THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE YONG JAE KIM Abstract. In order to study the geometries of a hyperbolic plane, it is necessary to understand the set of transformations that map from the space to itself. This paper shows that in the Poincaré half-plane model, this group of transformations is the general Möbius group. This is done by showing that the general Möbius group is a group of homeomorphisms that map circles to circles on the extended complex plane Ĉ, and that all homeomorphisms that send circles to circles on Ĉ are Möbius transformations. ontents 1. Introduction 1 2. The Half-Plane Model 2 3. Hyberbolic Lines as ircles on the Riemann Sphere Ĉ 4 4. Homeomorphisms on Ĉ 5 5. Möbius Transformations as Homeomorphisms 7 6. Transitivity Properties of Möb + 9 7. The General Möbius Group 10 Acknowledgments 12 References 12 1. Introduction A model of the hyperbolic plane is constructed by defining points and lines that are not necessarily consistent with the parallel postulate of Euclidean geometry. The parallel postulate states that in two-dimensional space, for any line L and point p not in L, there exists exactly one line through P that is parallel with L, where the term parallel is understood to indicate that the two lines do not intersect. There are multiple geometries that can be considered hyperbolic, but the model to be constructed in this paper is the Poincaré half-plane model. The goal of this paper is to find the automorphism group on this particular geometric construct. We will see that the set of Möbius transformations is this group. Using the notation to be elaborated later in this paper, this result can be expressed in the form Homeo (Ĉ) = Möb. Date: July 2013. 1

2 YONG JAE KIM 2. The Half-Plane Model The model that is relevant in this paper is the Poincaré half-plane model. The construction of this model starts by defining the upper half-plane H. Definition 2.1. H = {z Im(z) >0 } The half-plane H inherits the notions of points from the complex plane. Once the notion of hyperbolic lines is defined, we also prove that the half-plane also inherits a property of lines such that for a pair of points p and q in H, there exists a unique hyperbolic line through p and q. First, we define hyperbolic lines. Hyperbolic lines on H are defined in terms of Euclidean objects in. Definition 2.2. Lines in H are defined as one of the following: (1) The intersection of half-plane H and Euclidean lines orthogonal to the real axis R or (2) The intersection of half-plane H and Euclidean half-circles with center on the real axis R Figure 1. Some examples of lines in H With this definition, we note a property of these two types of hyperbolic lines which will later be used in justifying the Möbius transformations as a reasonable set of transformations that send circles to circles on the hyperbolic plane. Proposition 2.3. Hyperbolic lines can be expressed as solutions to the equations (2.4) βz + β z + γ = 0 where γ R and β, or (2.5) αz z + βz + β z + γ = 0 where α, γ R, α 0 and β Proof. A Euclidean line ax+by+c = 0 and a Euclidean circle (x h) 2 +(y k) 2 = r 2 can both be expressed on in terms of z = x + iy. Since x = Re(z) = 1 2 (z + z) and y = Im(z) = i 2 (z + z), ax + by + z = a bi (z + z) + 2 2 (z + z) + γ = 1 2 (a + ib)z + 1 (a ib) z + γ = 2 βz + β z + γ = 0

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE 3 and (x h) 2 + (y k) 2 = r 2 (Re(z) h) 2 + (Im(z) k) 2 = r 2 Re(z) 2 + Im(z) 2 2hRe(z) 2kIm(z) + h 2 + k 2 = r 2 z 2 h(z + z) ik(z z) + γ = 0 αz z + βz + β + γ = 0 We see that all hyperbolic lines of the first category can be represented as solutions of equations of the form βz + β z + γ = 0 where γ R and β, while all hyperbolic lines of the latter category can be represented as solutions of equations of the form where α, γ R, α 0 and β αz z + βz + β z + γ = 0 As said earlier, with the notion of hyperbolic lines clearly defined, we can now check that for a pair of distinct points p and q in H, there exists a unique hyperbolic line l through p and q. Proposition 2.6. For a pair of distinct points p and q in H, there exists a unique hyperbolic line l in H through p and q. Proof. If Re(p) = Re(q), the desired hyperbolic line l is the intersection between the half-plane H and the Euclidean line L = {z Re(z) = Re(p)} If Re(p) Re(q), we need to construct a Euclidean circle centered on the real axis R passing through p and q. For points p and q on H, we can construct a circle with center on the intersection of the real axis and the perpendicular bisector of the Euclidean line segment connecting p and q and with radius z. The perpendicular bisector intersects H since Re(p) Re(q). The resulting Euclidean circle is unique due to the uniqueness of the components used in constructing it. Where the hyperbolic model begins to differ from conventional Euclidean geometry is with the introduction of the notion of parallel lines. Parallel lines are defined in the same manner as in Euclidean geometry. Figure 2. A line in H (left) and lines parallel to it Definition 2.7. Two hyperbolic lines in H are parallel if they are disjoint.

