multiplicity r j in the notation of theorem 1.1. These are the "numerical invariants" of a transformation. The "spatial invariants" of a transformation are the corresponding orthogonal decomposition of the space, and the signatures of the quadratic form of the summands. Roughly speaking the "spatial invariants" define the z-class.
We determine the centralizers of elements and z-classes in §4, and also count their number. Theorem 1.3. For u a natural number, let p(u) denote the Eulerian partition function, ) where ǫ = 1 or 2 according as n is even or odd, z-classes of elliptic elements. For this count identity is considered as an elliptic element.
Each z-class is a manifold, and so each component has a well-defined dimension. It is said to be generic if it has the maximum dimension ( = dim I(Q)). Theorem 1.4. i) Let n be even, n = 2n ′ . The generic z-classes are the z-classes of n ′ -rotatory elliptics, (n ′ − 1)-rotatory hyperbolics, and (n ′ − 1)-rotatory hyperbolic inversions, all with pairwise distinct rotation angles.
In §5 we give some simple criteria to detect the type of an isometry of I(H n ) to be hyperbolic, resp. elliptic, resp. parabolic. For example, T is hyperbolic iff χ oT (1) < 0. T is parbolic, resp. elliptic, iff χ oT (1) > 0 and (x − 1) 2 divides (resp. does not divide) the minimal polynomial of T . There are also criteria in terms of the trace of T .
Previously Chen and Greenberg [4], Kulkarni [9] analysed the conjugacy classes of hyperbolic isometries. The criterion for conjugacy provided in Theorem 1.2 appears to be new. Ahlfors, [1], suggested the use of Clifford algebras to study Möbius transformations in higher dimensions, cf. [2], [5], [13], [14] for elaboration of this theme. The present paper presents an alternate approach. It appears that Clifford-algebraic approach has not yet been worked out for orientation-reversing transformations.
Also for v ∈ V c,λ and w ∈ V c,µ we have < v, w >=< T v, T w >=< T s v, T s w >= λµ < v, w > .
So unless λµ = 1 we have V c,λ and V c,µ are othogonal with respect to Q c . Let ⊕ denote the orthogonal direct sum, and + the usual direct sum of subspaces. We have 2) Now let T be in I(Q). Note that a unipotent transformation must preserve {v ∈ V|Q(v) ≥ 0}, so T u and hence T s , are also in I(Q).
Suppose that T has a time-like eigenvector v. Since < v, v > = 0, we see that the corresponding eigenvalue is equal to 1. T leaves the orthogonal complement of v invariant, and the metric is negative definite on this complement. So T must be semisimple. The characteristic polynomial χ T (x) is of the form Here a j 's are distinct. We have l ≥ 1. The minimal polynomial m T (x) divides the characteristic polynomial, it has the same factors as the characteristic polynomial, and for a semisimple element it is a product of distinct factors. So We have agreed to call such T a k-rotatory elliptic if m is even, and a k-rotatory elliptic inversion if m is odd.
3) Let T be in I(Q) and suppose that r is a real eigenvalue = 1. By 1) the corresponding eigenvector is light-like. Now T preserves {v ∈ V| < v, v >≥ 0}. So r must be positive. By 1) r −1 must also be an eigenvalue, since Q is non-degenerate. The dimension of a subspace on which Q = 0 is at most 1.
