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The class of non-Desarguesian projective planes is Borel complete


Author: Gianluca Paolini
Journal: Proc. Amer. Math. Soc. 146 (2018), 4927-4936
MSC (2010): Primary 03E15, 05B35, 22F50, 54H05
DOI: https://doi.org/10.1090/proc/14137
Published electronically: August 7, 2018
MathSciNet review: 3856159
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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.


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Additional Information

Gianluca Paolini
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel

DOI: https://doi.org/10.1090/proc/14137
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
Article copyright: © Copyright 2018 American Mathematical Society