Lecture 14 Poincaré Extension of Möbius Transformations

Given sphere preserving (Möbius) transformations in n-dimensional Euclidean space one can use the Poincaré extension to obtain sphere preserving transformations in a half-space of n+1 dimensions. The Poincaré extension is usually provided either by an explicit formula or by some geometric construction. We investigate its algebraic background and describe all available options. The solution is given in terms of one-parameter subgroups of Möbius transformations acting on triples of quadratic forms. To focus on the concepts, this paper deals with the Möbius transformations of the real line only.

It is not surprising that we will arrive to the three possible situation—elliptic, parabolic, hyperbolic—considered above. The interesting feature of this chapter is what we completely avoid usage of complex, dual and double numbers which were are permanent companions so far.

14.1 Introduction

It is known, that Möbius transformations on ℝⁿ can be expanded to the “upper” half-space in ℝⁿ⁺¹ using the Poincaré extension [24]*§ 3.3 [273]*§ 5.2. An explicit formula is usually presented without a discussion of its origin. In particular, one may get an impression that the solution is unique. This paper considers various aspects of such extension and describes different possible realisations. Our consideration is restricted to the case of extension from the real line to the upper half-plane. However, we made an effort to present it in a way, which allows numerous further generalisations.

14.2 Geometric construction

We start from the geometric procedure in the standard situation. The group SL₂(ℝ) consists of real 2× 2 matrices with the unit determinant. SL₂(ℝ) acts on the real line by linear-fractional maps:

A pair of (possibly equal) real numbers x and y uniquely determines a semicircle C_xy in the upper half-plane with the diameter [x,y]. For a linear-fractional transformation M (1), the images M(x) and M(y) define the semicircle with the diameter [M(x),M(y)], thus, we can define the action of M on semicircles: M(C_xy)=C_M(x)M(y). Geometrically, the Poincaré extension is based on the following lemma, see Fig. 14.1(a) and more general Lem. 18 below:

Elementary geometry of right triangles tells that a pair of intersecting intervals [x,y], [x′,y′], where x<x′<y<y′, defines the point

An alternative demonstration uses three observations:

1. the scaling x↦ ax, a>0 on the real line produces the scaling (u,v) ↦(au,av) on pairs (2);
2. the horizontal shift x↦ x+b on the real line produces the horizontal shift (u,v) ↦ (u+b,v) on pairs (2);
3. for the special case y=−x⁻¹ and y′=−x′⁻¹ the pair (2) is (0,1).

Finally, expression (2), as well as (3)–(4) below, can be calculated by the specialised CAS for Möbius invariant geometries [186, 211].

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(a)   (b) 
(c)   (d) 
(e)   (f) 

Figure 14.1: Poincaré extensions: first column presents points defined by the intersecting intervals [x,y] and [x′,y′], the second column—by disjoint intervals. Each row uses the same type of conic sections—circles, parabolas and hyperbolas respectively. Pictures are produced by the software [211].

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This standard approach can be widened as follows. The above semicircle can be equivalently described through the unique circle passing x and y and orthogonal to the real axis. Similarly, an interval [x,y] uniquely defines a right-angle hyperbola in ℝ² orthogonal to the real line and passing (actually, having her vertices at) (x,0) and (y,0). An intersection with the second such hyperbola having vertices (x′,0) and (y′,0) defines a point with coordinates, see Fig. 14.1(f):

where x<y<x′<y′. Note, the opposite sign of the product under the square roots in (2) and (3).

If we wish to consider the third type of conic sections—parabolas—we cannot use the unaltered procedure: there is no a non-degenerate parabola orthogonal to the real line and intersecting the real line in two points. We may recall, that a circle (or hyperbola) is orthogonal to the real line if its centre belongs to the real line. Analogously, a parabola is focally orthogonal (see [198]*§ 6.6 for a general consideration) to the real line if its focus belongs to the real line. Then, an interval [x,y] uniquely defines a downward-opened parabola with the real roots x and y and focally orthogonal to the real line. Two such parabolas defined by intervals [x,y] and [x′,y′] have a common point, see Fig. 14.1(c):

where D=±√(x−x′)(y−y′)(y−x)(y′−x′). For pencils of such hyperbolas and parabolas respective variants of Lem. 1 hold.

