Intersection configurations in free and free times free-abelian groups

TL;DR

This paper investigates intersection configurations of subgroups in free and free times free-abelian groups, establishing conditions for realizability and providing algorithms for finitely generated intersections.

Contribution

It characterizes when intersection configurations are realizable in free and FTFA groups and introduces an algorithm to decide and compute finitely generated intersections.

Findings

The Howson property is the only obstruction in free groups.

FTFA groups can realize any intersection configuration.

An algorithm to decide and compute finitely generated intersections in FTFA groups.

Abstract

In this paper we study intersection configurations -- which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability -- in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots , H_k \leqslant G$ realizing it. It is well known that free groups $\mathbb{F}_n$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups $\mathbb{F}_n \times \mathbb{Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a `basis' for it. We finally prove that any intersection configuration is realizable in a FTFA group $\mathbb{F}_n \times \mathbb{Z}^m$, for $n\geq 2$ and big enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.