Intersection configurations in free and free times free-abelian groups

In this paper we delve into the behavior of multiple intersections of subgroups of free and free times free-abelian (FTA) groups with respect to finite generability.

We first show that a multiple version of Howson property still holds in $\Fn$, and that, in fact, this is the only obstruction for multiple intersection configurations within $\Fn$.

Then, we study multiple intersections within FTFA groups, and we give an algorithm to decide whether the intersection of fintely many finitely generated subgroups \R{$H_1,\ldots ,H_k\leqslant \FTA$} is again finitely generated and, in the affirmative case, compute a basis for it. Finally, we show that any intersection configuration is realizable in a FTA of enough abelian rank. As a consequence, we exhibit a finitely presented group where every intersection configuration is realizable.