Subgroup and Coset Intersection in abelian-by-cyclic groups
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We consider two decision problems in infinite groups. The first problem is Subgroup Intersection: given two finitely generated subgroups ⟨𝒢⟩, ⟨ℋ⟩ of a group G, decide whether the intersection ⟨𝒢⟩∩⟨ℋ⟩ is trivial. The second problem is Coset Intersection: given two finitely generated subgroups ⟨𝒢⟩, ⟨ℋ⟩ of a group G, as well as elements g, h ∈ G, decide whether the intersection of the two cosets g ⟨𝒢⟩∩ h ⟨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 ∈ℤ∖{0}). We also point out some obstacles for generalizing these results from abelian-by-cyclic groups to arbitrary metabelian groups.
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