Focally orthogonal parabolas make the angle 45^∘ with the real line. This suggests to replace orthogonal circles and hyperbolas by conic sections with a fixed angle to the real line, see Fig. 14.1(b)–(e). Of course, to be consistent this procedure requires a suitable modification of Lem. 1, we will obtain it as a byproduct of our study, see Lem. 18. However, the respective alterations of the above formulae (2)–(4) become more complicated in the general case.

The considered geometric construction is elementary and visually appealing. Now we turn to respective algebraic consideration.

14.3 Möbius transformations and Cycles

The group SL₂(ℝ) acts on ℝ² by matrix multiplication on column vectors:

A linear action respects the equivalence relation (

) ∼ (

), λ≠ 0 on ℝ². The collection of all cosets for non-zero vectors in ℝ² is the projective line Pℝ¹. Explicitly, a non-zero vector (

)∈ℝ² corresponds to the point with homogeneous coordinates [x₁:x₂]∈ Pℝ¹. If x₂≠ 0 then this point is represented by [x₁/x₂:1] as well. The embedding ℝ → Pℝ¹ defined by x ↦ [x:1], x∈ℝ covers the all but one of points in Pℝ¹. The exceptional point [1:0] is naturally identified with the infinity.

The linear action (5) induces a morphism of the projective line Pℝ¹, which is called a Möbius transformation. Considered on the real line within Pℝ¹, Möbius transformations takes fraction-linear form:

This SL₂(ℝ)-action on Pℝ¹ is denoted as g: x↦ g· x. We note that the correspondence of column vectors and row vectors i: (

) ↦ (x₂, −x₁) intertwines the left multiplication L_g (5) and the right multiplication R_g⁻¹ by the inverse matrix:

We extended the map i to 2× 2-matrices by the rule:

Two columns (

) and (

) form the 2× 2 matrix M_xy=(

). For geometrical reasons appearing in Cor. 3, we call a cycle the 2× 2-matrix C_xy defined by

We note that detC_xy=−(x−y)²/4, thus detC_xy=0 if and only if x=y. Also, we can consider the Möbius transformation produced by the 2× 2-matrix C_xy and calculate:

Thus, points [x:1], [y:1]∈ Pℝ¹ are fixed by C_xy. Also, C_xy swaps the interval [x,y] and its complement.

Due to their structure, matrices C_xy can be parametrised by points of ℝ³. Furthermore, the map from ℝ²→ ℝ³ given by (x,y)↦ C_xy naturally induces the projective map (Pℝ¹)² → Pℝ² due to the identity:

Conversely, a zero-trace matrix (

) with a non-positive determinant is projectively equivalent to a product C_xy (8) with x,y=a±√a²+bc/c. In particular, we can embed a point [x:1]∈ Pℝ¹ to 2× 2-matrix C_xx with zero determinant.

The combination of (5)–(8) implies that the correspondence (x,y)↦ C_xy is SL₂(ℝ)-covariant in the following sense:

To achieve a geometric interpretation of all matrices, we consider the bilinear form Q: ℝ²× ℝ²→ ℝ generated by a 2× 2-matrix (

):

Due to linearity of Q, the null set

factors to a subset of Pℝ¹× Pℝ¹. Furthermore, for the matrices C_xy (8), a direct calculation shows that:

The map i appearance in (13) is justified once more by the following result.

Alternatively, the identities C_xy(x)=C_xy(y)=0 follows from (9) and the fact that i(z) is orthogonal to z for all z∈ℝ². Also, we note that:

Motivated by Lem. 2, we call ⟨ C_xy,C_x′y′ ⟩:=−tr(C_xy C_x′y′) the pairing of two cycles. It shall be noted that the pairing is not positively defined, this follows from the explicit expression (13). The sign is chosen in such way, that

Also, an immediate consequence of Lem. 2 or identity (11) is

From (13), the null set (12) of the form Q=C_xy can be associated to the family of cycles {C_x′y′  ∣  ⟨ C_xy,C_x′y′ ⟩=0, (x′,y′)∈ℝ²× ℝ²} which we will call orthogonal to C_xy.

