publication . Article . Preprint . 2017

The Class of Non-Desarguesian Projective Planes is Borel Complete

Paolini, Gianluca;
Open Access English
  • Published: 02 Jul 2017
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 [15] on the realizability of every group as the group of collineations of some projective plane. Fin...
Subjects
arxiv: Mathematics::Group TheoryAstrophysics::High Energy Astrophysical Phenomena
free text keywords: Mathematics - Logic, 51A35, 03E15, 54H05, 05B35, 22F50

[1] Joseph E. Bonin and Joseph P. S. Kung. Every Group is the Automorphism Group of a Rank-3 Matroid. Geom. Dedicata 50 (1994), no. 3, 243246.

[2] Riccardo Camerlo and Su Gao. The Completeness of the Isomorphism Relation for Countable Boolean Algebras. Trans. Amer. Math. Soc. 353 (2001), no. 2, 491-518.

[3] Henry H. Crapo and Giancarlo Rota. On the Foundations of Combinatorial Theory: Combinatorial Geometries. M.I.T. Press, Cambridge, Mass, 1970.

[4] Henry H. Crapo. Single-Element Extensions of Matroids. J. Res. Nat. Bur. Standards Sect. B 69B (1965), 55-65.

[5] Harvey Friedman and Lee Stanley. A Borel Reducibility Theory for Classes of Countable Structures. J. Symbolic Logic 54 (1989), no. 3, 894-914.

[6] Robert Frucht. Herstellung von Graphen mit Vorgegebener Abstrakter Gruppe. Compositio Math. 6 (1939), 239-250.

[7] Robert Frucht. Graphs of Degree Three with a Given Abstract Group. Canadian J. Math. 1 (1949), 365-378.

[8] Su Gao. Invariant Descriptive Set Theory. Pure and Applied Mathematics (Boca Raton), 293. CRC Press, Boca Raton, FL, 2009.

[9] Frank Harary, Mike J. Piff, and Dominic J. A. Welsh. On the Automorphism Group of a Matroid. Discrete Math. 2 (1972), 163-171.

[10] Greg Hjorth and Alexander S. Kechris. Analytic Equivalence Relations and Ulm-Type Classifications. J. Symbolic Logic 60 (1995), no. 4, 1273-1300.

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 [15] on the realizability of every group as the group of collineations of some projective plane. Fin...
Subjects
arxiv: Mathematics::Group TheoryAstrophysics::High Energy Astrophysical Phenomena
free text keywords: Mathematics - Logic, 51A35, 03E15, 54H05, 05B35, 22F50

[1] Joseph E. Bonin and Joseph P. S. Kung. Every Group is the Automorphism Group of a Rank-3 Matroid. Geom. Dedicata 50 (1994), no. 3, 243246.

[2] Riccardo Camerlo and Su Gao. The Completeness of the Isomorphism Relation for Countable Boolean Algebras. Trans. Amer. Math. Soc. 353 (2001), no. 2, 491-518.

[3] Henry H. Crapo and Giancarlo Rota. On the Foundations of Combinatorial Theory: Combinatorial Geometries. M.I.T. Press, Cambridge, Mass, 1970.

[4] Henry H. Crapo. Single-Element Extensions of Matroids. J. Res. Nat. Bur. Standards Sect. B 69B (1965), 55-65.

[5] Harvey Friedman and Lee Stanley. A Borel Reducibility Theory for Classes of Countable Structures. J. Symbolic Logic 54 (1989), no. 3, 894-914.

[6] Robert Frucht. Herstellung von Graphen mit Vorgegebener Abstrakter Gruppe. Compositio Math. 6 (1939), 239-250.

[7] Robert Frucht. Graphs of Degree Three with a Given Abstract Group. Canadian J. Math. 1 (1949), 365-378.

[8] Su Gao. Invariant Descriptive Set Theory. Pure and Applied Mathematics (Boca Raton), 293. CRC Press, Boca Raton, FL, 2009.

[9] Frank Harary, Mike J. Piff, and Dominic J. A. Welsh. On the Automorphism Group of a Matroid. Discrete Math. 2 (1972), 163-171.

[10] Greg Hjorth and Alexander S. Kechris. Analytic Equivalence Relations and Ulm-Type Classifications. J. Symbolic Logic 60 (1995), no. 4, 1273-1300.

Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue
publication . Article . Preprint . 2017

The Class of Non-Desarguesian Projective Planes is Borel Complete

Paolini, Gianluca;