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.

翻译：本文研究无限群中的两个判定问题。第一个问题是子群交：给定群$G$的两个有限生成子群$\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$，判定其交$\langle \mathcal{G} \rangle \cap \langle \mathcal{H} \rangle$是否为平凡子群。第二个问题是陪集交：给定群$G$的两个有限生成子群$\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$及元素$g, h \in G$，判定两个陪集$g \langle \mathcal{G} \rangle \cap h \langle \mathcal{H} \rangle$的交集是否为空。我们证明这两个问题在有限生成阿贝尔-循环群中是可判定的。具体而言，我们将它们归约为移位单项式成员问题（即整数系数洛朗多项式环的理想是否包含形如$X^z - f,\; z \in \mathbb{Z} \setminus \{0\}$的元素）。同时指出将这些结果从阿贝尔-循环群推广到任意亚交换群时存在的若干障碍。