Graphs and Separability Properties of Groups

Abstract

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.

Hall’s Theorem for Limit Groups

“…Three general theorems have been proved about subgroup separability of graphs of groups. One concerns the amalgamation of a subgroup separable group and a free group [6]. Another concerns graphs of free groups [33].…”

“…In [6], Gitik uses a notion of pre-cover; our pre-covers are analogous to hers.ΓΓ Figure 2: The diagram shows the covering spaceΓ from Figure 1. A finite subgraphΓ ofΓ is drawn with a thick line.…”

“…If we don't require π 1 (Ŷ d ) to retract onto δ then Lemma 3.6 can be significantly strengthened for residually free groups. Indeed, whenever the fundamental group of a space Y is residually torsion-free nilpotent, the cover Y d can be taken to be normal, so every elevation of δ toŶ d is of degree d (see Remark 3.1 of [6]). By a theorem of W. Magnus, free groups (and hence residually free groups) are residually torsion-free nilpotent.…”

Groups and semigroups: connections and contrasts

“…Similar techniques are used by Gitik [48], [49] to study separability properties and quasiconvexity in hyperbolic groups and by Steinberg [135], [136], [137] to study monoid presentations and profinite topologies on free groups. Methods related to Stallings foldings, making use of 2-complexes instead of graphs, have been employed by McCammond and Wise [97] to study coherence in groups and by Schupp [127] to study surface groups and Coxeter groups of extra large type.…”

Subgroup separability of graphs of abelian groups

“…Since then many people have worked on the subject. For the families of groups that are known to satisfy subgroup separability the reader should consult the papers of Gitik [8] and Wise [19] and the references cited there.…”