Subgroup and Coset Intersection in abelian-by-cyclic groups

Abstract

We consider two decision problems in infinite groups. The first problem is Subgroup Intersection: given two finitely generated subgroups $\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$ of a group $G$, decide whether the intersection $\langle \mathcal{G} \rangle \cap \langle \mathcal{H} \rangle$ is trivial. The second problem is Coset Intersection: given two finitely generated subgroups $\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$ of a group $G$, as well as elements $g, h \in G$, decide whether the intersection of the two cosets $g \langle \mathcal{G} \rangle \cap h \langle \mathcal{H} \rangle$ is empty. We show that both problems are decidable in finitely generated abelian-by-cyclic groups. In particular, we reduce them to the Shifted Monomial Membership problem (whether an ideal of the Laurent polynomial ring over integers contains any element of the form $X^z - f,\; z \in \mathbb{Z} \setminus \{0\}$). We also point out some obstacles for generalizing these results from abelian-by-cyclic groups to arbitrary metabelian groups.