POINCARE'S´ THEOREM FOR FUCHSIAN GROUPS JAMES BUCHANAN Abstract. We present a proof of Poincar´e'sTheorem on the existence of a Fuchsian group for any signature (g; m1, ..., mr), where g, m1, ..., mr provide an admissable solution to the formula describing the hyperbolic area of the group's quotient space. Along the way we elucidate relevant concepts in hyperbolic geometry and the theory of Fuchsian groups. Contents 1. Introduction 1 2. Hyperbolic Geometry and the Group P SL(2; R) 2 3. Fuchsian Groups 5 4. Fundamental and Dirichlet Regions 7 5. Quotient Spaces of Fuchsian Groups 10 5.1. Proof of Poincar´e'sTheorem 11 Acknowledgments 13 References 13 1. Introduction Our goal is to prove, as well as provide appropriate context for, the following theorem. Theorem 1.1 (Poincar´e'sTheorem). If g ≥ 0, r ≥ 0, mi ≥ 2 (1 ≤ i ≤ r) are integers and if r X 1 (2g − 2) + 1 − ≥ 0 m i=1 i then there exists a Fuchsian group with signature (g; m1, ..., mr). We first examine the relevant hyperbolic geometry, followed by an introduction to Fuchsian groups. Our chief reference is Svetlana Katok's Fuchsian Groups [1], in which the main definitions and results of this paper may be found; I have strived to maintain Katok's notation throughout, to facilitate cross-referencing with her work. Basic knowledge of algebra and topology will be assumed (e.g. terms such as \genus" and \quotient group"). Date: August 23, 2011. 1 2 JAMES BUCHANAN 2. Hyperbolic Geometry and the Group P SL(2; R) The set H = fz 2 CjIm(z) > 0g is called the hyperbolic (half) plane. Notice that H does not contain the real line. If we wish to consider these points as well, we will find it fruitful to append a point labeled 1 to the \end" of the real line, such that the real line together with 1 is homeomorphic to the circle { the ends of this line are \joined" at infinity. The set H~ = H [ f1g is the Euclidean closure of H, since it is the closure in C [ f1g of H with the standard Euclidean metric. The set of fractional linear transformations from C [ f1g to C [ f1g specified by fz 7! (az + b)=(cz + d)ja; b; c; d 2 R, ad − bc = 1g, together with the operation of function composition, form a group referred to as P SL(2; R). Each of these transformations is associated with two distinct 2 × 2 matrices a b −a −b ; c d −c −d both of whose determinants are 1. This gives us a ready definition for the trace of an element of P SL(2; R), namely the absolute value of the trace of either of its corresponding matrices. The group of all 2 × 2 real matrices is called SL(2; R), and we immediately see that P SL(2; R) is isomorphic to SL(2; R)=f ± 12g, where 12 is the 2 × 2 identity matrix. We can discriminate the elements of P SL(2; R) into three different types based on the values of their traces. Those with trace < 2 are the elliptic elements, those with trace = 2 are parabolic, and those with trace > 2 are hyperbolic. This geometric terminology stems from the action of SL(2; R) on R2: A matrix in SL(2; R) is hyperbolic (resp. elliptic) if and only if its invariant curve in R2 is a hyperbola (resp. ellipse), and the parabolic transformations are thought of as \intermediate" between the hyperbolic and elliptic transformations. Finally, we expand our consideration to the group PS*L(2; R) comprised of the fractional linear transformations from C to C of the form z 7! (az + b)=(cz + d), ad − bc = ±1; P SL(2; R) is a subgroup of PS*L(2; R) of index 2. We now consider two representations of plane hyperbolic geometry, both embed- ded in the complex plane. The set H together with the metric pdx2 + dy2 (2.1) ds = y form our first representation, the half-plane model. The function f : C ! C defined by zi + 1 (2.2) f(z) = z + i is a bijective map that sends H to U = fz 2 C jzj < 1g, a (Euclidean) circle of radius 1 centered at z = 0. In a sense, f shrinks the “infinitely large circle" H down to a circle of radius 1 whose boundary Σ is the set fz 2 C jzj = 1g, referred to as the principal circle. U together with its principal circle Σ is U~, the Euclidean closure of U. To obtain the appropriate metric ds to use in this model, we first examine the metric on H and how hyperbolic distances are transformed by the function f. For POINCARE'S´ THEOREM