## 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, NN

Difficulty level: EML 1 (for courageous students), EML 2 or EML 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.]
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