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Pappus’s theorem and the modular group. (English) Zbl 0806.57004

The author treats Pappus’s theorem in the real projective plane \({\mathcal P} (\mathbb{R})\) as a dynamical system defined on so-called marked boxes. The latter are essentially quadrilaterals with distinguished top and bottom edges to be taken as initial data for the iterative application of the Pappian theorem. The resulting orbit may be indexed by the modular group \(\mathbb{Z}/2* \mathbb{Z}/3\). This group also occurs as the group of projective symmetries of an orbit: Varying the initial box one has two (commuting) modular group actions on the space of all marked boxes. If the marked boxes are convex then certain distinguished points of the boxes of an orbit form a set which is dense in some subspace of \({\mathcal P} (\mathbb{R})\) homeomorphic to the unit circle \(S^ 1\). Dual statements for distinguished lines hold true as well. These circles give rise to a subset \({\mathcal N}\) of the projective tangent bundle; furthermore the modular group has a representation \(\overline{M}\) in the Lie group of symmetries of the tangent bundle. From this the author finally gets a quotient space of \({\mathcal N}/ \overline{M}\) which is a three-dimensional analytic manifold with the homotopy type of the trefoil knot complement in \(S^ 3\).

MSC:

57M99 General low-dimensional topology
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
51A30 Desarguesian and Pappian geometries
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References:

[1] A. Beardon,The Geometry of Discrete Groups, Springer-Verlag, 1983. · Zbl 0528.30001
[2] Hilbert andCohn-Vossen,Geometry and the Imagination, Chelsea Publishing Company, 1950.
[3] W. Thurston,The Geometry and Topology of Three-Manifolds, Princeton University Lecture Notes, 1978. · Zbl 0399.73039
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