The algebra and geometry of continued fractions with integer quaternion coefficients
Date
2015-05-06
Authors
Mennen, Carminda Margaretha
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
We consider continued fractions with coe cients that are in K, the quaternions. In particular
we consider coe cients in the Hurwitz integers H in K. These continued fractions are
expressed as compositions of M¨obius maps in M
R4
1
that act, by Poincar´e extension, as
isometries on H5.
This dissertation explores groups of 2 2 matrices over K and two particular determinant
type functions acting on these groups. On the one hand we find M
R4
1
, the group of
orientation preserving M¨obius transformations acting on R4
1 in terms of a determinant D
[19],[38]. On the other hand K may be considered as a Cli ord algebra C3 based on two
generators i and j, or more generally i1 and i2, where i j = k or i1i2 = k. It is shown this
group of matrices over C4 defined in terms of a pseudo-determinant [1],[37] can also be
used to establish M
R4
1
. Through this relationship we are able to connect the determinant
D to the pseudo-determinant when acting on the matrices that generate M
R4
1
.
We explore and build on the results of Schmidt [30] on the subdivision of a Farey simplex
into 31 Farey simplices. These results are reinterpreted in H5 with boundary K1 using the
group of M¨obius transformations on R4
1 [19], [38]. We investigate the unimodular group
G = PS DL(2;K) with its generators and derive a fundamental domain for this group in H5.
We relate this domain to the 24-cells PU and r that tessellate K. We define the concepts
of Farey neighbours, Farey geodesics and Farey simplices in the Farey tessellation of H5.
This tessellation of H5 by a Farey pentacross under a discrete subgroup G of M
R4
1
is
analogous to the Farey tessellation by Farey triangles of H2 under the modular group [31].
The result in Schmidt [30], that for each quaternion there is a chain of Farey simplices that
converge to , is reinterpreted as a continued fraction, with entries from H, that converges
to . We conclude with a review of Pringsheim’s theorem on convergence of continued
fractions in higher dimensions [5].
Description
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. 2015.