FOR FUCHSIAN GROUPS 3 a path γ through H parametrized by t 2 [0; 1], its hyperbolic length h(γ) is given by q 1 1 dx 2 dy 2 Z ds Z ( dt ) + ( dt ) (2.3) h(γ) = dt = 0 dt 0 y(t) and the hyperbolic distance ρ(z; w) between points z; w 2 H is taken to be the infimum of h(γ) over all γ joining z and w. For z; w 2 U, the hyperbolic distance ρ*(z; w) between them is thus equal to ρ(f −1z; f −1w). A brief calculation confirms that for z = x + iy 2 H, 2jdf(z)=dzj 1 = ; 1 − jf(z)j2 Im(z) which in turn equals 1 pdx2 + dy2 ds = = y ypdx2 + dy2 dz so 2jdf(z)j 2jdz*j (2.4) ds = = 1 − jf(z)j2 1 − jz*j2 for z* 2 U, which is the metric we were looking for. This model of hyperbolic geometry will be referred to as the unit disk model. Returning to our consideration of PS*L(2; R) and its subgroup P SL(2; R), we now look at its action on H. As it turns out, PS*L(2; R) is isomorphic to the group Isom(H) of hyperbolic isometries of H [Katok, Thm. 1.3.1]. For the remainder of this paper, we will convern ourselves only with the subgroup P SL(2; R). Theorem 2.5. P SL(2; R) is the group of orientation-preserving isometries of H. Proof. An isometry is orientation-preserving if it is composed only of rotations and translations, and not reflections, in the hyperbolic plane. A reflection would be some rotations and translations combined with the operation z 7! −z¯, but the latter operation does not belong to P SL(2; R) - therefore, P SL(2; R) contains only orientation-preserving isometries. Combining the above operation with any elements from P SL(2; R) results in a transformation whose determinant is −1, and recalling that P SL(2; R) is of index 2 in PS*L(2; R), we see that the demarcation between elements whose determinant is plus or minus one is precisely the same as that between transformations which are orientation-preserving or -reversing, and that the orientation-reversing isometries are those with negative determinant. One can similarly show that the orientation-preserving isometries of U are the fractional linear transformations from C to C of the form z 7! (az +c ¯)=(cz +a ¯) where a; c 2 C, aa¯ − cc¯ = 1, which together form their own group isomorphic to P SL(2; R). Remark 2.6. Parabolic and hyperbolic transformations have one and two fixed points, respectively, which are located in R [ f1g. Elliptic transformations have one fixed point, in H. We wrap up our introduction to hyperbolic geometry with a discussion of the hyperbolic version of the polygon. First we investigate the hyperbolic equivalent of a straight line: A geodesic in hyperbolic space is the shortest curve (with respect to the appropriate metric) joining two points. 4 JAMES BUCHANAN Theorem 2.7. The geodesics in H are segments of either straight lines perpendic- ular to the real line, or of semicircles with their ends touching the real line whose tangents at the points of intersection are perpendicular to it. Proof. First consider two points z1 = ia; z2 = ib 2 H, with b > a. For any path γ(t) = (x(t); y(t)) joining z1 and z2, Z 1 p(dx=dt)2 + (dy=dt)2 Z 1 jdy=dtj h(γ) = dt ≥ dt 0 y(t) 0 y(t) Z 1 dy=dt Z b dy b ≥ dt = = ln ; 0 y(t) a y a meaning ln (b=a) ≤ ρ(z1; z2). Meanwhile, the particular path ζ(u(t); v(t)) where u(t) = 0 and v(t) = a + t=(b − a), i.e. the segment of the imaginary axis joining z1 and z2, has hyperbolic length Z 1 p(du=dt)2 + (dv=dt)2 Z 1 dv=dt h(ζ) = dt = dt 0 v 0 v(t) Z b dv b = = ln a v a so ln (b=a) ≥ ρ(z1; z2), and hence ln (b=a), the hyperbolic length of the straight vertical line joining z1 and z2, is also the hyperbolic distance between them. Any straight line can be mapped to the imaginary axis by the transformation z ! z + b for some b 2 R, and all such transformations are elements of P SL(2; R) and hence isometries. One can also show that, for a semicircle as described above which meets the real line at α and β, the transformation z ! (γz−(αγ+1))=(z−α) for γ = (β −α)−1 has determinant 1 and maps the semicircle to the imaginary axis. Thus the hyperbolic length of the line or semicircle segment joining two points in H is the hyperbolic distance between their images under the isometry that sends them to the imaginary axis, and so is the hyperbolic distance between the points themselves.