Graphs and Separability Properties of Groups

Ž . A group G is LERF locally extended residually finite if for any finitely generated subgroup S of G and for any g f S there exists a finite index subgroup S of G which contains S but not g. Using graph-theoretical methods we give 0 algorithms for constructing finite index subgroups in amalgamated free products of groups with good separability properties. We prove that a free product of a free group and a LERF group amalgamated over a cyclic subgroup maximal in the free factor is LERF. The maximality condition cannot be removed, because adjunction of roots does not preserve property LERF. We also give short proofs of some old theorems about separability properties of groups, including a theorem of Brunner, Burns, and Solitar that a free product of free groups amalgamated over a cyclic subgroup is LERF. Q 1997 Academic Press