A Universal Homogeneous Simple Matroid of Rank $3$

We construct a $\wedge$-homogeneous universal simple matroid of rank $3$, i.e. a countable simple rank~$3$ matroid $M_{\ast }$ which $\wedge$-embeds every finite simple rank $3$ matroid, and such that every isomorphism between finite $\wedge$-subgeometries of $M_{\ast }$ extends to an automorphism of $M_{\ast }$. We also construct a $\wedge$-homogeneous matroid $M_{\ast }(P)$ which is universal for the class of finite simple rank $3$ matroids omitting a given finite projective plane $P$. We then prove that these structures are not ${\mathrm{\aleph }}_0$-categorical, they have the independence property, they admit a stationary independence relation, and that their automorphism group embeds the symmetric group $Sym(\omega )$. Finally, we use the free projective extension $F(M_{\ast })$ of $M_{\ast }$ to conclude the existence of a countable projective plane embedding all the finite simple matroids of rank $3$ and whose automorphism group contains $Sym(\omega )$, in fact we show that $Aut(F(M_{\ast }))\cong Aut(M_{\ast })$.