We have agreed to call such T a k-rotatory hyperbolic if m is even, and a k-rotatory hyperbolic inversion if m is odd. 4) Now let T be in I(Q), and λ a non-real complex eigenvalue. Let v ∈ V c be a corresponding eigenvector. Write v = v 1 + iv 2 where v 1 resp. v 2 are real, and λ = a + ib, where a, b are real and b = 0. We have T v = λv, so Since b = 0, we see that both v 1 , v 2 are non-zero, and they are linearly independent. Let It is indeed a minimal invariant subspace. So Q| W must be either 0, or it must be non-degenerate, for otherwise the nullspace of W would be 1-dimensional and would be T -invariant. Also Q| W cannot have signature (1, 1), for an elementary fact from 2-dimensional Lorentzian geometry is that the eigenvalues of T | W would be real. So Q| W must be negative definite. But then the eigenvalues of T on W ⊗ R C are (λ,λ), and must be of the form (e iθ , e −iθ ). By interchanging λ withλ if necessary, we may take 0 < θ < π. By making this argument on V c,λ we see that V c,λ + V c,λ is a complexification of a real subspace on which Q is negative definite. Let us denote this real subspace V λ,λ , and call it the real trace of the eigenspace of V c,λ + V c,λ . 5) Suppose −1 is an eigenvalue of T and v a corresponding eigenvector. Since T preserves the set {u ∈ V|Q(u) ≥ 0}, v must be outside this set. In other words, v is space-like. Now combining 3), 4) we see that 1 is the only possible eigenvalue which is mixed.
6) V is said to be orthogonally indecomposable with respect to T , or T -orthogonally indecomposable for short, if V is not an orthogonal direct sum of proper T -invariant subspaces. Given T , we can decompose V into T -orthogonally indecomposable subspaces W, Q| W is non-degenerate. So far we know three types of T -orthogonally indecomposable subspaces.
i) Dim W = 1: Here W is spanned by an eigenvector of non-zero length.
ii) Dim W = 2: Here T | W has no real eigenvalue and Q| W is negative definite. The eigenvalues of T are e ±iθ , where 0 < θ < π.
iii) Dim W = 2: Here W = V r + V r −1 , where both V r and V r −1 are spanned by eigenvectors of length 0.
We now describe the fourth, and last, type of T -orthogonally indecomposable subspaces.
Then T is unipotent and dim W = 3.
Proof. Let dim W = m. From previous discussion we see that T | W must be unipotent.
Let v be an eigenvector. From 3) and 4) we must have v of length 0, and eigenvalue 1. Let C = {u ∈ W|Q(u) = 0}. This is a right circular cone. Since T v = v, T must preserve the tangent space τ at v to C. Now dim τ = m − 1, and Q| τ is degenerate. Q| τ has nullity 1, spanned by v, and otherwise its signature is (0, m − 2). Let τ → τ / < v > be the canonical projection. Q induces the canonical symmetric quadratic formQ on τ / < v >. T induces a linear transformationT on τ / < v >, and it preservesQ. Now Q has signature (0, m − 2), so it is negative definite. T is unipotent, so isT . SoT is both semisimple and unipotent, so it must be I, the identity. It is now easy to see that since we assumed W is orthogonally indecomposable with respect to T , we must have dim τ / < v >= m − 2 = 1. So dim W = 3.

7)
Let now T be a non-semisimple element of I(Q). Let T = T s T u , T u = I be the Jordan decomposition of T .
Let λ be a non-real eigenvalue of T s . In 4) we have seen that V c,λ + V c,λ is a complexification of its trace V λ,λ on which Q is negative definite. Now T u commutes with T s . So T u keeps the eigenspace of T s in V c invariant. It follows that T u leaves V c,λ + V c,λ , and hence Let −1 be an eigenvalue of T s . By 5) the corresponding eigenvector is space-like. It follows that Q restricted to V −1 is negative definite. By the above argument T u restricted to V −1 is identity also.
Let r be a (necessarily positive) real eigenvalue = 1 of T s . By 4), T is hyperbolic. So T = T s and T u = identity. By our hypothesis, this is not the case. So this case cannot occur.
So T u can be different from identity only on the eigenspace V 1 with eigenvalue 1 of T s . But T s being semisimple, it must be itself identity on V 1 .
We agree to call such T parabolic. The characteristic polynomial of such T has the form .
In a more refined way, we call T a k-rotatory parabolic if m is even, and a k-rotatory parabolic inversion if m is odd. Notice also that m Tu (x) = (x − 1) 3 .
This completes the proof.

Conjugacy Classes
If the transformations are conjugate their characteristic and minimal polynomials are the same, as indeed this is true for elements of GL(V). The interesting point is the converse.
