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Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

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HYPERBOLIC GEOMETRY: ISOMETRY GROUPS OF HYPERBOLIC SPACE ILYAS KHAN Abstract. The goal of this paper is twofold. First, it consists of an intro- duction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that pre- serve orientation, and deals with the topological properties of their discrete subgroups. In the final section, we show that if one of these Fuchsian groups satisfies certain conditions, then its quotient space is a 2-manifold for which the hyperbolic plane is a covering space. Contents 1. Hyperbolic Geometry and PSL(2,R) 1 2. Geodesics 5 3. Discrete Isometry Groups and Proper Discontinuity 8 4. Topological Properties of Fuchsian Groups 12 Acknowledgments 15 References 16 1. Hyperbolic Geometry and PSL(2,R) There are several models of hyperbolic space, but for the purposes of this paper we will restrict our view to the Lobachevski upper half plane. Definition 1.1. The upper half plane is the subset H = fz 2 C : Im(z) > 0g of C with the metric pdx2 + dy2 ds = y where z = x + iy. We can use this metric to find the length of a differentiable path γ through hyperbolic space by integrating over its domain. The hyperbolic length, h(γ) of the differentiable path γ : [0; 1] ! H where γ(t) = (x(t); y(t)) is q 1 dx 2 dy 2 1 dz Z ( dt ) + ( dt ) Z j j h(γ) = dt = dt dt 0 y(t) 0 y(t) q dx 2 dy 2 dz since ( dt ) + ( dt ) is the Euclidean norm of the vector dt . Date: August 24, 2012. 1 2 ILYAS KHAN Definition 1.2. The hyperbolic distance ρ(z; w) between z; w 2 H is given by ρ(z; w) = inf h(γ) where γ is an element of the set of all differentiable paths connecting z and w. Definition 1.3. An isometry of H is a function g : H ! H such that for any z; w 2 H, ρ(z; w) = ρ(g(z); g(w)). This paper will focus on the properties of special groups of isometries of H. Definition 1.4. A group is a set G with an operation ∗ which satisfies the axioms G1 through G3 (G1) ∗ is associative. (G2) There is an element e in G such that a ∗ e = a and e ∗ a = a for every element a in G. (G3) For every element a in G, there is an element a−1 in G such that a∗a−1 = e and a−1 ∗ a = e. One group of isometries (of R2) particularly important to our aims is the special linear group SL(2,R). SL(2,R) is the set of 2 by 2 matrices with determinant 1 under the group operation of matrix multiplication. That SL(2,R) is a group is easy to show using the multiplicity of determinants, the existence of an identity matrix, and the inversion formula for 2 by 2 matrices. Definition 1.5. The elements of the projective special linear group PSL(2,R) are az+b the M¨obiustransformations, rational functions from C to C of the form z 7! cz+d with ad − bc = 1. We now want to show that PSL(2,R) is a group. We see immediately that the set az+b PSL(2,R) is isomorphic to SL(2,R) modulo {±Idg, because for any element cz+d a factor of −1 in the numerator will cancel with a factor of −1 in the denominator. Hence, we set up a surjective identification a b az + b g( ) = c d cz + d from SL(2,R) to PSL(2,R). Definition 1.6. If G and H are groups, a homomorphism from G to H is a function f : G ! H such that for any two elements a and b in G f(ab) = f(a)f(b): Let the group operation on PSL(2,R) be function composition. A simple calcu- lation confirms that composition of two M¨obiustransformations corresponds to the multiplication of the corresponding 2 by 2 matrices in SL(2,R). Furthermore, the inverse matrices and identity matrices of SL(2,R) on the subset SL(2,R) ={±Idg correspond to the inverse and identity elements of PSL(2,R). We conclude that PSL(2,R) is a group and that our identification g is a surjective group homomor- phism. Hence, we can say that PSL(2,R) is a homomorphic image of SL(2,R). PSL(2,R) plays an