14.4 Extending cycles

Since bilinear forms with matrices C_xy have numerous geometric connections with Pℝ¹, we are looking for a similar interpretation of the generic matrices. The previous discussion identified the key ingredient of the recipe: SL₂(ℝ)-invariant pairing (13) of two forms. Keeping in mind the structure of C_xy, we will parameterise¹ a generic 2× 2 matrix as (

) and consider the corresponding four dimensional vector (n,l,k,m). Then, the similarity with (

) ∈SL₂(ℝ):

corresponds to the linear transformation of ℝ⁴, cf. [198]*Ex. 4.15:

In particular, this action commutes with the scaling of the first component:

This expression is helpful in proving the following statement.

Proof. Let T be 4× 4 a matrix from (15), if a SL₂(ℝ)-invariant pairing is defined by a 4× 4 matrix J=(j_fg), then T′JT=J, where T′ is transpose of T. The equivalent identity T′J=JT⁻¹ produces a system of homogeneous linear equations which has the generic solution:

with four free variables j₁₁, j₄₂, j₄₃ and j₄₄. Since a solution shall not depend on a, b, c, d, we have to put j₄₂=j₄₄=0. Then by the homogeneity of the identity T′J=JT⁻¹, we can scale j₄₃ to 1. Thereafter, an independent (sign-preserving) scaling (16) of n leaves only three non-isomorphic values −1, 0, 1 of j₁₁.

The appearance of the three essential different cases σ_c =−1, 0 or 1 in Lem. 5 is a manifestation of the common division of mathematical objects into elliptic, parabolic and hyperbolic cases [185] [198]*Ch. 1. Thus, we will use letters “e”, “p”, “h” to encode the corresponding three values of σ_c.

Now we may describe all SL₂(ℝ)-invariant pairings of bilinear forms.

Note that we can explicitly write Q_τ for Q= (

) as follows:

In particular, Q_h=−Q⁻¹ and Q_p=1/2(Q_e+Q_h). Furthermore, Q_p has the same structure as C_xy. Now, we are ready to extend the projective line Pℝ¹ to two dimensions using the analogy with properties of cycles C_xy.

If a form Q= (

) has k≠ 0 then we can scale it to obtain k=1, this form of Q is called normalised. A normalised τ-isotropic form is completely determined by its diagonal values: (

). Thus, the set of such forms is in a one-to-one correspondence with points of ℝ². Finally, a form Q= (

) is e-orthogonal to the τ-isotropic form (

) if:

that is the point (u,v)∈ℝ² belongs to the quadratic curve with coefficients (k,l,n,m).

14.5 Homogeneous spaces of cycles

Obviously, the group SL₂(ℝ) acts on Pℝ¹ transitively, in fact it is even 3-transitive in the following sense. We say that a triple {x₁,x₂,x₃}⊂ Pℝ¹ of distinct points is positively oriented if

where we agree that the ideal point ∞∈ Pℝ¹ is greater than any x∈ℝ. Equivalently, a triple {x₁,x₂,x₃} of reals is positively oriented if:

Also, a triple of distinct points, which is not positively oriented, is negatively oriented. A simple calculation based on the resolvent-type identity:

shows that both the positive and negative orientations of triples are SL₂(ℝ)-invariant. On the other hand, the reflection x↦ −x swaps orientations of triples. Note, that the reflection is a Moebius transformation associated to the cycle

A significant amount of information about Möbius transformations follows from the fact, that any continuous one-parametric subgroup of SL₂(ℝ) is conjugated to one of the three following subgroups²:

where t∈ℝ. Also, it is useful to introduce subgroups and N′ conjugated to A and N respectively:

Thereafter, all three one-parameter subgroups , N′ and K consist of all matrices with the universal structure

We use the notation H_τ for these subgroups. Again, any continuous one-di­men­sio­nal subgroup of SL₂(ℝ) is conjugated to H_τ for an appropriate τ.