4 YONG JAE KIM Theorem 2.8. For a hyperbolic line l and point p both in H where p / l, there exist infinitely many distinct hyperbolic lines through p parallel to l. Proof. First, if l is contained in a Euclidean line L, then a line K that passes through p and is orthogonal to the real axis R is parallel to l. More parallel lines can be drawn by taking a point q on R between L and K and drawing a Euclidean circle A with its center on R and passing through p and q. H A is a hyperbolic line parallel to l. If l is contained in a Euclidian circle A, then there exists a concentric Euclidean circle B that passes through p. Similarly, by taking a point q on the real axis R between A and B, we can construct a Euclidean circle D that passes through p and q that is disjoint from A. H D is a hyperbolic line parallel to l. In this way, the half-plane model constructs a system of points and lines defined on H that is non-euclidean. 3. Hyberbolic Lines as ircles on the Riemann Sphere Ĉ The two different types of hyperbolic lines defined in terms of Euclidean lines and Euclidean circles can be unified into one object by defining an extended complex plane which is the Riemann Sphere Ĉ. Definition 3.1. Ĉ = { } 0 Figure 3. A visualization of the Riemann sphere To understand more properties of the extended plane Ĉ, we define openness of subsets in Ĉ. In the conventional complex plane, a subset X is open if for each z X, there exists an ε > 0 such that U ε (z) X, where U ε (z) = {w Ĉ w z < ε} Also, a set X in is closed if its complement X in is open and a set X in is bounded if there exists some constant ε such that X U ε (0). A notable example of an open subset in is the half-plane H. For each point z in H, U Im(z) (z) H, so H is open in.

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE 5 The extended complex plane Ĉ is the complex plane with the addition of a single point at { }. Therefore, to extend the above definitions for Ĉ we need only define U ε ( ) and retain the above definitions for all other z Ĉ. We define U ε( ) as the following: U ε ( ) = {w w > ε} { } Definition 3.2. A set A in Ĉ is open if for each point a A, there exists some ε > 0 such that U ε (a) A With this definition, we see that an open set in must also be open in Ĉ. For example, H is open in, so we know that it is also an open subset of Ĉ. In other words, is not distorted in being viewed as a subset of Ĉ. Naturally, we can also extend the definition of closedness to Ĉ. Definition 3.3. A set X in Ĉ is closed if its complement Ĉ X in Ĉ is open. With one more definition, we can unify the two different types of hyperbolic lines defined in terms of Euclidean lines and Euclidean circles on. Definition 3.4. The closure X of X in Ĉ is the set X = {z Ĉ U ε(z) X for all ε > 0} Definition 3.5. A circle in Ĉ is either a Euclidean circle in or the union of a Euclidean line in with { }. Hyperbolic lines are circles on the Riemann sphere, and the circles on the Riemann sphere are the closure of the Euclidean lines and Euclidean circles on that they are derived from. Furthermore, we can describe the circles on Ĉ as the sets of solutions of equations in Ĉ. Specifically, we recall equations 2.4 and 2.5 and note that all Euclidean circles in can be considered sets of solutions to equations of the form αz z + βz + β z + γ = 0 where α, γ R and β, and that every Euclidean line in can be seen as the set of solutions of equation βz + β z + γ = 0 where γ R and β. We can combine these to say that every circle on the Riemann sphere can be described as the solutions in Ĉ of equations that have the form αz z + βz + β z + γ = 0 where α, γ R and β. 4. Homeomorphisms on Ĉ In this section, we will define homeomorphisms and find an especially well behaved class of homeomorphisms on Ĉ. First, to define homeomorphisms, we must first define the notion of continuous functions on Ĉ.

