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Preprint Number 1256
1256. Gianluca Paolini
A Universal Homogeneous Simple Rank 3 Matroid (Old title: The Random Plane and its Automorphism Group.
Submission date: 17 July 2017
We construct a countably infinite simple rank 3 matroid M_* which ∨-embeds every finite simple rank 3 matroid, and such that every isomorphism between finite ∨-subgeometries of M_* extends to an automorphism of M_*. We then prove that Aut(M_*) is not oligomorphic, it has the strong small index property, it is complete, it admits ample generics and it embeds the symmetric group Sym(ω). Finally, we use the free projective extension F(M_*) of M_* to conclude the existence of a countably infinite projective plane embedding all the finite simple rank 3 matroids as subgeometries and whose automorphism group satisfies all the properties listed above.
Mathematics Subject Classification: 03E15, 54H05, 05B35, 22F50
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