Construction of vector fields and Riccati foliations associated to groups of projective automorphisms

Our main result states that given a finitely generated subgroup $G$ of $\operatorname {Aut}(\mathbb {C} P (2))$, there is an algebraic foliation $\mathcal {F}$ on a complex projective $3$-manifold $M^3$ with a bundle structure over $\mathbb {C} P(1)$ and fiber $\mathbb {C} P(2)$, such that $\mathcal {F}$ is transverse to almost every fiber of the bundle and with global holonomy conjugate to $G$.