Lectures on Dessins d'Enfants

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.

Homework Assignment 3 Download

Problem 1.

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 .

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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