The class of non-Desarguesian projective planes is Borel complete
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Abstract:
For every infinite graph $\Gamma$ we construct a non-Desarguesian projective plane $P^*_{\Gamma }$ of the same size as $\Gamma$ such that $Aut(\Gamma ) \cong Aut(P^*_{\Gamma })$ and $\Gamma _1 \cong \Gamma _2$ iff $P^*_{\Gamma _1} \cong P^*_{\Gamma _2}$. Furthermore, restricted to structures with domain $\omega$, the map $\Gamma \mapsto P^*_{\Gamma }$ is Borel. On one side, this shows that the class of countable non-Desarguesian projective planes is Borel complete, and thus not admitting a Ulm type system of invariants. On the other side, we rediscover the main result of [J. Geometry 2 (1972), pp. 97-106] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.References
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Additional Information
- Gianluca Paolini
- Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
- MR Author ID: 1110693
- Received by editor(s): October 15, 2017
- Received by editor(s) in revised form: November 29, 2017, January 11, 2018, February 6, 2018, and February 20, 2018
- Published electronically: August 7, 2018
- Additional Notes: This research was partially supported by European Research Council grant 338821.
- Communicated by: Heike Mildenberger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4927-4936
- MSC (2010): Primary 03E15, 05B35, 22F50, 54H05
- DOI: https://doi.org/10.1090/proc/14137
- MathSciNet review: 3856159