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Klein’s discrete model: The case of odd order. (English) Zbl 1022.51011

Consider the conic \(\mathcal C\) with the equation \(f(x) = x_{\hskip-1pt 1}^2- x_2x_0=0\) in the projective plane \(\mathcal P \) over the field \({\mathbb F}_q\) of odd order \(q\). By definition, each tangent to \(\mathcal C\) contains \(q\) outer points, all other lines are incident with equally many inner and outer points. A point \(a \notin {\mathcal C}\) is an outer point if, and only if, \(f(a) \in \square \), the subgroup of squares in \({\mathbb F}_q^{\times}\). Klein’s discrete model (of a hyperbolic plane) consists of the set \({\mathcal I}\) of inner points together with all non-trivial intersections of \({\mathcal I}\) with lines of \({\mathcal P}\).
The authors discuss the hyperbolic motion group in terms of matrices.
Reviewer’s remarks: Several inconsistencies in the notation mark the paper. Only references to classical texts are given, but none to other papers dealing with finite hyperbolic planes [e.g., C. W. L. Garner, “Motions in a finite hyperbolic plane”, The geometric vein, The Coxeter Festschr., 485-493 (1982; Zbl 0505.51002) or S. Bruno and G. Tallini, Ric. Mat. 19, 48-78 (1970; Zbl 0197.46603)].

MSC:

51F05 Absolute planes in metric geometry
51M09 Elementary problems in hyperbolic and elliptic geometries
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