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 $⟨\mathcal{G}⟩,⟨\mathcal{H}⟩$ of a group $G$, decide whether the intersection $⟨\mathcal{G}⟩\cap ⟨\mathcal{H}⟩$ is trivial. The second problem is Coset Intersection: given two finitely generated subgroups $⟨\mathcal{G}⟩,⟨\mathcal{H}⟩$ of a group $G$, as well as elements $g,h\in G$, decide whether the intersection of the two cosets $g⟨\mathcal{G}⟩\cap h⟨\mathcal{H}⟩$ 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.

25 pages