6 YONG JAE KIM Definition 4.1. A function f : Ĉ Ĉ is continuous at z Ĉ if for each ε > 0, there exists δ > 0 (depending on z and ε) such that w U δ (z) implies that f(w) U ε (f(z)). A function f : Ĉ Ĉ is continuous on Ĉ if it is continuous at every point z Ĉ This definition allows us to apply the same proofs as with functions from R to R to show the continuity of functions from Ĉ to Ĉ. That is, we can show that constant functions, products and quotients, sums and differences and compositions of functions from Ĉ to Ĉ are continuous (when defined) with the same proofs as for functions from R to R. There are, however, some differences between functions from R to R and functions from Ĉ to Ĉ that arise from the existence of the point. For example the function J : Ĉ Ĉ defined by J(z) = 1 z for z Ĉ {0}, J(0) = 0, and J( ) = 0 is continuous on Ĉ. With this, on Ĉ, a homeomorphism is defined as the following. Definition 4.2. A function f : Ĉ Ĉ is a homeomorphism if f is a bijection and both f and f -1 are continuous. For convenience, we introduce a bit of notation and say that Homeo(Ĉ) = {f : Ĉ Ĉ f is a homeomorphism} We see that by definition, the inverse of a homeomorphism is a homeomorphism. Since the composition of bijections is a bijection and the composition of continuous functions is also continuous, we can also see that the composition of two homeomorphisms is also a homeomorphism. Lastly, the identity homeomorphism f : Ĉ Ĉ is also a homeomorphism. From these facts, we therefore note that Homeo(Ĉ) is a group. At this point we also introduce two functions: f(z) = az + b and g(z) = 1 z. We propose that they are both homeomorphisms on Ĉ. Proposition 4.3. The function F : Ĉ Ĉ defined by F (z) = az + b for z Ĉ and f( ) = where a, b and a 0 is a homeomorphism. Proof. Function F is a homeomorphism if it is a bijection and both F and F 1 are continuous. We can prove this simply by writing an explicit expression for F 1 F 1 (z) = 1 a (z b) for z and F 1 ( ) = we see that F is a bijection and that F 1 is continuous. Proposition 4.4. The function J : Ĉ Ĉ defined by J(z) = 1 z for z Ĉ {0}, J(0) =, and J( ) = 0 is a homeomorphism on Ĉ.

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE 7 Proof. Suppose that there exist points z and w for which J(z) = J(w). Since J J(z) = z for all z Ĉ, z = J(J(z)) = J(J(w)) = w. So we know that J is injective. Since z = J(J(z)) for all z Ĉ, we see that J is surjective. Since J is injective and surjective, J is bijective. Furthermore, since J 1 (z) = J(z) for all z Ĉ and J is continuous, J 1 is continuous. Since J is bijective and continuous, J is a homeomorphism. Since all hyperbolic lines on H are contained in circles in Ĉ, we are especially interested in homeomorphisms of Ĉ that send circles on Ĉ to circles on Ĉ. Again, for convenience, we introduce another notation and let Homeo (Ĉ) be the subset of Homeo(Ĉ) that have this property. Note that we do not yet know that the inverse of elements of Homeo (Ĉ) are in Homeo (Ĉ), and so do not yet know whether Homeo (Ĉ) is a group. 5. Möbius Transformations as Homeomorphisms Definition 5.1. A Möbius Transformation is a function m : Ĉ Ĉ of the form m(z) = az + b cz + d where a, b, c, d, ad bc 0 and a 0 = lim a w 0 w. Let Möb + denote the set of all Möbius transformations a With this, we see that lim w 0 w = in Ĉ. However, the expressions or 0 0 are still ambiguous. Similarly, we define the image of under the Möbius transformation m(z) = az+b cz+d by continuity. That is, we set az + b m( ) = lim z cz + d = lim a + b z z c + d z The two types of Homeo(Ĉ) mentioned in the previous section are also elements of Homeo (Ĉ). = a c Proposition 5.2. The element F of Homeo(Ĉ) defined by F (z) = az + b for z Ĉ and f( ) = where a, b and a 0 is an element of Homeo (Ĉ). Proof. Every circle A on the Riemann sphere can be described as the set of solutions in Ĉ to equations of the form αz z + βz + β z + γ = 0 where α, γ R and β. We first consider the case in which A is a Euclidean line in. In this case, the A can be represented in the following manner. A = {z βz + β z + γ = 0}

