The geometry of the hecke groups acting on hyperbolic plane and their associated real continued fractions.

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dc.contributor.authorMaphakela, Lesiba Joseph
dc.date.accessioned2014-06-12T08:29:27Z
dc.date.available2014-06-12T08:29:27Z
dc.date.issued2014-06-12
dc.description.abstractContinued fractions have been extensively studied in number theoretic ways. In this text we will consider continued fraction expansions with partial quotients that are in Z = f x : x 2 Zg and where = 2 cos( q ); q 3 and with 1 < < 2. These continued fractions are expressed as the composition of M obius maps in PSL(2;R), that act as isometries on H2, taken at 1. In particular the subgroups of PSL(2;R) that are studied are the Hecke groups G . The Modular group is the case for q = 3 and = 1. In the text we show that the Hecke groups are triangle groups and in this way derive their fundamental domains. From these fundamental domains we produce the v-cell (P0) that is an ideal q-gon and also tessellate H2 under G . This tessellation is called the -Farey tessellation. We investigate various known -continued fractions of a real number. In particular, we consider a geodesic in H2 cutting across the -Farey tessellation that produces a \cutting sequence" or path on a -Farey graph. These paths in turn give a rise to a derived -continued fraction expansion for the real endpoint of the geodesic. We explore the relationship between the derived -continued fraction expansion and the nearest - integer continued fraction expansion (reduced -continued fraction expansion given by Rosen, [25]). The geometric aspect of the derived -continued fraction expansion brings clarity and illuminates the algebraic process of the reduced -continued fraction expansion.en_ZA
dc.identifier.urihttp://hdl.handle.net10539/14760
dc.language.isoenen_ZA
dc.subject.lcshContinued fractions.
dc.subject.lcshGeometry, hyperbolic.
dc.subject.lcshHecke operators.
dc.titleThe geometry of the hecke groups acting on hyperbolic plane and their associated real continued fractions.en_ZA
dc.typeThesisen_ZA
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