We note, that matrices from A, N and K with t≠ 0 have two, one and none different real eigenvalues respectively. Eigenvectors in ℝ² correspond to fixed points of Möbius transformations on Pℝ¹. Clearly, the number of eigenvectors (and thus fixed points) is limited by the dimensionality of the space, that is two. For this reason, if g₁ and g₂ take equal values on three different points of Pℝ¹, then g₁=g₂.

Also, eigenvectors provide an effective classification tool: g∈SL₂(ℝ) belongs to a one-dimensional continuous subgroup conjugated to A, N or K if and only if the characteristic polynomial det(g−λ I) has two, one and none different real root(s) respectively. We will illustrate an application of fixed points techniques through the following well-known result, which will be used later.

Proof. Often, the statement is quickly demonstrated through an explicit expression for φ, cf. [26]*Thm. 13.2.1. We will use properties of the subgroups A, N and K to describe an algorithm to find such a map. First, we notice that it is sufficient to show the Lemma for the particular case y₁=0, y₂=1, y₃=∞. The general case can be obtained from composition of two such maps. Another useful observation is that the fixed point for N, that is ∞, is also a fixed point of A.

Now, we will use subgroups K, N and A in order of increasing number of their fixed points. First, for any x₃ the matrix g′=(

)∈ K such that cott=x₃ maps x₃ to y₃=∞. Let x₁′=g′x₁ and x₂′=g′x₂. Then the matrix g″=(

)∈ N, fixes ∞=g′ x₃ and sends x′₁ to y₁=0. Let x″₂=g″x₂′, from positive orientation of triples we have 0<x″₂<∞. Next, the matrix g‴=(

)∈ A with a=√x″₂ sends x″₂ to 1 and fixes both ∞=g″g′x₃ and 0=g″g′ x₁. Thus, g=g‴g″g′ makes the required transformation (x₁,x₂,x₃)↦(0,1,∞).

We will denote by φ_XY the unique map from Lem. 8 defined by triples X={x₁,x₂,x₃} and Y={y₁,y₂,y₃}.

Although we are not going to use it in this paper, we note that the following important result [240]*§ III.1 is an immediate consequence of our proof of Lem. 8.

In particular, we note that it is not a coincidence that the subgroups appear in the Iwasawa decomposition SL₂(ℝ)=ANK in order of decreasing number of their fixed points.

14.6 Triples of intervals

We change our point of view and instead of two ordered triples of points consider three ordered pairs, that is—three intervals. For them we will need the following definition.

Aligned triples determine certain one-parameter subgroups of Möbius transformations as follows:

Proof. Consider the one-parameter subgroup of ψ(t)⊂ SL₂(ℝ) such that φ_XY=ψ(1). Note, that ψ(t) and φ_XY have the same fixed points (if any) and no point x_j is fixed since x_j≠ y_j. If the number of fixed points is less than 2, then ψ(t)x₁, t∈ℝ produces the entire real line except a possible single fixed point. Therefore, there are t₂ and t₃ such that ψ(t₂)x₁=x₂ and ψ(t₃)x₁=x₃. Since ψ(t) and φ_XY commute for all t we also have:

If there are two fixed points x<y, then the open interval (x,y) is an orbit for the subgroup ψ(t). Since all x′₁, x′₂ and x′₃ belong to this orbit and C_xy commutes with φ_XY we may repeat the above reasoning for the dashed intervals [x′_j,y′_j]. Finally, x″_j=C_xyx′_j and y″_j=C_xyy′_j, where C_xy commutes with φ and ψ(t_j), j=2, 3. Uniqueness of the subgroup follows from Lemma 13.

The group SL₂(ℝ) acts transitively on collection of all cycles C_xy, thus this is a SL₂(ℝ)-homogeneous space. It is easy to see that the fix-group of the cycle C_−1,1 is  (21). Thus the homogeneous space of cycles is isomorphic to SL₂(ℝ)/.

