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.

Homework Assignment 2 Download

**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.

**Problem 2.** Define the modular group as the set

More generally, fix an integer , and define the Hecke group as the set

Observe that . Consider the map which sends , , and

- 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 .

**Problem 5.** Assume that is an odd prime. The set is that compact, connected Riemann surface formed by gluing the triangles associated with the triangle group .

- Show that the Euler characteristic is the integer

*Hint:* Using , show .

- The genus is that nonnegative integer such that . Show that .

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## About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.

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