(This is joint work with J. Delgado and M. Roy) For two subgroups of a group, $H_1, H_2\leq G$, we can look at the eight possibilities for the finite/non-finite generability of $H_1$, $H_2$, and $H_1\cap H_2$. For example, all eight are possible in a free non-abelian group except one of them, expressing the well-known fact that free groups are Howson: intersection of two finitely generated subgroups is again finitely generated. A group $G$ is called intersection-saturated when, for every $k$, each of the $2^{2^k-1}$ such $k$-configurations are realizable by appropriate subgroups $H_1,\ldots ,H_k\leq G$.
In this talk we construct explicit finitely presented intersection-saturated groups. We also show that the Howson property is the only restriction for realizability in free groups: a $k$-configuration is realizable in a free non-abelian group if and only if it respects the Howson property.
If time permits I will explain some ideas to dualize the situation and be able to realize quotient $k$-configurations (this is still work in progress by the same authors).