Lecture 10 Conformal Unit Disk

The upper half-plane is a universal object for all three EPH cases, which was obtained in a uniform fashion considering two-dimensional homogeneous spaces of SL₂(ℝ). However, universal models are rarely well-suited to all circumstances. For example, it is more convenient, for many reasons, to consider the compact unit disk in ℂ rather than the unbounded upper half-plane:

“…the reader must become adept at frequently changing from one model to the other as each has its own particular advantage.” [24]*§ 7.1

Of course, both models are conformally-isomorphic through the Cayley transform.

We derive similar constructions for parabolic and hyperbolic cases in this chapter. However, we shall see that there is not a “universal unit disk”. Instead, we obtain something specific in each EPH case from the same upper half-plane. As has already happened on several occasions, the elliptic and hyperbolic cases are rather similar and, thus, it is the parabolic case which requires special treatment.

10.1 Elliptic Cayley Transforms

In the elliptic and hyperbolic cases, the Cayley transform is given by the matrix Y_σ=(

), where σ=ι² and detY_σ =2. The matrix Y_σ produces the respective Möbius transform ℝ_σ→ ℝ_σ:

The same matrix Y_σ acts by conjugation g_Y=1/2Y_σgY_σ⁻¹ on an element g∈SL₂(ℝ):

The connection between the two forms (1) and (2) of the Cayley transform is g_Y Y_σw= Y_σ(gw), i.e. Y_σ intertwines the actions of g and g_Y.

The Cayley transform in the elliptic case is very important [240]*§ IX.3 [321]*Ch. 8, (1.12), both for complex analysis and representation theory of SL₂(ℝ). The transformation g↦ g_Y (2) is an isomorphism of the groups SL₂(ℝ) and SU(1,1). Namely, in complex numbers, we have

The group SU(1,1) acts transitively on the elliptic unit disk. The images of the elliptic actions of subgroups A, N, K are given in Fig. 10.3(E). Any other subgroup is conjugated to one of them and its class can be easily distinguished in this model by the number of fixed points on the boundary—two, one and zero, respectively. A closer inspection demonstrates that there are always two fixed points on the whole plane. They are either

• one strictly inside and one strictly outside of the unit circle,
• one double fixed point on the unit circle, or
• two different fixed points exactly on the circle.

Consideration of Fig. 7.1(b) shows that the parabolic subgroup N is like a phase transition between the elliptic subgroup K and hyperbolic A, cf. (2). Indeed, if a fixed point of a subgroup conjugated to K approaches a place on the boundary, then the other fixed point moves to the same place on the unit disk from the opposite side. After they collide to a parabolic double point on the boundary, they may decouple into two distinct fixed points on the unit disk representing a subgroup conjugated to A.

In some sense the elliptic Cayley transform swaps complexities. In contrast to the upper half-plane, the K-action is now simple but A and N are not. The simplicity of K orbits is explained by diagonalisation of the corresponding matrices:

These diagonal matrices generate Möbius transformations which correspond to multiplication by the unimodular scalar e^{2i φ}. Geometrically, they are isometric rotations, i.e. they preserve distances d_e,e (2) and length l_ce.

10.2 Hyperbolic Cayley Transform

A hyperbolic version of the Cayley transform was used in [170]. The above formula (2) in ℝ_h becomes

with some subtle differences in comparison to (3). The corresponding A, N and K orbits are given in Fig. 10.3(H). However, there is an important distinction between the elliptic and hyperbolic cases similar to the one discussed in Section 8.2.

Recall from Section 8.2 that we defined ℂ′ to be the two-fold cover of the hyperbolic point space ℝ_h consisting of two isomorphic copies ℝ_h⁺ and ℝ_h⁻ glued together, cf. Fig. 8.3. The conformal version of the hyperbolic unit disk in ℂ′ is, cf. the upper half-plane from (1),

We call ℂ′ the hyperbolic unit cycle in ℝ_h. Figure 8.3(b) illustrates the geometry of the hyperbolic unit disk in ℂ′ compared to the upper half-plane. We can also say, rather informally, that the hyperbolic Cayley transform maps the “upper” half-plane onto the “inner” part of the unit disk.

One may wish that the hyperbolic Cayley transform diagonalises the action of subgroup A, or some conjugate of it, in a fashion similar to the elliptic case (4) for K. Geometrically, it corresponds to hyperbolic rotations of the hyperbolic unit disk around the origin. Since the origin is the image of the point ι in the upper half-plane under the Cayley transform, we will use the isotropy subgroup . Under the Cayley map (7), an element of the subgroup becomes

where e^{є t} = cosht +є sinht. The corresponding Möbius transformation is a multiplication by e^{2є t}, which obviously corresponds to isometric hyperbolic rotations of ℝ_h for distance d_h,h and length l_ch. This is illustrated in Fig. 10.1(H:A’).

