Residually finite groups / Lim Hui Min - Lim, Hui Min (2012)
In this thesis, we will study a stronger residually finite property called weak potency. More precisely, we aim to study the weak potency of HNN extensions and generalised free products of weakly potent groups and the main tools we used are lters. First we study the weak potency of HNN extensions by introducing the concept of h-filters and then use it to prove the main criterion. Then we prove several characterisations for the weak potency of certain HNN extensions with cyclic associated subgroups as well as a characterisation for the Baumslag-Solitar groups. Next, we will also apply our results to HNN extensions of finitely generated nilpotent groups. We shall give characterisations for certain HNN extensions of characteristically weakly potent groups with finitely generated central associated subgroups and HNN extensions of free abelian groups of finite rank to be weakly potent. In the last part we study the weak potency of generalised free products. Werst introduce w-filter and prove a criterion for generalised free products to be weakly potent. By using it, we then give characterisations for the weak potency of generalised free products with cyclic amalgamated subgroups and with central amalgamated subgroups. Then we extend the results to tree products of finitely many groups. Finally we show that certain one-relator groups with torsion are weakly potent.