Recall that the collection of Möbius transformations is the group
while the (ordinary) triangle group is the set
This homework set is meant to discuss the relationship with these two sets. This assignment is due Friday, September 20, 2013 at the start of class.
Problem 1. Fix an integer . Upon denoting the positive real number in terms of a primitive -th root of unity , consider the matrices and .
- Show that . Hint: Verify that
- Show that for any . Conclude that has infinite order.
- Show is a well-defined group homomorphism which is injective but not surjective.
- Denote the extended complex numbers , , and . Show that , , and Conclude that the Hecke group tiles the extended upper half plane with triangles . (Observe that the sum of the angles is less than .)
Problem 3. When , the special linear group
is generated by and since .
- Consider the map defined by . Show that , , and .
- Show that this map induces a short exact sequence . That is, .
Problem 4. Let be an integer. The canonical projection extends to a surjection . Define the principal congruence subgroup of level as that subgroup of which makes the following diagram commute:
(Contrary to Wikipedia, is not the same as in general.)
- Show that . Conclude that .
- Show that . Conclude that , , , and .
- Show that the Euler characteristic is the integer
Hint: Using , show .
- The genus is that nonnegative integer such that . Show that .