Graphs and Separability Properties of Groups Original Research Article

A groupGis LERF (locally extended residually finite) if for any finitely generated subgroupSofGand for anyg negated set membership Sthere exists a finite index subgroupS0ofGwhich containsSbut notg. Using graph-theoretical methods we give 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.