Suppose T is elliptic. Then V is an orthogonal direct sum of 1-or 2-dimensional indecomposable T -invariant subspaces. The metric on all 2-dimensional subspaces is negative definite. On one 1-dimensional subspace it is positive definite, and on the others it is negative definite. Given T ′ conjugate to T , V is a similar orthogonal direct sum of 1or 2-dimensional indecomposable T ′ -invariant subspaces. There is an isometry in O(Q) carrying one decomposition into another. The isometry may be chosen to lie in The proof in the hyperbolic case is similar. In this case V is an orthogonal direct sum of one 2-dimensional subspace on which Q has signature (1, 1). The rest is similar to the elliptic case.
Suppose T is parabolic. Then V is an orthogonal direct sum of one 3-dimensional T -indecomposable subspace on which the metric has the signature (1, 2), and the other 1 or 2-dimensional T -invariant subspaces on which the metric is negative definite. Now the proof is similar to the elliptic case.
Notice that if we know a transformation is not parabolic then the characteristic polynomial itself determines the conjugacy class. 4. Centralizers, and z-classes 1) Let T be an elliptic transformation. First let us note some invariants of T from a dynamic viewpoint. Let Λ be the set of fixed points of T . Then Λ is the eigenspace of T with eigenvalue 1. Now Q| Λ is non-degenerate, cf. part 1) of §2. Let Π be the subspace orthogonal to Λ. Then V = Λ ⊕ Π. The centralizer Z(T ) in I(Q) leaves each eigenspace, or a trace of a complex eigenspace, of T invariant. So it is clear that Z(T | Λ ) = I(Q| Λ ).
Note that Q| Π is negative definite. In Π let Π −1 be the eigenspace with eigenvalue −1, and Π o be its orthogonal complement. Let Here a j 's are distinct. Let Π j = ker (T 2 − a j T + 1) r j . Then Π = ⊕ s j=1 Π j , and dim Π j = 2r j .
Let v be a vector in Π j , < v, v >= −1, and T v = v 1 . Then span {v, v 1 } is Tinvariant. Choose orientation on span {v, v 1 } so that the angle between v and v 1 is θ, is independent of the choice of v. Thus this process in fact defines J j as a T -invariant complex structure on Π j . It follows that the Z(T | Π j ) is isomorphic to the unitary group U(r j ). On the orthogonal complement of W, the metric is negative definite. Now the analysis is similar to the elliptic case. We have Z(T ) isomorphic to 3) Now let T be a parabolic transformation, T = T s T u . We know that T s and T u are polynomials in T . So if A commutes with T , it commutes with T s and T u as well.
Conversely if A commutes with T s and T u it commutes with T . In other words we have We have a decomposition V = W ⊕ U, where T u | U = identity, dim W = 3, and Q| W has signature (1, 2). Also T s | W = identity. On U the metric is negative definite. Decompose U into U 1 ⊕ U −1 ⊕ U 2 , where U 1 (resp. U −1 ) are the eigenspaces with eigenvalue 1 (resp. −1), and U 2 is the orthogonal sum of the real traces of the complex eigen-spaces. Since is a direct product of O(U −1 ) and various unitary groups. This leaves Z(T | W⊕U 1 ) = Z(T u | W⊕U 1 ). To compute Z(T u | W⊕U 1 ) it is convenient to use the upper half space model of the hyperbolic space, where ∞ is the fixed point of T u . Let dim W ⊕ U 1 = n ′ + 1. Take the upper half space, which we conveniently again denote by H n ′ , with orthogonal coordinates (x 1 , x 2 , . . . , x n ′ ), with x n ′ ≥ 0. The (n ′ − 1)-planes x n ′ = c > 0, are parts of the horo-spheres. We may further assume that the coordinates are so chosen that T u is given by (x 1 , x 2 , . . . , x n ′ ) → (x 1 + 1, x 2 , x 3 , . . . , x n ′ ). Now the orthogonal group in coordinates (x 2 , x 3 , . . . , x n ′ −1 ) commutes with T u , as also the translations in the planes x n ′ = c. The structure of the group may be described as follows. It is a direct product of two groups.
