The Geometry of Möbius Transformations John Olsen University of Rochester Spring 2010 Contents 1 Introduction 2 2 Affine Transformations 2 3 The Stereographic Projection 4 4 The Inversion Map 9 5 Möbius Transformations 11 6 The Cross Ratio 14 7 The Symmetry Principle and Maps of the Unit Disk and the Upper Halfplane 17 8 Conjugacy Classes in M(C ) 23 ∞ 9 Geometric Classification of Conjugacy Classes 26 10 Möbius Transformations of Finite Period 28 11 Rotations of C 28 ∞ 12 Finite Groups of Möbius Transformations 31 13 Bibliography 34 14 Problems 35 1 Introduction The purpose of these notes is to explore some basic properties of Möbius transforma- tions (linear fractional transformations) which are one-to-one, onto and conformal (angle preserving) maps of the socalled extended complex plane. We will develop the basic properties of these maps and classify the one-to-one and onto conformal maps of the unit disk and the upper half plane using the symmetry principle. The one-to-one, onto and conformal maps of the extended complex plane form a group denoted P SL2(C). We will study the conjugacy classes of this group and find an explicit invariant that determines the conjugacy class of a given map. We finish with a classification of the finite subgroups of P SL2(C). The theory of Möbius Transformations is developed without any use of and only one reference to complex analysis. This point of view certainly requires more work, but I feel the effort is worth it, since it allows somebody with no knowledge of complex analysis to study the subject. The prerequisite is some basic knowledge of group theory, which is certainly met if the students have taken an undergraduate algebra course. If not, a couple of lectures at the beginning of the course where one introduces the basics of group theory should suffice. I would like to thank John Harper and the students in the course Introduction to Geometry (MTH 250, Spring 2010) for valuable comments and proofreading. 2 Affine Transformations Let us briefly recall a few basic properties of the complex numbers. If z C, then we can write z = r(cos(θ)+ i sin(θ)), where r is the modulus, z , of z ∈and θ the argument, arg(z), of z. We will denote the real and imaginary part| | of z by (z) and < (z), respectively. = We have the following five basic maps, which we will study in the following: 1. z cz, c Ê, scaling 7→ ∈ 2. z z + A, A C, translation 7→ ∈ 3. z Az, A = eiθ, rotation 7→ 4. z z, complex conjugation 7→ 5. z 1 , inversion. 7→ z Definition 1. A direct affine transformation is a combination of (1), (2) and (3), i.e. a map of the form T (z)= Az + B. 2 Remark 1. Since we have T (z) T (z0) = (Az + B) (Az0 + B) = A z z0 , | − | | − | | || − | we see that a direct affine transformation is an isometry if and only if A =1. | | Lemma 2.1. The direct affine transformation T (z)= Az + B is a translation if and B only if A =1. If A =1, A =1 then T is a rotation about 1 A by an angle arg(A). | | 6 − Proof. If A = 1, then T (z) = z + B is a translation. If A = 1 and A = 1, we let F be a fixed point of T , i.e. a point where T (F )= F . Then| | we have F =6 AF + B, B which implies F = 1 A . Now − Az A2z + B BA B T (z) F = − − − − 1 A − Az A2z BA = − − 1 A Az(1 −A) B = − − 1 A = A(z −F ). − This shows that if A = 1 and A = 1 then T is a rotation about F by an angle | | 6 arg(A). Proposition 2.1. The set of direct affine transformations form a group under com- position, which is denoted by Aff(C). Proof. The identity map I(z)= z is the unit in the group Aff(C). Since composition of functions is always associative, we only need to check that the set is closed and that inverses exist. Let T (z)= Az + B and S(z)= A0z + B0, then we have S T = A0(Az + B)+ B0 = A0Az + A0B + B0, ◦ which is again an affine transformation. 