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, Luca Notarnicola
Difficulty level: EML 1 (for courageous students), EML 2 or EML 3.
Tools: Any computer language.
Results: [to be completed at the end of the project.]