From this description of centralizers we see that to each centralizer there is attached a certain orthogonal decomposition of the space, and its isomorphism type is determined by it. Given any two such decompositions of the same type (meaning that the dimensions and the signatures of the quadratic form for the corresponding summands are the same) there exists an element of I(Q) carrying one decomposition into the other. This element will conjugate one centralizer into the other. Thus there are only finitely many z-classes. Now the centralizers are described in terms of group-structure itself. This fact is interpreted as accounting for the finiteness of "dynamical types".
The actual count of z-classes requires more work, and involves counting certain types of partitions of n + 1. Let T be in , and distinct. We associate to T the partition n + 1 = l + m + s j=1 r j , or n + 1 = 2 + l + m + s j=1 r j . Notice that knowing that T is elliptic, resp. hyperbolic, resp. parabolic, and the partition of n + 1 determine its z-class uniquely. It is easy to see that distinct partitions (and knowing that T is elliptic, resp. hyperbolic, resp. parabolic) are associated to distinct z-classes. In considering z-classes we consider identity as elliptic. There are some restrictions on which partitions can actually occur.
First observe that in any group the center forms a single z-class. In other words g must be identity. Now suppose g has a fixed point on the boundary. But I(Q) is transitive also on the boundary. So g must fix all the boundary. But then g fixes the convex hull of the boundary. So again g has to be identity.
Case 1: Let T be an elliptic element. Let n + 1 = l + m + 2 s j=1 r j be the associated partition. We have l ≥ 1 and |a j | < 1. Let k ′ be the number of rotation angles lying in (0, π), k ′ = s j=1 r j . We assume, as we may, that the rotation angles are ordered in increasing order. The rotation angle θ j is repeated as many times as its multiplicity r j . The contribution of this part to the centralizer is the product of unitary groups U(r j ). Thus there are p(k ′ ) choices of distinct rotation angles counting with multiplicities. Since l ≥ 1, we have [ n 2 ]+1 choices for k ′ (including 0). For a fixed k ′ , there are n+1−2k ′ choices for l. Thus setting p(0) = 1, there are (n + 1)p(0) + (n − 1)p(1) + (n − 3)p(2) + . . .+ ǫp([ n 2 ]) where ǫ = 1 or 2 according as n is even or odd, z-classes of elliptic elements.
Case 2: Let T be a hyperbolic element. Rewrite the associated partition of n + 1 as n + 1 = 2 + l + m + 2 s j=1 r j , where the first "2" stands for a pair of real eigenvalues = ±1, and l (resp. m) is the multiplicity of the eigenvalue 1 (resp. −1), and |a j | < 1. In this case we see that if l and m are interchanged we get the same z-class. So for counting z-classes we may further assume l ≤ m. Case 3: Let T be a parabolic element. Rewrite the associated partition of n + 1 as n + 1 = 3 + l + m + 2 s j=1 r j , where 3 + l (resp. m) is the multiplicity of the eigenvalue 1 (resp. −1). Now there are [ n−2 2 ] + 1 choices for k ′ (including 0), and for a fixed k ′ , we have n − 1 − 2k ′ choices for l (including 0). Thus we have (setting p(0) = 1) (n − 1)p(0) + (n − 3)p(1) + . . . + ǫp([ n−2 2 ]) where ǫ = 1 or 2 according as n is even or odd, z-classes of parabolic elements.
We have completely proved theorem 1.3.
Lastly note that each z-class is a manifold, so one can talk about its dimension for each of its components. A z-class may not be connected but there is actually a single dimension associated to a z-class. It can be read from theorem 2.1 of [8]. From that description we see that in a semisimple Lie group, the z-class is generic, i.e. the dimension of a z-class is maximal, iff the centralizer is abelian. The stated z-classes in theorem 1.4 are precisely the z-classes with this property. This completes the proof of Theorem 1.4.