8 YONG JAE KIM where β and γ R. We want to show that for every z that satisfies the equation, w = az + b also satisfies the equation. Since w = az + b, we have z = 1 a (z b). Substituting this into the equation for A, we get β(z) + β z + γ = β 1 a (w b) + β 1 (w b) + γ a = β ( ) β α w + w β α α b β α b + γ = 0 ( ) since β α b β α b = 2Re β α b is real and the coefficients of w and w are complex conjugates, w also satisfies the equation of a Euclidean line. Therefore, F sends Euclidean lines in to Euclidean lines in. Similarly, we consider the case in which A is a Euclidean circle in. Again, using the equation derived earlier, we can represent the line A to be A = {z αz z + βz + β z + γ = 0} Again setting w = az+b and substituting z = 1 a (w b) into αz z+βz+ β z+γ = 0, we get αz z = α 1 a (w b) 1 a (w b) + β 1 a (w b) + β 1 (w b) + γ a = α a 2 (w b)(w b) + β ( ) β (w b) + w b + γ α α = α a 2 w + βa 2 α b + γ β 2 α = 0 Since this is again the equation for a Euclidean circle on, we see that F sends Euclidean lines on to Euclidean lines on and Euclidean circles on to Euclidean circles on. onsequently, F sends circles on Ĉ to circles on Ĉ. F is an element of Homeo (Ĉ) Proposition 5.3. The function J : Ĉ Ĉ defined by J(z) = 1 z for z Ĉ {0}, J(0) = 0, and J( ) = 0 is an element of Homeo (Ĉ). Proof. As in the proof of Proposition 4.5, we let A be a circle in Ĉ given by the equation αz z + βz + β z + γ = 0 where α, γ R and β. Set w = 1 z so that z = 1w. Substituting into the equation for A, we get α 1 w 1 w + β 1 w + β 1 w + γ = 0

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE 9 Multiplying through by w w, we get the equation α + β w + βw + γw w = 0 and this again is the equation of a circle on the Riemann sphere. J is an element of Homeo (Ĉ) The elements of Homeo (Ĉ) discussed above are the basis of Möbius transformations. We will soon prove that all Möbius transformations are the compositions of these homeomorphisms of Ĉ that send circles to circles on the Riemann sphere. Theorem 5.4. Let m(z) = az+b cz+d, where a, b, c, d and ad bc 0 be a Möbius transformation. If c = 0, then m(z) = a d z + b d, or If c 0, then m(z) = f(j(g(z))), where g(z) = c 2 z +cd and f(z) = (ad bc)z + a c for z and f( ) = = g( ). Proof. We prove this theorem through simple calculation. If c = 0, the result is obvious. If c 0, Since ad bc 0, we have that m(z) = acz + bc c 2 + cd m(z) = az + b (az + b)c azc + bc = = cz + d (cz + d)c c 2 z + cd. = acz + ad (ad bc) c 2 z + cd = a c where g(z) = c 2 z + cd and f(z) = (ad bc)z + a c. ad bc c 2 z + cd = f(j(g(z))) From this we see that a Möbius transformation is a composition of homeomorphisms. An immediate consequence of Theorem 5.2, then, is that every Möbius transformation is a homeomorphism. Furthermore, we have proved earlier in propositions 4.5 and 4.6 that the homeomorphisms on Ĉ whose compositions are the set of Möbius transformations have the property of sending circles to circles on the Riemann sphere Ĉ. From these facts, we can derive an important theorem: Theorem 5.5. Möb + Homeo (Ĉ). 6. Transitivity Properties of Möb + One of the defining characteristics of Möbius transformations is that they act uniquely triply transitively on Ĉ. This means that there exists a unique element m of Möb + such that for any two triples (z 1, z 2, z 3 ) and (w 1, w 2, w 3 ) of distinct points on Ĉ, m(z 1) = w 1, m(z 2 ) = w 2, m(z 3 ) = w 3. To prove this, we consider a lemma concerning fixed points. Definition 6.1. A fixed point of the Möbius transformation m is a point z of Ĉ satisfying m(z) = z. Lemma 6.2. If m(z) is a Möbius transformation with three distinct fixed points on Ĉ, m is the identity transformation. That is, m(z) = z for every z in Ĉ.

