SL2(R) AND FRACTIONAL LINEAR TRANSFORMATION general linear group and projective line For a general field F general linear group is a a b GL ( ) = : ad − bc 6= 0 2 F c d 2 The action of GL2(F) on F is the matrix multiplication of 2 × 2−matrices with 1 × 2−matrices. The special linear group is an a b SL ( ) = : ad − bc = 1 2 F c d Definition 1. (a; b) 2 F2 is called an admissible pair if there exists c; d 2 F such a b that 2 GL ( ): c d 2 F Definition 2. we define an equivalence relation on F2 nf(0; 0)g as follows: (x; y) ∼ (u; v) if there is a non-zero λ 2 F such that x = λu and y = λv: Definition 3. P(F) = f[x : y]:(x; y) is admissibleg is called the projective line over F: The projective line over F is the extension of a usual line by a point called a point at infinity. Mobius¨ transformation A linear map F2 ! F2 is a class invariant for ∼: Thereby, the linear transforma- tion of F2 produces the map P(F) ! P(F) as follows: a b [x : y] = [ax + by : cx + dy]; ad − bc 6= 0; c d where a; b; c; d 2 F: a b Definition 4. For some A = 2 SL ( ): The map T :( ) ! ( ) is c d 2 F A F F defined by TA([x : y]) = [ax + by : cx + dy]:TA is called a M¨obiustransformation. Proposition 5. the set of all M¨obiustransformations is isomorphic to SL2(F)={∓Ig Lemma 6. TA([u]) = [u] if and only if u is an eigenvector of A: The eigenvectors of A (the fixed points of TA) is [x : y] = [(a − d) ± p(d + a)2 − 4 : 2c]: Then we can classify M¨obiusmaps through eigenvalues of A: If F = R or C then Date: October 15, 2017. 1 (1) A has two different complex-conjugated eigenvalues if and only if 0 ≤ tr2(g) < 4: That means, g fixes two distinct complex-conjugated points in P(C) and fixes no point in P(R): Such a map is called elliptic. (2) A has a double eigenvalue if and only if tr2(g) = 4: That means, g fixes a double point. Such a map is called parabolic. (3) A has two distinct real eigenvalues if and only if tr2(g) > 4: That means, g fixes two distinct points. Such a map is called hyperbolic. (4) For K = C; there is an extra class as follows. A has two distinct non-real eigenvalues if and only if tr2(g) 2= [0; 1): In other words, A has two distinct complex eigenvalues if Im(ptr2(g)) 6= 0: That means, g fixes two distinct complex points. Such a map is called strictly loxodromic. The last type of transformation is not possible for K = R: The class, which contains the classes of hyperbolic and strictly loxodromic maps, is called the class of loxodromic maps. Proposition 7. Let TA be a M¨obiustransformation and suppose TA leaves three points fixed. Then TA is the identity map. Proposition 8. Fix w0; w − 1; w2 distinct. Then, for each distinct z0; z1; z2, there is exactly one TA with TA(wi) = zi; for i = 0; 1; 2: w1−w2 w−w0 Let f(w) = : f(w0) = 0; f(w1) = 1; f(w2) = 1: w1−w0 w−w2 one-parameter subgroup Based on the number of fixed points, there are only, up to similarity and rescal- ing, the following three types of non-trivial one-parameter subgroups of SL2(R) . cosh t sinh t 1 0 cos t − sin t A(t) = ;N(t) = and K(t) = : sinh t cosh t t 1 sin t cos t a b Let H = : a; b 2 : Every element in H fixes [1 : 0]: 0 1 R Let p : SL2(R) ! P(R) be a function defined by p(g) = [b : d] where g = a b : c d Let s : P(R) ! SL2(R) be a function defined by: 8 ! > 1 x > if x 2 R; <> 0 1 s(x) = ! > 0 −1 > if x = 1: :> 1 0 The third important map here is r : SL2(R) ! H which is defined by: s(p(g))r(g) = g: Therefore the action of SL2(R) on P(R) is g : x ! p(g ∗ s(x)): The action of SL2(R) on the projective line is a M¨obiustransformation. The action of SL2(R) on the upper half plane (unit disk) is equivalent to Cayley transform which is given 1 −i by C = : −i 1 2 double and dual numbers The commutative ring with identity of all double numbers is the triple (O; +; ·); where O = fo = a + bj : a; b 2 Rg; and for all a = a1 + a2j; b = b1 + b2j in O addition and multiplication are defined as follows: (1) a + b = (a1 + a2j) + (b1 + b2j) = (a1 + b1) + (a2 + b2)j: (2) a · b = (a1 + a2j) · (b1 + b2j) = (a1b1 + a2b2) + (a2b1 + a1b2)j: In particular j2 = 1. The commutative ring with identity of all dual numbers is the triple (D; +; ·); where D = fd = a + b : a; b 2 Rg; and for all a = a1 + a2, b = b1 + b2 in D addition and multiplication are defined as follows: (1) a + b = (a1 + a2) + (b1 + b2) = (a1 + b1) + (a2 + b2). (2) a · b = (a1 + a2) · (b1 + b2) = a1b1 + (a2b1 + a1b2). 2 In particular = 0: Let A be one of O; D: GL2(A) is the set of all invertible 2×2−matrices. SL2(A) is the set of all invertible 2×2−matrices with determinant equal to 1. P(O) = O [ f1; σ1; σ2g [ fa!1; a!2 : a 2 Rg: P(D) = D [ f1; a! : a 2 Rg: References [1] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR1393195 [2] Vladimir V. Kisil, Geometry of M¨obiustransformations, Imperial College Press, London, 2012. Elliptic, parabolic and hyperbolic actions of SL2(R), With 1 DVD-ROM. MR2977041 [3] Barry Simon, Szeg¨o'stheorem and its descendants, M. B. Porter Lectures, Princeton Univer- sity Press, Princeton, NJ, 2011. Spectral theory for L2 perturbations of orthogonal polyno- mials. MR2743058 3.