Classes of separable two-generator free subgroups of 3-manifold groups
A subgroup \(H\) of a group \(G\) is called separable if for any element \(g\) in \(G\) lying outside \(H\), there exists a subgroup of \(G\) of finite index containing \(H\) and avoiding \(g\). The author shows that certain subgroups of free products with amalgamation are separable. In particular, suppose \(M_1\), \(M_2\) are compact orientable 3-manifolds, each with at least one torus on its boundary. Denote by \(T_1\), \(T_2\) these tori. Suppose the interiors of the \(M_i\) admit complete hyperbolic structures of finite volume. Let \(M\) denote the 3-manifold obtained by identifying \(T_1\) and \(T_2\) by a homeomorphism. If \(\alpha \in \pi_1(M_1)\) and \(\beta \in \pi_1(M_2)\) are loxodromic, then the subgroup of \(\pi_1(M)\) generated by \(\alpha\) and \(\beta\) is a separable subgroup of \(\pi_1(M)\).
scientific article; zbMATH DE number 1955996
scientific article