10 YONG JAE KIM Proof. Suppose that m fixes three points on Ĉ but is not the identity transformation. We have that for m = az+b cz+d, m( ) = a c and consequently that m( ) = if and only if c = 0. If c = 0, then m(z) = a d z + b d, and the fixed point of m in is the solution to the equation m(z) = a d z + b d = z. If a d = 1, then b 0 since m is not the identity. In this case, there is no solution in. If a d 1, z = b d a is the unique solution in. In particular, if c = 0, m has one or two fixed points. If c 0, then m( ), and the fixed points of m are the solutions in of the equation m(z) = az+b cz+d = z. These can be represented otherwise as the solutions to the quadratic equation cz 2 + (d a)z b = 0. In this case again, m has either one or two fixed points. These observations allow us to conclude that if an element m of Möb + fixes three points in, it is the identity transformation. With this lemma, we can now prove the proposition. Proposition 6.3. Möb + acts uniquely triply transitively on Ĉ. Proof. We start the proof of this fact by first showing uniqueness. Given two triples (z 1, z 2, z 3 ) and (w 1, w 2, w 3 ) of distinct points in Ĉ, suppose that there are two Möbius transformations m and n satisfying m(z 1 ) = w 1 = n(z 1 ), m(z 2 ) = w 2 = n(z 2 ) and m(z 3 ) = w 3 = n(z 3 ). Since m 1 n fixes three distinct points of Ĉ, we know by Lemma 6.2 that it is the identity. This means that m = n and proves uniqueness. To show the existence of a Möbius transformation taking (z 1, z 2, z 3 ) to (w 1, w 2, w 3 ), we need only show that there exists an element of Möb + m such that m(z 1 ) = 0, m(z 2 ) = 1 and m(z 3 ) =. If we show the existence of such an m, we can also find n such that n(w 1 ) = 0, n(w 2 ) = 1 and n(w 3 ) = and n 1 m is the desired transformation taking (z 1, z 2, z 3 ) to (w 1, w 2, w 3 ). We can find such a transformation quite simply by writing m(z) = z z 1 z z 3 z 2 z 3 z 2 z 1 = (z 2 z 3 )z z 1 (z 2 z 3 ) (z 2 z 1 )z z 3 (z 2 z 1 ) 7. The General Möbius Group The general Möbius group Möb is a very natural extension of the existing group of Möbius transformations Möb +. We consider the simplest homeomorphism of Ĉ not already contained in Möb + to find a largest set of homeomorphisms on Ĉ that send circles to circles in Ĉ. This simple homeomorphism is the complex conjugation defined as (z) = z for z and ( ) = Definition 7.1. The general Möbius group Möb is the group formed from the composition of Möb + and. That is, every element p of Möb can be expressed as a composition p = m k m 1 for some k 1, where each m k is an element of Möb +.

