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Embeddings and Ambient Automorphisms of the Pappus Configuration

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Abstract

We classify embeddings (i.e., “labeled drawings”) of the Pappus configuration in projective planes over commutative fields, up to projective equivalence. Using pairs of field elements, we parameterize the space of classes of projectively equivalent embeddings, and then explicitly determine the group of ambient automorphisms (or dualities) for any given parameter pair, i.e., the subgroup of the group of all automorphisms (and dualities) of the abstract configuration that are induced by projective collineations (or dualities) leaving invariant the image under any embedding in the given class. It turns out that the existence of an ambient duality implies an ambient polarity. We show that these parameter pairs can be interpreted as pairs of cross ratios associated in a rather natural way with the embedded configuration. The number of equivalence classes of embeddings in a projective plane over a given finite field is determined. The groups that occur as full ambient groups are identified in the subgroup lattice of the full automorphism group of the abstract configuration. Finally, we use our results to understand embeddings of the Möbius–Kantor configuration.

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Notes

  1. In the German original ([10, p. 117]): “[...] so können wir sagen, daß der Satz von Pascal der einzig wesentliche Schnittpunktsatz der Ebene ist, daß also die Konfiguration \((9_3)_1\) die wichtigste Figur der ebenen Geometrie darstellt”.

  2. See [26, 3.1]. The order of \({\text {Aut}}_{}(\Pi )\) has already been determined by Schoenflies [25, p. 59]; its isomorphism type has been noted in Levi’s 1929 monograph [16, pp. 108 ff]. A different description of the group is given in [4, Section 4, pp. 261–266].

  3. For embeddings into \(\mathbb {P}_{2}(\mathbb {R})\) or \(\mathbb {P}_{2}(\mathbb {C})\), this has been observed already by Levi [16, pp. 116], see also [5, p. 276]. For embeddings into \(\mathbb {P}_{2}(\mathbb {R})\) such that no triad is collinear and no parallel class is confluent, Kommerell [14, p. 32] reports that he has not found any ambient automorphisms apart from those in \(\langle \tau \rangle \).

  4. The authors are grateful to Peter Müller at Würzburg for providing this example plus the theory necessary to understand it in context.

  5. The action of \(\overline{\Gamma }/\mathrm {Z}\) on \(\mathbb {A}_{2}(\mathbb {F}_3)\) may be helpful here, see 5.7.

  6. Some of these embeddings into \(\mathbb {P}_{2}(\mathbb {R})\) have been noted by Coxeter in [4, p. 267–269] and [5, p. 270–273].

  7. See https://archive.org/details/pappialexandrin01hultgoog/page/n409

    (accessed on October 15, 2019).

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  1. LExMath, Fakultät für Mathematik und Physik, Universität Stuttgart, 70550, Stuttgart, Germany

    Norbert Knarr & Markus J. Stroppel

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Correspondence to Markus J. Stroppel.

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Knarr, N., Stroppel, M.J. Embeddings and Ambient Automorphisms of the Pappus Configuration. Results Math 74, 196 (2019). https://doi.org/10.1007/s00025-019-1115-7

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