important part in our overall goal to classify the isometries of H. We must first prove that the elements of PSL(2,R) are in fact isometries, and then we will examine what properties distinguish PSL(2,R) from the rest of the isometries of H. HYPERBOLIC GEOMETRY: ISOMETRY GROUPS OF HYPERBOLIC SPACE 3 Lemma 1.7. Every M¨obiustransformation φ in PSL(2,R) is a homeomorphism of H. az+b Proof. First show that φ maps H into H. Let φ(z) = cz+d where ad − bc = 1 and z 2 H. (az + b)(cz + d) acjzj2 + adz + bcz + bd φ(z) = = (cz + d)(cz + d) jcz + dj2 φ(z) − φ(z) adz + bcz − adz − bcz Im(φ(z)) = = 2i 2ijcz + dj2 (ad − bc)(z − z) Im(z) = = 2ijcz + dj2 jcd + zj2 The imaginary component of z is greater than 0, so clearly Im(φ(z)) > 0, which implies that φ(z) is in H. Because φ is a rational function with a nonzero denominator, continuity is clear. The existence of an inverse follows from the fact that φ−1 is also an element of PSL(2,R). By the argument above, it is continuous and maps into H. Therefore, φ is a homeomorphism of H. Theorem 1.8. PSL(2,R) ⊂ Isom(H). Proof. Let γ : [0; 1] ! H be a piecewise differentiable path in H. Let γ be given by z(t) = (x(t); y(t)) and w(t) = T (z(t)) = u(t) + iv(t) By the quotient rule, dw a(cz + d) − c(az + b) acz − caz + ad − bc 1 = = = dt (cz + d)2 (cz + d)2 (cz + d)2 Im(z) dw v Since Im(T (z)) = jcz+dj2 = v and y = Im(T (z)) by Lemma 1.7, j dt j = y . From the definition of hyperbolic length, we obtain the following equation. v(t) Z 1 j dw j Z 1 j dw dz j Z 1 j dz j Z 1 j dz j h(T (γ)) = dt dt = dz dt dt = y(t) dt dt = dt dt = h(γ): 0 v(t) 0 v(t) 0 v(t) 0 y(t) The hyperbolic distance ρ(j; k) is the infimum of the differentiable paths γ be- tween j and k, so since each γ is invariant under T, ρ(j; k) is invariant under T. Thus, T 2 PSL(2,R) is an isometry. [1] There is one further property that classifies PSL(2,R) as a subset of Isom(H), which is that PSL(2,R) is the set of all orientation preserving isometries of H. A linear operator on a vector space is orientation preserving if its determinant is positive. The equivalent notion for M¨obiustransformations is the following: Definition 1.9. Let φ be a map H ! H. Then φ is orientation preserving if det(Dφz) > 0 for all z 2 H. Definition 1.10. The tangent space at z in H, written TzH, is the vector space generated by the collection of vectors based at z tangent to a curve through z. The derivative of φ at z Dφz is a linear map from TzH to Tφ(z)H. To show that PSL(2,R) is orientation preserving, we will establish an inner product on TzH. Ob- serve that each point in H has two linearly independent tangent vectors associated 2 ∼ 2 with it and that H is a two dimensional subset of R , which imply that TzH = R . This observation motivates the following definition. 4 ILYAS KHAN Definition 1.11. The inner product of v; w 2 TzH is given by 1 < v; w > = < v; w > 2 z (Im(z))2 R 2 2 where < v; w >R is the standard dot product for R . Definition 1.12. A function φ : H ! H is an isometry if for all z 2 H and v; w 2 TzH < v; w >z=< Dφz(v); Dφz(w) >φ(z) : Remark 1.13. It follows from a simple calculation that this definition is equivalent to Definition 1.3. Lemma 1.14. If σ and φ preserve orientation, then φ ◦ σ preserves orientation. Proof. This follows immediately from the multiplicity of determinants. D(φ◦σ)z = Dφσ(z) ◦ Dσz by the chain rule. Then, det D(φ ◦ σ)z = det(Dφσ(z) ◦ Dσz) = det Dφσ(z) · det Dσz > 0, since det Dφσ(z) > 0 and det Dσz > 0 since φ and σ are orientation preserving. Lemma 1.15. Any isometry of H of the form z 7! yz + x where y; x 2 R; y 6= 0 and z 2 H preserves orientation. Proof. Let z = r + is, or equivalently z = (r; s) 2 C where r; s 2 R. Let f(r; s) = (yr + x; ys). @f1 @f1 y 0 Df = @r @s = (r;s) @f2 @f2 0 y @r @s 2 det(Df(r;s)) = y > 0, so f preserves orientation. Theorem 1.16.
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