The Class of Non-Desarguesian Projective Planes is Borel Complete

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 [15] 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.