# Homework Assignment 3

Recall that the upper half plane consists of complex numbers with . Let denote the closed unit disk as the union of the unit disk with its boundary . This assignment is meant to explain the pictures at the Wolfram Demonstrations site as well as on Wikipedia:

This assignment is due Friday, October 18, 2013 at the start of class.

Problem 1.

• Let denote the collection of those real matrices with nonzero determinant. Prove the Iwasawa Decomposition:

in terms of the real numbers , , , and .

• Let denote those real matrices with positive determinant. Consider the map defined by . Show that is surjective, whereas
.

Problem 2. Show that . Hint: Use the Iwasawa Decomposition.

Problem 3. Define the the modular curve in terms of

The normalized hyperbolic area differential on the upper half plane is given by .

Problem 4. Let be a complex number with . In the Poincare Disk Model, a “line” is given by .

• Identifying the affine complex line with the affine real plane , show that is a circle of radius centered at which intersects the boundary at the two distinct complex points . Hint: Solve the simultaneous equations .

• Show that is perpendicular to at the points where they intersect. Hint: Consider the inner product of the vectors and which are normal to and , respectively.

Problem 5.Using the isomorphism

show that the intersection of the unit disk and the “line” is in bijection with the intersection of the hyperboloid and the plane . Hint: Show that points in map to .

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