Some simple Criteria
In this section we note some simple criteria for detecting the type of isometries of the hyperbolic n-space, using the linear model. Let T be in I(Q), and let χ oT (x) be its reduced characteristic polynomial, that is the polynomial obtained after factoring all factors of the type x ± 1. Note that the factors x 2 − 2ax + 1 for |a| < 1 always takes positive values for real x. The factors (x − r)(x − r −1 ) for r > 1 are present only in the case of hyperbolic transformation. The criteria involve only detecting the factors x ± 1 of the characteristic or minimal polynomial of T . So they are defined over any subfield of R which contains the coefficients of the characteristic polynomial of T , in particular on any subfield of R generated by the coefficients of the matrix of T with respect to a suitable basis.
If T is parabolic then the minimal polynomial m T (x) must be divisible by (x − 1) 3 . On the other hand, an elliptic transformation is semisimple.
• Note that the above two statements provide a finite algorithm to test whether T is hyperbolic, parabolic or elliptic.
Some other simple criteria for recognizing the type can be obtained by just looking at the trace. Note that for elliptic and parabolic transformations all eigenvalues have absolute value 1. So • If trace T > n + 1, then T is hyperbolic.
The eigenvalues of T u are u-th powers of the eigenvalues of T . Only the hyperbolic transformation has eigenvalue r > 1, and its multiplicity is 1. So for large u the eigenvalue r u of T u will dominate the others. So • T is hyperbolic iff for all sufficiently large u, trace T u > n + 1.
Proof. Let T be hyperbolic. Then it has a pair of real eigenvalues {λ, λ −1 }, λ > 1. All other eigenvalues have absolute value 1. Let the other eigenvalues be u 1 , ...., u n−1 . So we have, As λ > 1 and |u i | = 1 for i = 1, 2, .., n − 1, we have for any d ∈ N, . Since λ > 1, we can find an u ∈ N such that λ u > 2n. i.e. trace T u > n + 1. On the other hand, if T is parabolic or elliptic, then |u i | = 1 for i = 1, 2, ..., n + 1. Hence for any d, For the cases n = 2, 3 we have very neat criteria. Compare the usual criteria given in the P SL(2, R) model of I o (H 2 ), or the P SL(2, C) model of I o (H 3 ) cf. [3], [11], or appendix A below. Note that the orientation-reversing case, either for n = 2 or for n = 3, is usually not treated in the literarature.
Let T be in I(Q), T = identity.
In these dimensions, by remarkable isomorphisms, the group I(H 2 ), resp. I o (H 3 ) can be identified with the group P GL(2, R), resp. P GL(2, C). In the appendix A we give algebraic criteria using these isomorphisms. We have treated the orientation-reversing case also.
• In higher dimensions although it is hard to compute individual eigenvalues of T , the trace T and trace T u are easily computable by using the well-known Newton's identities.
In our case, a n+1 is +1 or −1 according as T is orientation preserving or orientation reversing.
Let p k be the sum of k-th powers of eigenvalues of T . Since the eigenvalues of T k are the k-th powers of eigenvalues of T , we have p k = trace T k . The Newton's identities (cf. [10], [12]) express p k 's in terms of the coefficients of the characteristic polynomial of T as follows: 2) For k > n + 1, p k = a 1 p k−1 − a 2 p k−2 + ... + (−1) n−1 p k−n .
So we have, trace T = a 1 , .. If either of these numbers is > n + 1, then T is hyperbolic.
• Suppose T is hyperbolic and has eigenvalue λ such that λ ≥ a > 1. Then for any real r > 0, and m > ln (n−1+r) ln a we have trace T m > r.
Proof. We have for any positive integer k, Since a > 1, choose m such that a m − (n − 1) > r, or, m > ln (n−1+r) ln a .

Corollary 1. An isometry T of H n is hyperbolic if and only if the sequence {trace T k } is divergent.
Appendix A. Alternative criteria for I(H 2 ) and I(H 3 ), a Lie group theoretic and dynamic perspective In low dimensions there are two remarkable Lie theoretic isomorphisms. The group SO o (2, 1) is isomorphic to P SL(2, R) and SO o (3, 1) is isomorphic to P GL(2, C). These allow us to identify I o (H 2 ) and I o (H 3 ) with the groups P SL(2, R) and P GL(2, C) respectively.