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE 11 For a circle A given by the equation αz z +βz + β z +γ = 0, if w = (z) = z then w satisfies the equation αw w + βw + β w + γ = 0. Therefore, the fuction : Ĉ Ĉ lies in Homeo (Ĉ). ombining this with the fact that Möb+ Homeo (Ĉ), we can propose the following theorem for the general Möbius group. Theorem 7.2. Möb Homeo (Ĉ). With Theorem 7.2 established, we now only need to show that Homeo (Ĉ) Möb in order to show that Möb = Homeo (Ĉ); or that the general Möbius group is the automorphism group on the Riemann sphere. Theorem 7.3. Möb = Homeo (Ĉ) Proof. As said above, to prove this theorem at this point, we need only show that Homeo (Ĉ) Möb then combine with theorem 7.2. Let f be an element of Homeo (Ĉ). Let p be the Möbius transformation taking the triple (f(0), f(1), f( )) to the triple (0, 1, ) such that p f satisfies p f(0) = 0, p f(1) = 1, and p f( ) =. As both p and f take circles to circles on Ĉ, p f takes circles in Ĉ to circles in Ĉ. R = R { } is the circle determined by the triple (0, 1, ) and p f( ) =. Therefore, it must be that p f(r) = R. Since p f fixes and the real line R, p f(h) is either the upper half plane or the lower half plane. We choose m = p and m = p respectively to find an element of Möb such that m f(0) = 0, m f(1) = 1, and m f( ) = and m f(r) = R We try to show that m f is the identity by constructing a dense set of points in Ĉ each of which is fixed by m f. For convenience, let us set Z = {z Ĉ m f(z) = z}, or the set of points fixed by m f in Ĉ. We have already established that 0, 1 and are in Z. Definition 7.4. A subset X of Ĉ is dense if ˆX = Ĉ. The composition m f takes Euclidean lines in to Euclidean lines in and takes Euclidean circles in to Euclidean circles in since it lies in Homeo () and fixes. We also note that for any two Euclidean lines X and Y on for which m f(x) = X and m f(y ) = Y the point z 0 at which they intersect is fixed since m f(z 0 ) = z 0. Using this, we create a dense set of fixed points by intersecting lines. For each s R, let V (s) be the vertical line in through s, and H(s) be the horizontal line in passing through the point is. Let A be the Euclidean circle with Euclidean center 1 2 and Euclidean radius 1 2. As V (0) and V (1) are tangent to A at 0 and 1, we see that m f(v (0)) and m f(v (1)) respectively are the tangent lines to m f(a). Since V (1) and V (0) are parallel, m f(v (1)) and m f(v (0)) must also be parallel, so it must also be that m f(v (1)) = V (1) and m f(v (0)) = V (0). Furthermore, to have parallel tangent lines, it must also be true that m F (A) = A. Using the same method with the lines H( 1 2 ) and H( 1 2 ), we find that m f(h( 1 2 )) = H( 1 2 ) and m f(h( 1 2 )) = H( 1 2 ). This provides more points to be included in Z, namely H( 1 2 ) V (0) = 1 2 i, H( 1 2 ) V (1) = 1+ 1 2 i, H( 1 2 ) V (0) = 1 2 i and H( 1 2 ) V (0) = 1 1 2 i.

12 YONG JAE KIM Each pair of points in Z produces a new line that is taken to itself by m f. Every triple of points gives rise to a Euclidean circle that is also taken to itself by m f. The intersections of these lines and circles are more points in Z. By repeating this process we show that Z contains a dense set of points of Ĉ. Lemma 7.5. If X is dense in Ĉ and if f : Ĉ Ĉ is a continuous function for which f(x) = x for all x X, then f(x) = z for all z Ĉ Proof. For any z Ĉ, there exists a sequence {x n} in X converging to z since X is dense in Ĉ. Also, f is continuous. We know, therefore, that {f(x n)} converges to f(z). Since f(x n ) = x n, this gives that {x n } converges to both z and f(z), and so z = f(z). This means that m f is the identity and therefore Homeo (Ĉ) = Möb. This shows that the general Möbius group is the automorphism group on the Riemann Sphere. Acknowledgments. I would like to thank my mentor Bena Tshishiku for his patience and knowledge in guiding me through the process of writing this paper. Also, I thank Professor Peter May and everyone else involved for providing the oportunity to participate in such a program. References [1] J. W. Anderson. Hyperbolic Geometry. Springer. 2005