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I'm only a "contemplative" member of the gap-forum and I wonder, if GAP can

help to attack the following problem (to find a representation of a semigroup).

The following text I also sent to the newsgroup "sci.math".

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Subject: Gauss Numbers: how to generate them ?

I suppose, that the following problem is already solved, perhaps in another

context. I'm not a professional researcher, but only fascinated by the question.

Background:

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The Modular Group acts discontinously on the hyperbolic plane, thus

creating a tesselation of this space. In the boundary of the hyperbolic

plane - for the half space model this is the completed real axis - all

the rational numbers are created by the tessalation.

On this ground a process can be organized, which generates each positive

rational number exactly once.

This beautiful procedure is named "Stern-Brocot tree" in

R.L. Graham, D.E. Knuth, O. Patashnik: "Concrete Mathematics",

Reading Mass. 1989, p.116-123, p.291-292.

and it is demonstrated there by elementary means.

The homogeneous binary tree of Stern-Brocot corresponds to the free semigroup

of rank two, where the free generators are two Moebius transformations t1, t2,

represented by the matrices

m1: (1 1) m2: (1 0) (0 1) (1 1) .

Then each Moebius transformation from the generated semigroup maps the

number 1 to a different positive rational number; and each positive rational

number is obtained in this way (bijection).

Foreground:

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The Picard Group acts discontinously on the hyperbolic three-space, it is a

natural generalization of the Modular Group and it contains the Mod. Group.

Most works in a strong analogy: on the boundary of hyperbolic three-space

the Gauss numbers (as quotients of integer Gauss numbers) are created now

by the tessalation of the Picard Group. This analogy is discussed intensively

already in

R. Fricke, F. Klein: "Vorlesungen ueber die Theorie der automorphen

Funktionen", Leipzig 1897, Vol.1, p. 76-93.

Another reference is

W. Magnus, "Noneuclidean Tesselations and Their Groups", New York 1974

(there, on page 177, figure 18 visualizes the Stern-Brocot tree).

Problem:

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Is there a process, which generates each Gauss Number exactly once, in an

analogy to the Stern-Brocot tree ? (that is: "as free as possible")

This seems to be equivalent to the question:

Which semigroup is generated by the four Moebius transformations t1,t2,t3,t4

(without their inverses and) represented by the matrices m1, m2, m3, m4

(where: "i" = imaginary unit)m1: (1 1) m2: (1 0) m3: (1 i) m4: (1 0) (0 1) (1 1) (0 1) (i 1)

How can this semigroup be composed, e.g. as a direct product, a free product

with amalgamations or something else ?

There can be easily found some relations ("13" stands for "t1*t3", etc.): 13 = 31, 24 = 42, 143 = 432, 234 = 341, 343 = 434 (34)**3 = (43)**3 = identity

Any hints for a solution would be welcome!

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A further natural extension of the problem is:

- how to generate all rational quaternions ?

- how to generate all rational octaves (CAYLEY algebra) ?

as limit points of the hyperbolic spaces H^5 resp. H^9, created by the

reflections of the resp. regular polytope with all angles zero, corresponding

to the zero-angle tetrahedron in H^3.

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Toni Greil, Muenchen (Germany)

greil@guug.de

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