In particular, I o (H 3 ) is isomorphic to P GL(2, C). Let σ 0 be the reflection in the real line in C, i.e. σ 0 (z) =z. The coset P GL(2, C)σ 0 is isomorphic to the group of orientationreversing isometries of H 3 . Note that we consider P GL(2, C) rather than the usual P SL(2, C) in the literature. In general over fields F = C, the groups P GL(2, F) and P SL(2, F) are not always isomorphic. So we find it natural to consider the group P GL(2, C) and to avoid the "det = 1" normalization for formulating the criteria.
Before proceeding further we introduce a new terminology of "Möbius co-ordinates". The coordinate z in C, extended toĈ by setting z(∞) = ∞, is called a Möbius coordinate onĈ. It is not a complex coordinate in the sense the terminology is used in the theory of manifolds. Any P GL(2, C)-translate will also be termed as a Möbius coordinate.
Thus I(H 2 ) can be identified with the group P GL(2, R). Let GL + (2, R), resp. GL − (2, R), be the subgroup of all elements with positive, resp. negative, determinant in GL(2, R). Then the group I o (H 2 ) is isomorphic to P GL + (2, R), which can be identified with M + (1). The other component of the isometry group may be identified with GL − (2, R).
In the disk model, the group I o (H 2 ) consists of the usual holomorphic transformations of the unit disk D 2 . Recall that such a transformation is of the form The orientation-reversing isometries are given bȳ Let R + denote the group of all positive reals, and let SU(1, 1) = ac cā | |a| 2 − |c| 2 = 1 .
Then it follows from the above that I o (H 2 ) lifts to the subgroup R + ×SU(1, 1) of GL(2, C).
Observe that the co-efficient matrix M of f a,φ has the following decomposition: It turns out to be the polar decomposition, or the KP decomposition, of f a,φ .
A.3. The criteria. First note that the quotient map GL(2, C) → P GL(2, C) is surjective, and it maps the conjugacy class of an element A onto the conjugacy class of the corresponding elementÂ. Since the trace and determinant of A are the conjugacy invariants in GL(2, C), we can take (trace A) 2 det A as the conjugacy invariant forÂ in P GL(2, C).

Now suppose
A is an element in GL(2, C) and letÂ be the corresponding Möbius Then By Jordan theory we know the conjugacy classes in GL(2, C).
exists P in GL(2, C) such that D λ,µ = P AP −1 . ConsideringP : z → w as a Möbius change of co-ordinates onĈ, we haveÂ Further note that D λ,µ is conjugate to D µ,λ . So the size | λ µ | has no significance for the conjugacy problem. Now there are the following possibilities.
(a) λ µ is not a real number and | λ µ | = 1. then A acts as an 1-rotatory hyperbolic. Classically these are known as loxodromic.
(c) λ µ is a real number. Suppose | λ µ | = 1. If λ µ > 0, then A acts as a 0-rotatory hyperbolic. We agree to call it a stretch. If λ µ < 0, then A acts as an 1-rotatory hyperbolic with rotation angle π, and in this case we agree to call it a stretch half-turn.
Case (ii). Suppose A is conjugate to the upper-triangular matrix T λ = λ 1 0 λ . In this caseÂ takes the form w → w + 1 λ , and we see that A acts as a 0-rotatory parabolic. We agree to call it a translation.
