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) ∈ F2 is called an admissible pair if there exists c, d ∈ F such a b that ∈ GL ( ). c d 2 F

Definition 2. we define an equivalence relation on F2 \{(0, 0)} as follows: (x, y) ∼ (u, v) if there is a non-zero λ ∈ F such that x = λu and y = λv. Definition 3. P(F) = {[x : y]:(x, y) is admissible} 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 ∈ F. a b Definition 4. For some A = ∈ 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)/{∓I}

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) ∈/ [0, ∞). 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) = ∞. 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 ∈ . 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:  !  1 x  if x ∈ R,  0 1 s(x) = !  0 −1  if x = ∞.  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 = {o = a + bj : a, b ∈ R}, 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 = {d = a + b : a, b ∈ R}, 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 ∪ {∞, σ1, σ2} ∪ {aω1, aω2 : a ∈ R}. P(D) = D ∪ {∞, aω : a ∈ R}. 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

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