Proof. Since H is conjugated either to , N′ or K, the homogeneous space X=SL₂(ℝ)/H is isomorphic to the upper half-plane in double, dual or complex numbers [198]*§ 3.3.4. Orbits of one-parameter continuous subgroups in X are conic sections, which are circles, parabolas (with vertical axis) or equilateral hyperbolas (with vertical axis) for the respective type of geometry. Any two different orbits of the same type intersect at most at two points, since an analytic solution reduces to a quadratic equation.

Alternatively, we can reformulate Prop. 12 as follows: three different cycles C_x1y1, C_x2y2, C_x3y3 define a one-parameter subgroup, which generate either one orbit or two related orbits passing the three cycles.

We have seen that the number of fixed points is the key characteristics for the map φ_XY. The next result gives an explicit expression for it.

Proof. If a Möbius transformation (

) maps x₁↦ y₁, x₂↦ y₂, x₃↦ y₃ and s↦ s, then we have a homogeneous linear system, cf. [26]*Ex. 13.2.4:

A non-zero solution exists if the determinant of the 4× 4 matrix is zero. Expanding it over the last row and rearranging terms we obtain the quadratic equation for the fixed point s:

The value (23) is the discriminant of this equation.

If y=g₀· x for some g₀∈ H_τ, then for any g∈ H_τ we also have y_g=g₀· x_g, where x_g=g· x_g and y_g=g· y_g. Thus, we demonstrated the first part of the following result.

The family P_τ,t consists of the eigenvectors of the 4× 4 matrix from (15) with suitably substituted entries. There is (up to a factor) exactly one τ-isotropic form in P_τ,t, namely (

). We denote this form by ι. It corresponds to the point (0,1)∈ℝ² as discussed after Defn. 7. We may say that the subgroup H_τ fixes the point ι, this will play an important rôle below.

14.7 Geometrisation of cycles

We return to the geometric version of the Poincaré extension considered in Sec. 14.2 in terms of cycles. Cycles of the form (

) are τ-isotropic for any τ and are parametrised by the point x of the real line. For a fixed τ, the collection of all τ-isotropic cycles is a larger set containing the image of the real line from the previous sentence. Geometrisation of this embedding is described in the following result.

Proof. The first statement is verified by a short calculation. A form Q= (

) in the second statement may be calculated from the homogeneous system:

which has the rank 3 if x≠ y. The third statement can be checked by a calculation as well. Finally, the last item is a particular case of the more general statement as indicated. Yet, we can derive it here from the implicit derivative dv/du=ku−l/n of the function k(u²+τ v²)−2lu−2nv+m=0 (18) at the point (u,0). Note that this value is independent from τ. Since this is the tangent of the intersection angle with the real line, the square of the cosine of this angle is:

if ku²−2 u l+m=0.

Also, we note that, the independence of the left-hand side of (26) from x can be shown from basic principles. Indeed, for a fixed t the subgroup H_τ acts transitively on the family of triples {x, x+τ t/tx+1, ι}, thus H_τ acts transitively on all bilinear forms orthogonal to such triples. However, the left-hand side of (26) is SL₂(ℝ)-invariant, thus may not depend on x. This simple reasoning cannot provide the exact expression in the right-hand side of (26), which is essential for the geometric interpretation of the Poincaré extension.

To restore a cycle from its intersection points with the real line we need also to know its cycle product with the real line. If this product is non-zero then the sign of the parameter n is additionally required. At the cycles’ language, a common point of cycles C and S is encoded by a cycle Ĉ such that:

For a given value of τ, this produces two linear and one quadratic equation for parameters of Ĉ. Thus, a pair of cycles may not have a common point or have up to two such points. Furthermore, Möbius-invariance of the above conditions (26) and (27) supports the geometrical construction of Poincaré extension, cf. Lem. 1:

Summarising the geometrical construction, the Poincaré extension based on two intervals and the additional data produces two situations:

1. If the cycles C and C′ are orthogonal to the real line, then a pair of overlapping cycles produces a point of the elliptic upper half-plane, a pair of disjoint cycles defines a point of the hyperbolic. However, there is no orthogonal cycles uniquely defining a parabolic extension.
2. If we admit cycles, which are not orthogonal to the real line, then the same pair of cycles may define any of the three different types (EPH) of extension.