In the orientation-reversing case, lifting f : z → az+b cz+d to A has no dynamic significance for I(H 3 ), for the "correct" lift is Aσ 0 . Hence the conjugacy classes in GL(2, C) do not determine the conjugacy classes of the orientation-reversing isometries. However one dynamic invariant of f is f 2 , which corresponds to Aσ 0 Aσ 0 , and is an orientationpreserving transformation. The associated matrix to f 2 is given by B = AĀ. As we shall see in the next theorem, when B = λI, λ > 0, B determines A. The elusive case is when B = λI. It will turn out that in this case there are exactly two conjugacy classes. Either f is a 0-rotatory elliptic inversion, and the fixed point set of f is a circle, or f is an 1-rotatory elliptic inversion, and f is an antipodal map without fixed points. An antipodal map, by definition, is an orientation-reversing map of order 2 and without a fixed point onĈ. It turns out that these maps are conjugate to a : z → − 1 z . In the disk model of H 3 , these maps fix a unique point in the disk, and interchange the end points of a geodesic passing throgh the fixed-point. I. Suppose f is orientation-preserving.  (iv) If |c(B) − 2| = 2 and B = λI, then f acts as an inversion in a circle, or an antipodal map. In both cases f is an elliptic inversion. We detect these cases as follows.
If there exists a non-zero u in C, unique up to a non-zero real multiple, such that the matrix uA is of the form: a b c −ā , where b, c are reals, then f acts as an inversion in a circle, resp. antipodal map according as det uA < 0, resp. det uA > 0.
The inversion in a circle is a 0-rotatory elliptic-inversion and the antipodal map is a 1-rotatory elliptic inversion of H 3 .
Proof. I. Let the matrix A induce an orientation-preserving isometry. Let λ and µ be eigenvalues of A. So, (1) If c(A) is non-real, then the imaginary part of re iθ + 1 r e −iθ must be non-zero. This is possible if and only if r = 1 and θ = 0, π. Thus A acts as a 1-rotatory hyperbolic and is different from a stretch half-turn.
Let bc = 0. Then b must be a real multiple of c. Let b = uc = ure iθ , u ∈ R. Multiplying A by e −iθ , without loss of generality, we may assume A = a b c d , b, c ∈ R. The equation B = AĀ = λI also gives us, cā + dc = 0. Since c is a real, we haveā + d = 0, i.e.ā = −d.
Hence, A = a b c −ā . The induced Möbius transformation by Aσ 0 is Suppose c = 0. Note that f has a fixed point on C if and only if f (z) = z has a solution. We see that The subgroup N is normal in AN, and AN is in fact a semi-direct product of N by A, where A ≈ R >0 acts on N ≈ R n−1 by x → rx.
A general element of AN is g a • f r , which acts on H n by (x, x n ) → (rx + a, rx n ).
By the identification of AN with H n as indicated above, we see that the element g a • f r corresponds to the point (a, r).
Theorem B.1. 1. The non-identity conjugacy classes in H n are represented by f r , r = 1, and g a , a = 0.
i) The conjugacy class of f r , r = 1, in the above identification, corresponds to the hyperplane x n = r.
ii) The conjugacy class of g a , a = 0 corresponds to the ray (ra, 1), r > 0, in the hyperplane x n = 1.

2.
There are only two non-identity z-classes, which can be visualized as (i) the complement of the hyperplane x n = 1 in H n , and (ii) the hyperplane x n = 1, with puncture at (0, 1).
Proof. Let f = g a • f r , r = 1. Let x 0 = −(r − 1) −1 a. Then g x 0 f g −1 x 0 = f r . It is easy to see that f r is not conjugate to f s , r = s. Now suppose r = 1. Then f = g a . Now for any element h = g b • f s , we have hf h −1 = g sa .
Thus g a is conjugate to g sa , s > 0, and if b = ta for some t > 0, then g b is not conjugate to g a .
Hence the conjugacy classes in AN are represented by f r , for r > 0, and g a , where a is unique up to a multiplication by r > 0. In the above identification of AN with H n , the conjugacy class of f r corresponds to {(a, r) | a ∈ R n−1 }, and the conjugacy class of g a corresponds to {(x, 1) | x = ra for r > 0}.
For a = 0, the group Z(g a ) = N is normal. For r, s = 1, Z(f r ) = Z(f s ) = A. Since for r = 1, g a • f r is conjugate to f r , it follows that for all r = 1, all Z(f r ), and Z(g a • f r ) form a single conjugacy class of subgroups. Correspondingly there are exactly two non-identity z-classes in H n . The z-class corresponding to f r is given by The z-class of g a is given by {(x, 1)}| x = 0}.