## Continued fractions on Hecke triangle surfaces

**Goal:**

The group of 2x2-integer matrices with determinant 1, i.e. SL_2(Z), is of
fundamental importance in number theory. It is quite elementary to show that
it is generated by the matrices T=[[1,1],[0,1]] and S=[[0,1],[-1,0]]. In fact,
if one mods out scalar matrices, any matrix can be written in a unique
(subject to some obvious condition) way as a "word" in S and T.

Such a word can be interpreted as a continued fraction, and vice-versa, a
continued fraction (with a certain sign condition) gives rise to an element
of SL_2(Z).

Essentially the Euclidean algorithm can be used to compute the continued
fraction expansion of a rational number. This algorithm hence also gives a
way to compute the expression as a word in S and T of any matrix in
SL_2(Z). This algorithm also has a geometic interpretation.

In this project, we may also explore generalisations, such as (other) Hecke triangle surfaces, or surfaces arising from quaternion algebras.

The aim is to implement and visualise the interplay between continued fractions and geometry.

**Schedule:** To be determined.

**Supervisors:**
Lassina Dembélé Gabor Wiese

**Difficulty level:**
Any

**Tools:**
Any computer language.

**Literature:**

- David Rosen: A class of continued fractions associated with certain
properly discontinuous groups
- Stefan Müller-Stach, Jens Piontkowski: Elementare und algebraische
Zahlentheorie, Vieweg+Teubner

**Results:**
[to be completed at the end of the project.]