Subgroup Separability of HNN-Extensions with Abelian Base Group

Abstract

In the present paper we give a complete characterisation of subgroup separability of HNN-extensions with finitely generated abelian base group. We are also able to characterise subgroup separability for some families of free-by-cyclic groups and thus answer partially a question

Subgroup separability of graphs of abelian groups

“…Niblo and Wise in [14] show that the fundamental group of a graph manifold is subgroup separable if and only if it does not contain a subgroup isomorphic to L. The group L seems to be the key point for the results of the present paper also. In fact, summarising the results of the present work together with the results of [12], we have shown the following.…”

“…Use induction on the number n of edges X contains. If n = 1 the result is immediate from Theorem 2 in [12].…”

“…But by Theorem 2 in [12] there is a finite index subgroup of M l , say H l , such that H l is normalised by t l for every l ∈ {1, . .…”

“…In the second case, one can take the path in X λ that meets successively the two copies of G v , say A and B, associated by the stable letter. From the assumptions, A ∩ B has finite index in A (and necessarily in B) and there is a subgroup H of A ∩ B such that H is normal in A and B, so, by Theorem 2 in [12] is subgroup separable.…”

“…. , k and commensurable, one can work as in Lemma 6 or as in the proof of Theorem 2 in [12] to obtain a finite index subgroup H of some edge group G f such that H is normal G.…”