10.3 Parabolic Cayley Transforms

The parabolic case benefits from a larger variety of choices. The first, natural, attempt is to define a Cayley transform from the same formula (1) with the parabolic value σ=0. The corresponding transformation is defined by the matrix (

) and, geometrically, produces a shift one unit down.

However, within the extended FSCc, a more general version of the parabolic Cayley transform is also possible. It is given by the matrix

Here, σ_c=−1 corresponds to the parabolic Cayley transform P_e with an elliptic flavour, and σ_c=1 to the parabolic Cayley transform P_h with a hyperbolic flavour. Finally, the parabolic-parabolic transform P_p is given by the upper-triangular matrix mentioned at the beginning of this section.

Figure 10.3 presents these transforms in rows P_e, P_p and P_h, respectively. The row P_p almost coincides with Fig. 1.1 and the parabolic case in Fig. 1.2. Consideration of Fig. 10.3 by following the columns from top to bottom gives an impressive mixture of many common properties (e.g. the number of fixed points on the boundary for each subgroup) with several gradual mutations.

The description of the parabolic unit disk admits several different interpretations in terms of lengths from Definition 9.

The above descriptions (1 and 3) are attractive for reasons given in the following two exercises. Firstly, we note that K-orbits in the elliptic case (Fig. 10.1(E:K)) and -orbits in the hyperbolic case (Fig. 10.1(H:A’)) of the Cayley transform are concentric.

Secondly, we observe that Cayley images of the isotropy subgroups’ orbits in elliptic and hyperbolic spaces in Fig. 10.1(E:K) and (H:A) are equidistant from the origin in the corresponding metrics.

We will introduce linear structures preserved by actions of the subgroups N and N′ on the parabolic unit disk in Chapter 11.

Orbits of the isotropy subgroups , N′ and K from Exercise 20 under the Cayley transform are shown in Fig. 10.1, which should be compared with the action of the same subgroup on the upper half-plane in Fig. 3.1.

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Figure 10.1: Action of the isotropy subgroups of ι under the Cayley transform—subgroup K in the elliptic case, N′ in the parabolic and in the hyperbolic. Orbits of K and are concentric while orbits of N′ are confocal. We also provide orbits of N which are concentric in the parabolic case. The action of K, N′ and on the upper half-plane are presented in Fig. 3.1.

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10.4 Cayley Transforms of Cycles

The next natural step within the FSCc is to expand the Cayley transform to the space of cycles.

10.4.1 Cayley Transform and FSSc

The effect of the Cayley transform on cycles turns out to be a cycle similarity in the elliptic and hyperbolic cases only, the degeneracy of the parabolic case requires a special treatment.

The above extension of the Cayley transform to the cycle space is linear, but in the parabolic case it is not expressed as a similarity of matrices (reflections in a cycle). This can be seen, for example, from the fact that the parabolic Cayley transform does not preserve the zero-radius cycles represented by matrices with zero p-determinant.

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Figure 10.2: Cayley transforms in elliptic (σ=−1), parabolic (σ=0) and hyperbolic (σ=1) spaces. In each picture, the reflection of the real line in the green cycles (drawn continuously or dotted) is the blue “unit cycle”. Reflections in the solidly-drawn cycles send the upper half-plane to the unit disk and reflections in the dashed cycles send it to its complement. Three Cayley transforms in the parabolic space (σ=0) are themselves elliptic (σc=−1), parabolic (σc=0) and hyperbolic (σc=1), giving a gradual transition between proper elliptic and hyperbolic cases.

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Since orbits of all subgroups in SL₂(ℝ), as well as their Cayley images, are cycles in the corresponding metrics, we may use Exercises 2(p) and 2 to prove the following statements (in addition to Exercise 6).

Since the action of the parabolic Cayley transform on cycles does not preserve zero-radius cycles, one would be better using infinitesimal-radius cycles from Section 7.5 instead. Indeed, the Cayley transform preserves infinitesimality.

We recall a useful expression of concurrence with an infinitesimal cycle’s focus through f-orthogonality from Exercise 2. Some caution is required since f-orthogonality of generic cycles is not preserved by the parabolic Cayley transform, just like it is not preserved by cycle similarity in Exercise 2(p). A remarkable exclusion happens for infinitesimal cycles.

10.4.2 Geodesics on the Disks

We apply the advise quoted at the beginning of this chapter to the invariant distance discussed in Chapter 9. The equidistant curves and respective geodesics in ℝ_σ are cycles, they correspond to certain cycles on the unit disks. As on many previous occasions we need to distinguish the elliptic and hyperbolic cases from the parabolic one.

The appearance of straight lines as geodesics in the elliptic and hyperbolic disks is an illustration of its “own particular advantage” mentioned in the opening quote to this chapter.

We will look closer on isometric transformations the unit disks in the next chapter.

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Figure 10.3: EPH unit disks and actions of one-parameter subgroups A, N and K.

(E): The elliptic unit disk.

(Pe), (Pp), (Ph): The elliptic, parabolic and hyperbolic flavours of the parabolic unit disk.

(H): The hyperbolic unit disk.

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