1 To find an inverse to T (z)= Az + B, we guess that T − (z)= Cz + D. Then we 1 have T − T (z)= z which means that C(Az+B)+D = z. So we get CAz+CB+D = ◦ 1 1 z, which means that C = A− , and D = AB− . − Corollary 2.1. A direct affine transformation preserves circles and lines. Proof. A direct affine transformation, T (z)= Az+B, where A =1 is by Lemma 2.1 B | | a rotation about 1 A which clearly preserves circles and lines. A direct affine trans- − formation, T (z) = Az + B, where A = 1 can be written as T (z) = r(A0z + B0), | | 6 B0 where r is real and A0 =1. Again by Lemma 2.1 this map is a rotation about 1 A0 scaled by r, which preserves| | circles and lines. − 3 Remark 2. A map that preserves angles is called conformal. By the same argument as above, a direct affine transformation is conformal. 3 The Stereographic Projection b N P b C b O b z 2 3 Consider the unit sphere S in Ê , that is 2 3 2 2 2 S = (u,v,w) Ê u + v + w =1 . { ∈ | } We embed S2 such that the origin and the center of the sphere coincide. We identify 3 the complex plane with the equitorial plane. That is, for coordinates (u,v,w) of Ê , C is the plane where w =0. The North pole has coordinates N = (0, 0, 1). We will denote a complex number z by x+iy. Notice that the unit circle in C coincides with the equator of S2. For any point P S2, there is a unique line from N to P , which we denote by ∈ NP . This line intersects the complex plane in exactly one point z C. ∈ 2 Definition 2. The map from SP : S N C which assigns to a point P the point \ → C z C given by the intersection NP is called the stereographic projection. ∈ ∩ We will now derive the coordinate expressions for both stereographic projection and its inverse. 4 2 Proposition 3.1. For P S N and z C, P = (u,v,w) and z = x + iy ∈ \ ∈ 2 the coordinates of the stereographic projection SP : S N C and its inverse 1 2 \ → SP − : C S N are given by: → \ u v SP ((u,v,w)) = + i (1) 1 w 1 w − − and 2 2 1 2x 2y x + y 1 SP − (x + iy) = , , − (2) x2 + y2 +1 x2 + y2 +1 x2 + y2 +1 z + z i(z z) zz 1 = , − , − . (3) zz +1 zz +1 zz +1 Proof. N 1 w − L P ρ w O r z From the picture we see that NO = 1, LO = w, LN = 1 w, zO = r = x2 + y2. Set P L = ρ and note| that|ρ = √|u2 +| v2 + w|2 |w2 = −√u2 +| v2.| | | − Since the triangles NLP and NOZ are similar we have p ρ 1 w u v = − = = . r 1 x y from which is follows that 1 w u u − = x = 1 x ⇒ 1 w 1 w v −v − = y = . 1 y ⇒ 1 w − u v This proves that SP ((u,v,w)) = 1 w + i 1 w . For the second statement we observe− that− since z = x + iy we have u v u + iv z = + i = 1 w 1 w 1 w −u −v u − iv z = i = − 1 w − 1 w 1 w − − − 5 This implies that u + iv u iv x2 + y2 = zz = − = 1 w 1 w − − u2 + v2 (1 w)(1 + w) = = − = (1 w)2 (1 w)2 − − 1+ w 1 = = 1+ . 1 w − 1 w − − where we used u2 + v2 =1 w2 in the fourth equality. This gives − 2 x2 + y2 1 zz 1 1 w = w = − = − . − x2 + y2 +1 ⇒ x2 + y2 +1 zz +1 Since we have u + iv u + iv x + iy = 1 w = 1 w ⇒ − x + iy − and we get 2 2x +2iy u + iv = (1 w)(x + iy)= (x + iy)= . − x2 + y2 +1 x2 + y2 +1 Comparing real and imaginary parts gives 2x z + z 2y i(z z) u = = v = = − , x2 + y2 +1 zz +1 x2 + y2 +1 zz +1 which finishes the proof. C Definition 3. The extended complex plane is given by C = . ∞ ∪ {∞} Remark 3. If we identify, via stereographic projection, points in the complex plane with points in S2 N and further identify with N then we have a bijection between \ 2∞ 2 the extended complex plane C and S . Under this identification S is known as ∞ the Riemann sphere. 2 It is clear that stereographic projection is continuous as a map S N C with a continuous inverse, since both maps are given as fractions of polynomials\ → where the denominator is never zero. With the above identification of N with we get 2 1 ∞ 2 C a continuous map SP : S C with continuous inverse, SP − : S . A ∞ ∞ continuous map with continuous→ inverse is called a homeomorphism. → We note that since S2 is compact and stereographic projection is a homeomor- 1 C phism, C is compact as well. The space is a socalled one-point compactifica- ∞ ∞ tion of C. 1 There are other one-point compatifications of C. 6 The stereographic projection gives a way of mapping a region of the sphere onto a plane.