## 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, the analogous situation in so-called Hecke triangle groups
is studied, that is, subgroups of SL_2(R) generated by S and the matrix T_x,
where x=2cos(\pi/q) with q an integer at least 3, or x > 2 any real number.

The aim is to implement and visualise x-continued fractions and the word
decomposition in Hecke triangle groups.

**Schedule:** To be determined.

**Supervisors:**
Gabor Wiese, Alexander Rahm

**Difficulty level:**
EML 2 or 3.

**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.]