These peculiarities make the extension based on three intervals, described above, a bit more preferable.

14.8 Concluding remarks

Based on the consideration in Sections 14.3– 14.7 we describe the following steps to carry out the generalised extension procedure:

1. Points of the extended space are equivalence classes of aligned triples of cycles in Pℝ¹, see Defn. 11. The equivalence relation between triples will emerge at step 3.
2. A triple T of different cycles defines the unique one-parameter continues subgroup S(t) of Möbius transformations as defined in Prop. 12.
3. Two triples of cycles are equivalent if and only if the subgroups defined in step 2 coincide (up to a parametrisation).
4. The geometry of the extended space, defined by the equivalence class of a triple T, is elliptic, parabolic or hyperbolic depending on the subgroup S(t) being similar S(t)=gH_τ(t) g⁻¹, g∈SL₂(ℝ) (up to parametrisation) to H_τ (22) with τ=−1, 0 or 1 respectively. The value of τ may be identified from the triple using Prop. 14.
5. For the above τ and g∈SL₂(ℝ), the point of the extended space, defined by the the equivalence class of a triple T, is represented by τ-isotropic (see Defn. 7(2)) bilinear form g⁻¹ (

   1	τ
   1	1

   ) g, which is S-invariant, see the end of Section 14.6.

Obviously, the above procedure is more complicated that the geometric construction from Section 14.2. There are reasons for this, as discussed in Section 14.7: our procedure is uniform and we are avoiding consideration of numerous subcases created by an incompatible selection of parameters. Furthermore, our presentation is aimed for generalisations to Möbius transformations of moduli over other rings. This can be considered as an analog of Cayley–Klein geometries [339]*Apps. A–B [284].

It shall be rather straightforward to adopt the extension for ℝⁿ. Möbius transformations in ℝⁿ are naturally expressed as linear-fractional transformations in Clifford algebras [65] with a similar classification of subgroups based on fixed points [2, 343]. The Möbius invariant matrix presentation of cycles ℝⁿ is already known [65]*(4.12) [100] [208]*§ 5. Of course, it is necessary to enlarge the number of defining cycles from 3 to, say, n+2. This shall have a connection with Cauchy–Kovalevskaya extension considered in Clifford analysis [295, 310]. Naturally, a consideration of other moduli and rings may require some more serious adjustments in our scheme.

Our construction is based on the matrix presentations of cycles. This techniques is effective in many different cases [198, 208]. Thus, it is not surprising that such ideas (with some technical variation) appeared independently on many occasions [65]*(4.12) [100] [300]*§ 1.1 [163]*§ 4.2. The interesting feature of the present work is the complete absence of any (hyper)complex numbers. It deemed to be unavoidable [198]*§ 3.3.4 to employ complex, dual and double numbers to represent three different types of Möbius transformations extended from the real line to a plane. Also (hyper)complex numbers were essential in [198, 185] to define three possible types of cycle product (4), and now we managed without them.

Apart from having real entries, our matrices for cycles share the structure of matrices from [65]*(4.12) [100] [198] [185]. To obtain another variant, one replaces the map i (7) by

Then, we may define symmetric matrices in a manner similar to (8):

This is the form of matrices for cycles similar to [300]*§ 1.1 [163]*§ 4.2. The property (10) with matrix similarity shall be replaced by the respective one with matrix congruence: g· C_xy^t · g^t= C_x′y′^t. The rest of our construction may be adjusted for these matrices accordingly.

1

Further justification of this parametrisation will be obtained from the equation of a quadratic curve (18).

2

A reader may know that A, N and K are factors in the Iwasawa decomposition SL₂(ℝ)=ANK (cf. Cor. 10, however this important result does not play any rôle in our consideration.