Question

21-22 İmzi . İmz2 is invariant under Sl(2, R) Moebius transformations and therefore could be used as a measure of distance (even though the actual hyperbolic distance is some function of 8) Pick a point on the y-axis in the upper half plane and figure out the shape of the curve given by all points a fixed value of S away from that point. Sketch this curve. Repeat this for various other points on the y-axis using the same value of δ each time. Comment on what you find. You might like to google Escher Upper Half Plane" for inspiration.

Answer 1

"Moebius transformation is defined as follows: For gamma = (a c b d)elementof Sl(2, IR) z rightarrow gamma(z) = a z + b/c z + d if z elementof H we must show that gamma (z) elementof H Note that I_m (gamma(z)) = I_m[(a z + b)(c z^bar + d)/(c z + d)(c z^bar + d)] = I_m [a z z^bar c = I_m (a c z z^bar + a d z + b c z^bar + b d)/